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Computing K type multiplicities in standard representations after Vogan Peter E Trapa Notes from AIM workshops in July 2004 and 2005 The rst part of these notes is an updating and correction of Khat and is devoted to a paremetrization of the irreducible representations of the generally disconnected maximal compact subgroup of a real group in Harish Chandra s class The second part describes how to use that paremetrization of the rst part to compute K type multiplicities in standard modules By F robenius reciprocity this is equivalent to Blattner s formula and branching from K to K M which is complicated c y by the x me of the groups in question An interlude between the rst and second parts describes which K types are relevant for determining unitarity for irreducible Hermitian g K modules All of this material is intended as a report on ideas of David Vogan Part I A parametrization of irreducible representations of K Let G be a real reductive group in Harish Chandra s class It may be instructive and useful to weaken that hypothesis but we content ourselves with it here It certainly contains the class of groups obtained as the real points of a connected reductive algebraic group de ned over R Henceforth we shall call these groups algebraic Let K be the maximal compact subgroup of G The point of these notes is to recall a parametrization of K ie equivalence classes of irreducible representations of K due to David Vogan Note that even if G is algebraic the description of K is not covered by duCloux the group K need not belong to the class considered there For orientation one should consult branch Those notes provide provide a completely different perspective essentially that of Cartan Weyl and parametrize K in terms of irre ducible representations of a large Cartan subgroup By contrast these notes intricately use the fact that our K is the maximal compact subgroup of G Theorem 1 Let Etemw denote the set of irreducible tempered representations with real in nitesimal character Then the map tempp I obtained by taking lowest K types is a well de ned bijection More precisely if E Gtempt then 1 Y has a unique lowest K type 2 Two irreducible tempered representations with real in nitesimal character whose lowest K types coincide are necessarily isomorphic and 3 Each K type a E K arises as the lowest K type of an element of temw Thus K is parametrized by empquot A parametrization of this latter set in terms of more or less combinatorial data is given in Proposition 10 Putting them together we get the parametrization we seek Corollary 11 Sketch Fix a E K and let T denote a maximal torus in K Let to denote the Lie algebra of T We seek to nd a irreducible unique tempered representation with real in nitesimal character and lowest K type a This is a consequence of the Vogan Zuckerman classi cation The classi cation attaches an element a 6 it to a An algorithm to compute a is given in Vgr7 Proposition 533 see also De nition 664 and Lemma 665 of Vgr Later improvements of the algorithm due to Carmona are summarized in SV Let q denote the 0 stable parabolic subalgebra of g the complexi ed Lie algebra of G de ned by a and write L for the analytic subgroup of G corresponding to q E1 The group L is quasisplit Let aL denote the LO K type generated by a T highest weight space of a It is not immediately obvious that the highest weight space generates an irreducible L O K representation But it does Then aL is a ne L O K type for L The classi cation provides via cohomological induction using q a bijection between irreducible representations of G with lowest K type a and irreducible representations of L with lowest L O K type ML The bijection restricts to a bijection of tempered representations This is sketched in Vgr7 Section 677 although it should not be dif cult to prove it directly without reference to the Langlands classi cation In any event7 the bijection of the classi cation reduces matters to G quasisplit and a ne ie to the case that a is central More precisely7 it is enough to show that for each ne K type a there exists a unique tempered representation with real in nitesimal character which has a as its unique lowest K type This case is treated in Vgr7 Chapter 4 Here is a sketch Let MA denote a maximally split 0 stable Cartan subgroup in G Write MM61 H 61 2 This decomposition is multiplicity free since a is ne Choose an M type 6i appearing in the restriction Consider the principal series 61 indfIANQi 11 11 we will have no occassion to specify N Obviously 61 is tempered since it is induced from a discrete series By F robenius reciprocity and the fact that the restriction in 2 is multiplicity free7 it follows that a appears with multiplicity one in 61 Let Xa denote the constituent of 61 containing a The main results of Vgr7 Chapter 4 imply that Xa is well de ned independent of the choice of 61 and that the unique lowest K type of Xa is a This completes the case of la ne7 and hence the sketch of the theorem D The perspective offered by Theorem 1 has a number of wonderful advantages It appears to be the right kind of data structure77 from the point of View of du Cloux s existing software In Remark 14 we explain how to compute lowest K types of irreducible representations using this parametrization of We also remark that in the case that G is algebraic the computation of the character lattice of a large Cartan subgroup of K seems to be tractable using du Cloux s software This suggests the interesting and important auxilliary problem of implementing a translation between the parametrization of Theorem 1 and the Cartan Weyl parametrization Example 3 For orientation we include the example of G SL2lR Of course K 802 and K is parametrized by Z In the obvious notation write un E K The relevant observa tion is that for 2 2 un is the lowest K type of a tempered discrete series Meanwhile the two representations Mil arise as the lowest K types of the two tempered irreducible limits of discrete series We may also realize the pair Mil as the the lowest K types of the reducible nonspherical tempered principal series with in nitesimal character zero this is where the R group shenanigans rst appear Finally the trivial representation MO is the lowest K type of the tempered irreducible spherical principal series with in nitesimal character zero In particular we see that by passage to lowest K types we obtain a bijection from the set of irreducible tempered representation of G with real in nitesimal character to K D Example 4 It s also a good idea to keep GL2lR in mind here K is the disconnected orthogonal group 02 This time K is parametrized by strictly positive integers together with trivial and sgn representations which we denote poi For n gt O the K type un arises as the lowest K type of relative discrete series or in the case of 1 relative limits of discrete series To account for 0 note that there are four tempered principal series with in nitesimal character zero corresponding to the four characters of M Two of these principal series are isomorphic and isomorphic to a relative limit of discrete series so we have already accounted for them The other two are distinct one has lowest K type pa the other p3 See Example 12 below for a sharper treatment of GL2 Example 5 Next we consider U1 1 to illustrate how the parametrization behaves in the rank one case even when we restrict to connected groups In this case K U1 x U1 and a K type is thus a pair of integers a b E Z2 Suppose 1 A2 is the Harish Chandra parameter of a discrete series or a limit of discrete series S0 A E Z and there are two cases 1 2 A2 The lowest K type of the corresponding discrete series or limit is 1 1 pltx1 27 gt Hence we obtain all K types of the form a b E Z2 with a gt b 1 g kg The lowest K type of the corresponding discrete series or limit is 1 1 p A1 57A2 gt Hence we obtain all K types of the form a b E Z2 with a lt 1 Thus the discrete series and limits parameterize all K types of the form ab with a 7 b We are missing those with a b and according to the parametrization we need to look for them in representations induced from the Borel with real in nitesimal character The split Cartan H is naturally isomorphic to CX and the characters of Cgtlt that give real in nitesimal character are of the form Xnz for n E Z The lowest K types of the corresponding standard representations say X Xn are n2n2 ifn is even and n i12n I12 ifn is odd In the latter case the induced representation contains two limits of discrete series and we have already accounted for them Roughly speaking in the terminology introduced below XXn fails condition For 71 odd the lowest K types of the various XXn account for the missing K types of the form a a D Now we turn to parametrizing Gtempp To begin we need to discuss how to parametrize all irreducible admissible representations of G We are going to trot out pseudocharacters these are different from the parameters in duClouX but translating between the two is tractable This problem will be taken up in the future by Paul du Cloux and others The main point is that all the conditions we impose on our parameters also translate nicely into the framework of duClouX Write 9 for the Cartan involution of G Let be to 63 ac denote a 0 stable Cartan in go the Lie algebra of G As usual drop 0 subscripts to denote complexi cations Let H denote the centralizer of he in G The decomposition he to 63 ac implies that H TA where T H O K and A expuo is a vector group A regular pseudochamcter of H is a pair v 1 3 subject to the following conditions R1 P is an irreducible representation of H and 7y E Ff R2 Suppose 04 is an imaginary root of l in g Then 7y 04 is real and nonzero and hence 7 de nes a system of positive roots 1 making 7 dominant R3 If we write p ll for the half sum of the elements of 1 and pc l for the half sum of the compact ones then dr v pm 7 2M1 lt6 Write 73mgH for the set of regular pseudocharacters attached to H To each 39y Dy 6 73mg H we may build a standard module X y as follows Let L MA denote the centralizer in G of A Conditions R2 and R3 imply that 7m is the Harish Chandra parameter of a discrete series for M Since M may be disconnected the Harish Chandra parameter need not determine a single discrete series We may however specify such a discrete series say X M by requiring that its lowest M O K type have highest weight HT Next choose a real parabolic subgroup MN so the real part of 7y restricted to a is negative on the roots of a in n 7 Meanwhile write 1 for the character of the simply connected group A corresponding to 397 De ne XVind1IANXM V X 11 Then X 39y has in nitesimal character 7 and the condition in 7 guarantees that the pos sibly reducible Langlands subrepresentations occurs as a submodule The standard modules X 39y for 39y E 73mg are enough for some purposes but not enough for a classi cation For instance for SL2lR the only way to get the two limits of dis crete series is as the two constituents of the reducible nonspherical principal series with in nitesimal character zero Thus if we are interested in a map from our standard modules to irreducibles it must be multivalued To remedy this we must enlarge the class of stan dard modules by considering limit pseudocharacters in the SL2 case this amounts to including the two limits of discrete series as standard modules To make a bijection between standard and irreducibles we must then throw out some standard modules our by restrict ing to nal limit pseudocharacters in the SL2 case this amounts to throwing out the nonspherical principal series with in nitesimal character zero since their constituents are already accounted for by the addition of the limits of discrete series as standard modules In general we will throw out the standard modules corresponding to the less compact terms in a Hecht Schmid character identity We begin with the enlarged set of standard modules A limit pseudocharacter of H is a triple 7 37 P7 with the following properties L1 1 is a positive system for the imaginary roots of f in g P is an irreducible represen tation of H and 7y E Ff L2 lfa E 11 then 304 2 0 L3 dr v pm 7 2W Let P11mH denote the limit pseudocharacters for H Clearly PregH C 7311mH the regular pseudocharacter P 7y gets mapped to 11113 where 1 is speci ed by R2 above But the notion of limit pseudocharacter allows for more the in nitesimal character 7 can now be singular according to L2 In this case7 3 is not uniquely speci ed of course indeed in the connected case we should think of 3 specifying a chamber of discrete series from which we translate to in nitesimal character 7 of a limit of discrete series for M To each 39y E 7311mH7 we may de ne a standard module X39y as above this time XM is a limit of discrete series and the choice of N as in is a little messier Two issues present themselves the limit of discrete series XM may be zero and as in the case of SL2lR7 we need to rule out certain reducibilities among the standard modules X 39y accounted for by Hecht Schmid character identities The conditions F1 and F2 below are designed with these respective issues in mind7 and determine the standard modules we want to throw out A limit pseudocharacter is nal if F1 If 04 is a simple root in 3 such that 04739 07 then 04 is noncompact F2 If 04 is a real root of l in g and 04739 07 then 04 does not satisfy the Speh Vogan parity condition We write P nH for the set of nal limit pseudocharacters for H and 731311 for the union of all P nH Note that K acts on 731311 by componentwise conjugation To each 39y E 73 nH7 we attach a standard module X39y This time NT and 3 de ne a limit of discrete series XM of M which is nonzero by As above7 write 1 for the character of A determined by 39quo The associated standard module is given by parabolic induction7 XW indJTIANXM V X 11 Note that we have been somewhat sloppy and have not speci ed N precisely But we have no occasion to do so What matters for us is that the global character and K type spectrum of X39y are independent of the choice of N The condition F2 insures that X39y has a unique irreducible constituent that contains all of the lowest K types of X We write YW for that constituent Note that the de nition of NW does not depend on the choice of N Here is the sharpened version of the Langlands Knapp Zuckerman classi cation as inter preted by Vogan eg Vunit7 Section Theorem 8 Retain the notation aboue a Suppose 39thz E 731311 Then X 71 S X 72 if and only if V1 and 39y2 are conjugate by K 17 Suppose X is an irreducible Harish C handra module for G Then there exists y E 731311 such that Y z Consequently equiualence classes of irreducible Harish Chandra module for G are parametrized by K orbits on 731311 Here is where the tempered modules t in the above classi cation De nition 9 De ne PETPWH to be the subset of 39y E P nH such that the restriction of tempo temp 0 7y to a is identically zero Let 73 ll denote the union of all 73 ll Note again that K acts on PETE We say that Pimp o is the set of tempered nal limit pseudocharacters with real in nitesimal character The terminology is explained by the following result Proposition 10 Fix 39y E 731311 Then KW is tempered if and only if39y 6 73121311 In this case 7W X W that is the standard module corresponding to y E Pimp o is already irreducible Corollary 11 The set of equiualence classed of irreducible admissible representations ofK is parametrized by K orbits on Plaimp oH De nition 9 The parametrization takes an orbit K 39yH in gimp H to the lowest K type of X39y The corollary follows immediately from Proposition 10 and Theorem 1 Example 12 Let G GLnlR We begin with some structure theoretic facts Begin by noting that the K conjugacy classes of Cartan subalgebras in G are parametrized by partitions n a 2b so that if H denotes a corresponding representative of such a class H 2 KW x CXZ Under the Cartan decomposition H TA with T Z2 x 51 or perhaps more suggestively T 2 01 1 x so2b and A Rgt0 b Thus a character P of H is given by the data of an a tuple of signs 6 61 7 ea 6 Z2 and a b tuple of integers m1 7771b 6 Z177 and an a b tuple of complex numbers 11 Ila The imaginary roots of f E g in the standard coordinates for SO2b are simply i2517 i 2517 all of them are noncompact This assertion really amounts to one about GL2 where it is easy to check Now let 117 P 7y be a nal limit pseudocharacter with real in nitesimal character Def inition 9 with P a character of H We claim that such a pseudocharacter amounts to the following data a a pair of nonnegative integers a and b so that n a 21 b a b tuple of nonnegative integers ml7 7771 and c a single sign 6 if a 7 0 Here is a sketch The choice of H up to K conjugacy is the data of the partition in a Let P be given by the tuples mi7 Ej7 and 1 as described above The real in nitesimal character requirement that 7y restricted to a is zero together with L3 means that each 1 O The data of 1 in L1 together with the requirements of L2 means that we may take each mi 2 0 this is the data of Since there are no compact imaginary roots7 F1 is empty Meanwhile F2 means that all of the signs 6739 must all be or all be 7 If a pair of signs differ7 they specify a real root and we would be able to apply a Hecht Schmid identity to arrive at a more compact Cartan So indeed the data of the tuple 6 reduces to a single sign 6 this is the sign in Thus to the data of a c we may attach an irreducible tempered representation of G with real in nitesimal character What is the corresponding lowest K type Here is the answer Consider L so2b x 0a c K C71 So a representation of L is given by a b tuple of integers tensored with a representation of 0a Then the corresponding lowest K type u is the unique K type so that i The restriction of u to L contains the L representation speci ed by m11mb1 1 if e is l m11mb1 det if e is 7 and 11 t e restriction o to contains no we1g t 1g er t an m1 7mb h fp SO2l7 hh h h 1 1 The case of GL2 Example 4 is especially instructive D Remark 13 Suppose that 39y is a limit pseudocharacter that satis es F1 but not Then we may still form the standard module X39y as described above Loosely speaking7 the failure of F2 can be resolved after crossing a sequence of walls and each wall corre sponds to a Hecht Schmid character identity More precisely one may apply a sequence of multivalued inverse Cayley transforms to arrive at a set with each y E 733311 The number of possible inverse Cayley transforms is bounded by the rank of g and each can produce at most two new pseudocharacters so the cardinality of is at most 2 to the rank of g and typically much less The Hecht Schmid identities imply that lXv l ZlXWM 1 here the right hand side is effectively computable Thus if F2 fails for 39y the standard module X39y may be written as a sum of standard modules corresponding to elements in 73 Remark 14 As promised we conclude by describing how to compute lowest K types in terms of this parametrization Suppose X is any irreducible admissible representation of G Using the classi cation above we may write X X39y for some 39y E 731311 Then X is the lowest K type constituent of X39y Now modify 39y by making the 5quot factor trivial that is change 7 to 397 7y 7 1 let F F X 5 but leave 1 unchanged The resulting 39y is still a limit pseudocharacter which satis es the rst nal condition Unfortunately since we have changed the in nitesimal character F2 can fail lf F2 does hold then 39y E Pimp o and the lowest K type of X is simply the one parametrized according to Corollary 11 by 39y in particular the lowest K type is unique In the case that F2 fails we may apply the procedure of Remark 13 to arrive at 39y l y E Pimp o Then the lowest K types of X are precisely those parametrized by vi yf ie the lowest K types of the standard modules X39y This completes the calculation of the lowest K types of X in parametrization of Corollary 11 Interlude What Ktypes matter Retain the setting above Recall our eventual goal to compute signatures of invariant Hermitian forms on irreducible g7 K modules that possess them This computation can of course be handled K type by K type7 so it s evident that it is essential to be able to compute K isotypic components of irreducible g7 K modules The terminology computing isotypic components is vague7 but in the very least it requires computing multiplicities of K types in irreducible modules The Kazhdan Lustzig Vogan algorithm which appears reasonably close to being ef ciently implementable in Fokko s software makes the problem of computing K types in irreducibles equivalent to computing those in standard modules This is discussed more carefully in Part ll below But before turning to Part ll7 we recall that for applications to unitarity one need not be concerned with all K types7 but only certain small ones The modi er small may be quanti ed in the following de nition De nition 15 SalamancaVogan SV Fix G Recall the notation introduced in the proof of Theorem 1 We say that a K type a is unitarily small for G if a is contained in the convex hull of the Weyl group orbit of pa Here is the reason for the de nition Theorem 16 SalamancaVogan SV Suppose that X is an irreducible Hermitian rep resentation of G which does not contain a unitarily small K type Then there exists a proper 0 stable parabolic subgroup q 69 u of g and an irreducible Hermitian representation XL ofL containing a unitarily small L O K type so that X is unitary if and only if XL is unitary Hence the classi cation of the unitary dual is reduced by induction on rank to the classi cation of unitary representations of G which contain a unitarily small K type The above theorem is a consequence of a kind of bottom layer argument In particular examples7 that argument yields far more information and hence a more effective induction than provided for by the theorem Here is how to compute unitarity of representations of G which contain a unitarily small K type Conjecture 17 SalamancaVogan SV Suppose X is an irreducible g7 K module that possesses an inuariant Hermitian form lt Suppose in addition that X contains a unitarily small K type Write lt for Hermitian form obtained by restricting lt to the isotypic component of X Then lt is positiue de nite if and only if lt M is positiue de nite for all unitarily small K types Ia By our hypthesis on X this condition is not empty We recall evidence for the conjecture taken from Vunit De ne the length of u denoted H u H to be the length of u computed using a xed invariant bilinear form Then Vunit proves in the setting of Conjecture 17 that there exists a explicitly computable constant N so that lt is positive de nite if and only if lt gtM is positive de nite for each 1 such that H u Hg N The number of 1 such that u Hg N while nite is signi cantly larger than the number of unitarily small K types In practice the number of K types that ones needs to check is much less than those described in the conjecture In any any event to determine the unitarity of an irreducible Hermitian g K module for which Theorem 16 provides no reduction one need only to test that a nite number of the forms lt gtM are indeed positive Part II Computing Ktype multiplicities in standard modules As described at the beginning of the interlude what we really want to do is compute the multiplicity of suf ciently small K types in the sense of De nition 15 for instance in irreducible Hermitian Harish Chandra modules To describe a strategy to do so we need some additional notation Given a Harish Chandra module let X denote its class in the Grothendieck group of Harish Chandra modules Let denote the ring of formal integral linear combinations of K types ZHEH 27W Mfr MEI Given any virtual class Z there is an obvious notion of restriction to K that gives rise to an element of Write ZK for this restriction Given a nal limit pseudocharacter 39y consider the class X39y of the corresponding standard module Recall that we didn t really de ne X y in general in Part I we failed to specify the nilradical N precisely But in the Grothendieck group X39y is independent of the choice of N Fix an irreducible X Let denote the in nitesimal character of X Let 733D denote the set of nal limit pseudocharacters whose corresponding standard modules have in nitesimal character Then 7331 is nite In the Grothendieck group it is well known that we may write X Z mlevl7 7673 where the sum is of course nite The proof of the Kazhdan Lusztig conjecture due to Vogan implies that the coef cients m are explicitly computable for linear groups in Harish Chandra s class whose Cartans are all abelian Restricting to K we may write XlK Z m7XK7 7673 in Thus computing the coef cient of p in X K is equivalent given the Kazhdan Lusztig algorithm to computing the coef cient of u in each standard module X39yK According to the de nition of these modules F robenius reciprocity implies that this amounts to branching from K to K O M together with the Blattner formula for Since M and K O M are disconnected this is a dif cult branching problem The purpose of this section is an alternative approach that takes advantage of the fact that K is a maximal compact subgroup of M Vogan proposes to solve an inverse problem instead of computing the coef cient of p in a standard module he suggests that roughly speaking one ought to try instead to write a as a linear combination of standard modules in That is he suggests that one ought to look for expressions of the form Ml Z lXWHlK WEIEn for each suf cient small a of which there are only nitely many and then invert these expressions to compute the coef cient of a is a particular standard module X39y lK To make this precise we need to work within the framework of Part 1 Recall that Corollary 11 de nes a bijection between I and standard modules of the form X39y with 39y E 733311 The length of the in nitesimal character of X39y de nes a partial order on the latter set and using the bijection on Using this bijection and partial order we may de ne a matrix whose rows are indexed by I and whose columns are indexed by the standard modules with real in nitesimal character corresponding to tempered representations with entries gives as follows the entry in the X39y column and in the a row is the coef cient of a in X39yHK Here is a strengthened version of Corollary 11 taken again from Vunit Theorem 18 The in nite matrix say M de ned in the previous paragraph relating stan dard tempered modules with real in nitesimal character to K in is upper triangular with 1 s on the diagonal Hence it is invertible According to the lnterlude we are interested in only suf ciently small a Said differently we are only interested in some nite upper left corner of M More precisely if for all a so that a Hg N we nd expressions of the form lMl Z mleWHlK 19 7673 then after inverting a nite matrix we obtain for all a and 39y E Pimp o with 7y Hg N and a Hg N the multiplicity of a in X39yHK There is one last minor point to make Initially we were concerned with the multiplicity of suf ciently small a in an arbitrary standard representation X y with 39y E 731311 The previous paragraph describes a strategy to solve that problem for 39y E Pimp o But the former reduces to the latter as follows Given arbitrary 39y E 731311 recall from Part I that Xy indMANXM 1 11 It is obvious from F robenius reciprocity that lXvllK linMANXM 11 JUNK So let 39y be the pseudocharacter by taking 1 0 in 39y ie so that the right hand side of the previous displayed equation is the standard module corresponding to y Just as in Remark 14 it is easy to check that 39y is a limit pseudocharacter satisfying L17L3 but 39y is no longer nal F1 holds but F2 need not Using Remark 13 one may write lXv l ZlXWDL l with each 39y E 733311 Since we remarked above that X39ylK X39y lK it is indeed enough to determine expressions of the form of 19 for all suf ciently small u Summarizing the main problem is Problem A For all suf ciently small u obtain explicit expressions of the form 19 ex pressing p as a linear combination of standard tempered modules with real in nitesimal character Example 20 Let 311 C P n denote the subset of pseudocharacters with in nitesimal char acter p in the block of the trivial representation 110 of G For terminology relating to blocks see Vgr Chapter Then one version of the Zuckerman character formula is the following identity in the Grothendieck group of Harish Chandra modules Hal Z Elevl 7651 here 6 is zero if the character P of H which is part of the data of 39y is not trivial on the component group of H and otherwise 6 is i1 The precise sign 7 either plus or minus 7 is easy to determine Restricting to K we the following identity in ZHKH l Kl Z EleWlllK 76511 As in the discussion preceding Problem A we can take the 1 parameter of each standard module on the right hand side to be zero without affecting the formula in and then express the right hand side as a sum of standard tempered modules So this solves Problem A for u 11K We now describe Vogan s strategy to solve Problem A in more detail He suggest a method to nd expressions for u as a sum of various continuations of standards modules tempp These standard modules need not be of the form X39y for 39y E P n but as in Example 20 they can be converted into standard tempered modules in and hence lead to a solution of Problem A We rst de ne the relevant continued standard modules First we assume that G pos sesses a compact Cartan subgroup Then the relevant continued standard modules are simply coherent continuations of discrete series as originally considered by Schmid In more detail Let T denote a generally disconnected 0 stable compact subgroup of G Fix a choice of positive roots 110 for T in E and a choose positive roots 1 containing 14 30 for T in g 30 will be xed once and for all but 3 will vary in the discussion below Write pc for the half sum of the elements of 30 and likewise for p Write AT for the character lattice of T Recall that 2p 6 ito exponentiates to T but p need not We will be somewhat sloppy and write 2p AT for both the exponentiated character in AT and its differential in ito Let T denote the abelian double cover of T de ned by the squareroot of the 2p character and write AT for its lattice of genuine characters Recall that T comes equippedwith a genuine character whose differential is p Again we will be sloppy and write p E AT for both this character and its differential By tensoring with p we may canonically identify AT with AT X p but we avoid doing this at the moment Fix I E We claim that the character I together with the choice of 3 containing the xed 30 uniquely specify a unique K conjugacy class say K F 3 7y which is almost in Pimp o except possibly for the failure of F1 and L2 as explained below Here 3 in the pseudocharacter is the 3 we have already speci ed and we now explain how to de ne 7y and P Let liq teat denote the Borel subalgebra of G corresponding to 3 Then At0p de nes a character of T with differntial 72pc Meanwhile I X p also de nes a character of T De ne r lt1gt 29 p 29 NOW m a e AT a character of T The differential of P is dP 11 p7 2pc E ito De ne 7y 11 Since we have xed T the K conjugacy class of F 3quoty is de ned only up to the Weyl group of K But we have also xed 30 So the K conjugacy class of the triple F 3 7y is uniquely speci ed by 30 C 3 and I E AT It remains to determine whether 39y F 3quoty so de ned actually de nes a nal limit pseudocharacter Clearly L1 and L3 hold and F2 is empty since there are no real roots So it remains to investigate L2 and From the de nitions it is easy to see that L2 is equivalent to requiring that the differential dltIgt to be weakly dominant for 3 and unfortunately this may not always hold Meanwhile F1 is equivalent to requiring that if 04 E 3 is a simple root for which 01304 0 then 04 is not in 30 ie is noncompact In this case KW is a nonzero limit of discrete series with Harish Chandra parameter 11 and if dltIgt is actually 3 dominant and hence automatically 30 dominant KW is a discrete series with Harish Chandra parameter 11 But in general 39y need not satisfy L2 or 15 Even though F1 and L2 may fail for 39y we now de ne a virtual Harish Chandra module Q Q which we may regard as the image of a continued standard module attached to the pseudocharacter 39y We follow Vgr De nition 728 Let b t 63 u be the 0 stable Borel subgroup of g de ned by 11 Recall the derived Zuckerman functors R that on objects take representations of T to Harish Charish modules for G Here the normalization is arranged so that preserues in nitesimal character This is the normalization of Vorange and differs from the one in Let V4 denote the representation space for Q which recall need not be onedimensional Then RVqgt has in nitesimal character dQ 7y We de ne Q Q Euler characteristic of RVqgt 21 which is a virtual Harish Chandra module since has nite cohomological dimension It has in nitesimal character 7 and it is zero if F1 fails lf dQ is suf ciently dominant in the sense explained above then Q Q is simply the class of a discrete series or limit Corollary 7210 of Vgr explains the sense in which Q Q is a continuation of a discrete series That reference assumes T is a abelian so matters are slightly more complicated than indicated there Example 22 Suppose now that G is connected So T is connected and abelian and any element of AT is determined by its differential We will be sloppy and blur the distinction between a character and its differential For instance the canonical identi cation of AT with AT X p may now be written as MT Ap where we will just write A for AT Fix 110 as above and choose 1 containing Q0 and E A p viewed as a character of Let 39y denote the pseudocharacter de ned above and recall the virtual Harish Chandra modules Q A Note that the xed system 110 de nes a holomorphic structure on KT and hence on the line bundle 3 K gtltT Egg where ng denotes the one dimensional representation space corresponding to gt E A As explained below the following is a consequence of the Blattner formula Theorem 23 Assume G is connected with compact C artan subgroup T Fix 110 C 11 Let p p ll and use 110 to de ne a holomorphic structure on KT Let b 69 n denote the Borel subalgebra corresponding to 1 and write M for the span of the noncompact roots spaces in 11 Fix E A p Since 2p 6 A p de nes a character of T and hence we can consider Ap a holomorphic line bundle on KT Let N denote the set of T weights on Nun Then in mm Z1dejmKT jKT7 Ema Z Z 1quot91 7 WlK 24 j i 76M 16 Since the left hand side is up to sign an irreducible representation of K and since the right hand side may be expressed as a sum of standard tempered modules see Remark 30 Equation prouides a solution to Problem A if G is connected and equal rank Sketch There are three ingredients to the proof a Blattner type formula7 a formal iden tity in ZHKH and a fact about tensoring We start with the latter A basic fact about cohomological induction cf Vgr7 Lemma 729b implies that if ulT Zm h i then Ml 1 7 lK Zmiew MlK 25 in where the obVious notion of tensor product is used in Next we discuss the formal identity we need in Suppose V is a representation of K Let symV E denote the symmetric algebra of V Let MW 24quot N V 6 leKll i the signed exterior algebra of V The formal identity we need is symltvgt Aim 11 in 2m lt26 This is obVious for a T weight in ZHT we may compute directly sym WAX l rl W l2l39 l11Tl 7 W l rl 27 Because symV 69 W symV symW7 and MW ea W Aim 2 AiW7 it follows that 27 implies that the restriction of symV ARV from to ZHTH is simply 1T Hence 26 follows Finally we turn to the the Blattner type formula we need Let S denote the multiset of highest weights of irreducible representations of K appearing in symnn Then w W Z ZHVH lKI n7 lt28 73965 739 This is the usual Blattner formula coherently continued77 in case that is not suitably dominant To nish the sketch we take Equation 28 and multiply both sides by Aim in Combining the rst two facts gives conclusion of the theorem D To continue with the example we remark that something very close to Equation 24 holds if G ie K is disconnected But notice that the left hand side of Equation 24 need not be an irreducible representation of K if K is disconnected This is the obstacle to extending the solution of Problem A in the connected case to the disconnected case For instance in GL2 one nds that the span of the various 3 AlK in is spanned by the set 11K detulu2 that is we get formulas of the form 19 for each m but we cannot separate 11K from det So something new is needed But Example 20 provides a formula of the form 19 for 11K and tensoring it with det gives a formula for detK This kind of tensoring will be an ingredient in Vogan s general solution to Problem A This completes the example D We return to the general disconnected case and modify our notation slightly to more closely resemble that in the connected case In our initial discussion we were careful to distinguish I E from its differential 11 Nowwe will be sloppy and write A p for AT and write A E A p for a both character in AT and its differential This imprecision is customary and causes no confusion in practice Now we introduce more general continued standard modules Let H TA be a 0 stable Cartan subgroup of G and write MA for the Levi factor of the corresponding cuspidal parabolic Let 3 denote a choice of positive imaginary roots of T in m containing a xed compact system 30 Fix a character A E Ap recalling the new notational convention of the previous paragraph The discussion above de nes a pseudocharacter 39yM PM 3 3M for M for which L2 and F1 may fail Nonetheless as above we may consider the continued standard module M 3 A a virtual representation of M lf F1 fails it is zero if F1 holds it is nonzero De ne 3 A indfIAN M 3 A X 11 11 29 a virtual Harish Chandra module for G Here as usual the choice of N is irrelevant Just as we de ned the pseudocharacter 39yM we may also de ne a pseduocharacter 39y F 3 7y for the Cartan H TA of G 3 is the xed choice of positive imaginary roots P is de ned uniquely by requiring its retriction to A to be trivial and its restriction to T to be PM and 7y is determined by L3 One again 39y need not satisfy L2 or F1 and 3 A need not be of the form Xn for 7 6 Range Here is how to remedy that Remark 30 Fix 3 A as in the previous paragraph and let 39y denote the corresponding pseudocharacter So 39y is a pseudocharacter for an arbitrary 0 stable Cartan subgroup H TA satisfying all of the conditions to be a nal limit pseudocharacter except for L2 and F1 This is the situation we encountered in the previous paragraph and also just before Example 22 If F1 fails then we have already remarked that 3 A is zero So suppose F1 holds and suppose Oz 6 3 is an imaginary root for which 7 is not dominant If 04 is 18 compact then we may write 3 79 3 X this fact reduces to one about continued discrete series where it is due to Hecht Schmid see Vgr Proposition 843 for instance Let 39y be the pseudocharacter attached to 3 X The payoff here is that the number of roots in 3 for which L2 fails for 39y is strictly smaller than the corresponding number for 39y Next suppose that 04 is noncompact Then there is a Hecht Schmid identity of the form om ow X 9 see Vgr Proposition 845ll In this expression the number of roots in 3 for which L2 fails for 39y the pseudocharacter attached to 3 X as above is strictly smaller than the corresponding number for 39y Meanwhile 9 is an effectively computible sum of modules of the form 3 X whose corresponding pseduocharacters 39y are attached to a Cartan that is less compact than H Thus 3 is strictly smaller in cardinality that 3 and thus the number of of roots in 3 for which L2 fails for 39y is strictly smaller than the corresponding number for 39y We may thus proceed by induction on the number of roots in 3 for which L2 fails to conclude that that there is an effective algorithm to express 3 as a sum of standard tempered modules with real in nitesimal character The number of iterations that are needed in this algorithm is bounded by the number of roots in 3 for which L2 fails We conclude that in order to solve Problem A we may instead solve Problem B Find expression in of the form Ml Zmr 1 MlK 31 for each suf ciently small u This concludes the remark D Now we explore the tensoring idea in more generality Suppose that u is an arbitrary K type Consider a continued standard module 3 x induced from a continued discrete series for M arising from a 0 stable parabolic TA Since T C K we may restrict u to T and then pullbactho p 3 cover T of T Hence we can consider uT as a non genuine representation of T Since is a genuine representation of T the tensor product is also genuine and we can decompose it as MT X EmuV E Combined with Equation 25 we immediately have the following conclusion 11f G is nonlinear nonlinear Vgr Proposition 845 doesn t cover all the identities that are needed in this remark Proposition 32 Suppose there is an e ectiue algorithm to compute the restriction oflu E I to the compact part of an arbitrary 0 stable C artan for G Then by tensoring Zuckerman s formula for 11K Example 20 with a we obtain an e ectiuely computable expression for a of the form 31 and hence 19 The issue is of course disconnectedness of the compact part of an arbitrary Cartan This can be overcome for ne K types of quasisplit groups Then the machinery of cohomological induction and coherent continuation will ultimately reduce to that case This is Vogan s proposed solution to Problem A We begin with ne K types Suppose G is quasisplit Let HS TEAS denote a maximally split 0 stable Cartan Since G is quasisplit T9 M9 the centralizer of AS in G Our starting point is that we assume that the characters of MS are effectively computable This is part of what du Cloux has already done Now x a ne K type a Write MlM5169696k Since a is ne this decomposition amounts to computing the R group of say 61 We assume this decomposition is effectively computable and again du Cloux has essentially already implemented this In short our starting point is that we assume that the restriction of a to the maximally split Cartan is effectively computable Now we turn to general Cartans So let H TA denote an arbitrary 0 stable Cartan subgroup of G We recall how T is built Let 041 on denote a system of strongly orthogonal roots which via inverse Cayley transforms take MS to H Then the Lie algebra of T is given by the span t ltZ177Zrgt7 0 1 Zid gti lt71 0 here d i is the usual inclusion of 52lR into go Set where oi exp and m 0 Let I be the exponentiated be the exponentiation of in If G is linear In SL2lR a M each mi 6 M is necessarily of order 2 and the eigenvalues of Inmi are i1 In the nonlinear case the domain of I will be a cover of SL2 and the eigenvalues of Inmi will be roots of unity De ne Then T To M 20 This equality and the relations between To and M is algorithmically understood in du Cloux s software The restriction of a to To is effectively computable by say a version of the Kostant multiplicity formula and the description of to above Meanwhile since the restriction of a to M is effectively computable and since M is an explicitly de ned subgroup of M the restriction of a to M is effectively computable As the example of SL2 R indicates this is not quite enough to determine the restriction to T Nonetheless the remaining ambiguity is tractable We have Proposition 33 Suppose G is quasisplit a is ne and H TA is an arbitrary 0 stable C artan subgroup of G Then the restriction of to T is e ectiuely computable Hence by Proposition 32 we obtain an expression for a of the form 19 Finally we must reduce to the case that G is quasisplit and a is ne Suppose G is arbitrary and changing notation slightly let E denote an arbitrary K type Let T denote a maximal torus in K Let a denote a not necessarily unique highest weight of E this is an irreducible representation of T Then Vgr Section 53 attaches a 0 stable parabolic q e9 u to a The analytic subgroup L of G corresponding to IC q mg is quasisplit Moreover the a weight space of E generates an irreducible representation of L K say E L K Set ELmK E L K At0pu o E Then ELQK is ne As usual L O E gives a holomorphic structure to KL O K and so from ELmK we may form a holomorphic vector bundle ELQK K XLQK ELQK Proposition 34 We have that E demKL KKL m K ELM in particular the right hand side is irreducible Sketch The proposition amounts to the irreducibility assertion It is more or less obvious that the indicated cohomology say E is irreducible for Ktt Ko L O The tricky point is showing that the induction from Ktt to K is irreducible This amounts to showing that KK acts on the irreducible representation E of Ktt with no isotropy The next ingredient we need is a version of the Blattner formula for cohomological induction from q together with the way that coherent families behave under cohomological induction Proposition 35 Let q u be a 0 stable parabolic subalgebra of g Let L denote the analytic subgroup of G with Lie algebra q EL Suppose H TLAL C L is a 0 stable C artan subgroup of L and that L 1 L7L 21 is a continued standard module for L induced from the cuspidal parabolic subgroup corre sponding to H De ne IG IL U the imaginary roots of H in u Recall that L is a character of the pL couer of H hence L X pil is a character ofH and L 8031 PG L PU is a genuine character of the pa couer of H Set G L X 0Ll X At pu p7 36 a genuine character of the pa couer of H Then in 90 110 ACMK Euler characteristic of H39 KL O K L IIL LleK X symu O 13 Sketch Recall the cohomological induction functors R that appear around Equation 21 We claim that up to a sign 90 110G Euler characteristic of R L IIL AL 37 then the theorem follows from the Blattner formula for cohomological induction from q for instance page 376 of The Euler characteristic of the functors Ra take coherent families for L to coherent families for G this is the content of Vgr Corollary 7210 So to verify the equality in Equation 37 we need only verify it for L and hence AC suf ciently dominant In this case 90 110G and L IIL L are Langlands standard modules So the assertion of 37 follows from understanding how cohomological induction behaves with respect to standard modules This is the subject of KV Section X110 and in particular KV Theorem 11255 D Recall that Aiu p is the inverse of the symmetric algebra in Equation 26 After commuting tensoring with induction Proposition 35 then implies that 90 110G X Aiu plT Euler characteristic of H39 KL O K L IIL LleK Recall that ELQK is ne and L is quasisplit So Proposition 33 given an expression reverting to the notation aLmK for ELQK of the form lHLmKl ZmQL I L7 LlK Now apply the Euler characteristic of holomorphic induction from L O K to K to both sides of the above displayed equation Propositions 34 and 35 then apply to conclude again reverting to the notation a for E that M Z imxm a 1 07 0 AA m 13 lK This is the desired expression for a of the form given in 31 This completes the outline of Vogan s solution to Problem A 22 References duCloux F du Cloux Combinatorics for the representation theory of real reductive groups lKVl K hat SVl Vgrl Vunit Vorange branch notes from luctues at AlM July 2005 A Knapp D Vogan Cohomological Induction and Unitary Representations Princeton Mathematical Series 45 1995 Princeton University Press Prince ton P Trapa A parametrization of I after Vogan notes from a lecture at AlM July 2004 S Salamanca Riba D Vogan On the classi cation of unitary representations of reductive Lie groups Ann Math 2 148 1998 no 3 106771133 Lie D Vogan Representations of Real Reductiue in Math 151981 BirkhauserBoston Group s Progress D Vogan Unitarizability of certain series of representations Ann Math 2 120 1984 1417187 D Vogan Unitary representations of reductiue Lie groups Annals of Mathemat ical Studies 1181987 Princeton University Press Princeton D Vogan Branching laws for reductive groups notes from a lecture at AlM July 3 CHARACTERISTIC p METHODS AND TIGHT CLOSURE ANURAG K SINGH These are the notes of four lectures given at the Abdus Salam International Centre for Theoretical Physics Trieste in June 2004 on tight closure theory This theory was developed by Melvin Hochster and Craig Huneke in the paper An excellent account may be found in Huneke s CBMS lecture notes The lectures that follow are very far from a complete treatment but we hope they will be of some help to those who are new to the subject 1 TIGHT CLOSURE BASIC PROPERTIES By a ring we mean a commutative ring with a unit element A local ring denoted R m or R m K is a Noetherian ring with unique maximal ideal m and residue eld K Rmi If S is a subring of a ring R and there is an S linear map p R a S such that ps s for all s E S we shall say that S is a direct summand of Ri In this situation let a be an ideal of S and let 8 E aR 539 Then 8 Zairi where ai E a and ri E R Applying p to this equation we get MS ELM3a 6 a and since ps s it follows that aR S all Now let G be a group acting on a Noetherian ring Ri We use RC to denote the ring on invariants iiei RGrERgrrforallgEG If RC is a direct summand of R then aR RC a for all ideals a C RGi This has several strong consequences as we shall see in these lecturesias a start we observe that it implies RC is a Noetherian ring to see this consider a chain of ideals of the ring RC algaggaggmi Expanding these to ideals of R we have a chain of ideals c1le agRg agRg m Date June 17 2004 The author is supported in part by the National Science Foundation under Grant DMS 0300600 1 2 ANURAG K SINGH which stabilizes since R is Noetherian But aiR RC all7 so the original chain stabilizes as well Hilbert s fourteenth problem essentially asks whether RC is Noetherian whenever R is The answer turns out to be negative7 and the rst counterexamples were constructed by Nagata7 Nal We shall say more about these issues in Remark 18 For the moment7 we focus on the case where R is a polynomial ring over a eld K7 and G is a nite group acting on R by K algebra automorphismsithe subject of Benson7s book Ben In this case RC is always Noetherian7 see AM7 Exercise 75 If the order of the nite group G is invertible in the eld K7 consider the map szaRG givenby 1 W 16290 It is easily veri ed that p is an RGmodule homomorphism7 and that Ms s for all s 6 RC Hence RC is a direct summand of R whenever the characteristic of K does not divide the order of the nite group G Example 11 Let Sn be the symmetric group on n symbols acting on the polyno mial ring R Kzl7 7 In by permuting the variables Then the ring of invariants is RSquot Kel7 7 enl7 where ei is the elementary symmetric function of degree 239 in the variables 117 7 In Moreover7 R is a free RSquotmodule with basis m1 m2 mu 7 11 12 where 0gm7gz l7 see7 for example7 Art7 Chapter ILG Consequently R5quot is a direct summand of R7 and this is independent of the characteristic of the eld K Example 12 Let K be a eld of characteristic other than 2 For n 2 37 consider the alternating group An lt Sn acting on the polynomial ring R Kzl7 7177 by permuting the variables Let AHziizjER iltj Then 0A sgnaA for every permutation a E Sm so A is xed by even cycles It is not hard to see that RAquot Kel7 7 6777 A Since A2 is xed by all elements of Sm it must be a polynomial in the elementary symmetric functions 617 and so RAquot is a hypersurface with de ning equation of the form A2 7 fel7 7 en 0 It turns out that RAquot is a direct summand of R if and only if lAnl nl2 is invertible in K We examine the case p n 3 here7 and refer to Sill or SmL for details of the general case The ring of invariants is RAg K 17 27 637 A CHARACTERISTIC p METHODS AND TIGHT CLOSURE 3 where e1 11 12 137 eg 1112 1213 13117 e3 1112137 and A 11 7 12 12 7 zgzl 7 13 Since RA8 is a hypersurface with de ning equation A2 7 fe17e27e3 07 it follows that A e17e27e3RA3 On the other hand A 11 I213 1 62 E 61762753R7 so RA3 is not a direct summand of R For a nite group G7 when is RC a direct summand of R One answer comes from tight closure theory Theorem 13 Let K be a eld of positive characteristic and let G be a nite group acting on a polynomial ring R Kzl7 7 In by degree preserving Kalgebra automorphisms Then RC is a direct summand of R if and only RC is weakly F regular De nition 14 Let R be a ring of prime characteristic p and let R denote the complement of the minimal primes of R If R is a domain7 which will be the main case in these lectures7 then R is just the set of nonzero elements of R For an ideal a 1177zn of R and a prime power 4 p87 we use the notation alql I7 7137 The tight closure of a is atquot 2 E R l there exists c 6 R0 for which c2 1 6 alql for all q gt 0 which is an ideal of R containing a7possibly larger7 but a tight t nonetheless A ring R is weakly F regular if at a for all ideals a of R For a ring R of characteristic p gt 07 the map F R 7 R with Fr r17 is a ring homomorphism7 called the Frobenius homomorphism Note that R may be Viewed as an R module Via the Frobenius homomorphism or any iteration thereof For an ideal a 117 717 of R7 consider the exact sequence Rn R gt 391 gt 0 Applying Fe R R7 the right exactness of tensor gives us the exact sequence Rquot R mp R Ra 07 which shows that F8 R ER Ra E Ralql If R Zpzl7 7 rug is a polynomial ring7 the Frobenius F R 7 R may be identi ed with the inclusion Zpzzl777zZ CZpzl77zd The monomials in 11 with each exponents less than p form a basis for Zpzl7 7 10 as a Zpzzl77 7 Iglmodule7 hence the inclusion is free7 in particular7 at More generally7 we have 4 ANURAG K SINGH Proposition 15 Let R be a regular ring of prime characteristic p Then the Frobenius homomorphism is flat Proof The issue is local so we may assume R is a regular local ring lt suf ces to verify the assertion after taking the completion of R at its maximal ideal so by the structure theorem for complete local rings Theorem 51 of the Appendix we may assume R is a power series ring over a eld say R KHIl del By at descent we reduce to the case K Zp and then similar to the argument earlier the Frobenius F R a R may be identi ed with the inclusion ZszZl mg C Zszl de which is free hence at D The converse is a theorem of Kunz a ring R of positive characteristic is regular if and only if the Frobenius homomorphism F R a R is at Ku1 Her Note that for a nite group G acting on a ring R the extension RC Q R is integral since an element r E R is a root of the polynomial H I 7 90 6 Ralrl gEG The proof of Theorem 13 will be immediate from the three fundamental properties of weakly Fregular rings which we establish next Theorem 16 The following are true for rings of positive characteristic 1 Regular rings are weakly F regular 2 Direct summands of weakly F regular domains are weakly F regular 3 An excellent weakly F regular domain is a direct summand of every module nite extension domain Proof 1 The key point is the atness of the Frobenius homomorphism Proposi tion 15 Let R be a regular ring of characteristic p gt 0 Given an ideal a of R and an element 2 E R there is a short exact sequence OaRazz aRae RazR 0 Tensoring this with an iteration Fe R a R of the Frobenius and using that F8 is at we get a short exact sequence 0 Ra all Ralql Ra 2R 1lgt 0 which implies that alql 2 1 a zql for all g 09 If 2 6 at then by de nition there exists c E R such that czq 6 am for all q gt 0 But then c6 alql 2 1 a zql for all q gt 0 If 2 a there exists a maximal ideal m of R with a z 2 Q ml We then get ce ltazzgtmng mRmW qgtgt0 qgtgto CHARACTERISTIC p METHODS AND TIGHT CLOSURE 5 but then c E qgtgt0 mRmq which is zero by Krull7s intersection theoremi This contradicts the assumption that c E Roi 2 Let S be a direct summand of a weakly Fregular domain Ri For an ideal a C S and element 2 E S suppose that 2 E a Then czq E alql for all q gt 0 where c E S is a nonzero element This implies that czq 6 alqu for all q gt 0 and so 2 E aR aRi Since 5 is a direct summand of R we have aR S a and hence we get 2 E a 3 Let S Q R be a module nite extension of domains Then the fraction eld QR of R is a nite dimensional vector space over the fraction eld of 5 Choose an QSlinear map goo a with 00G 0 Since 00R is a nitely generated S submodule of QS there exists a nonzero element d E S such that go dgoo R a 5 Let c l E S which we note is a nonzero elementi Let a zliuzn be an ideal of S and let 2 E aR 5 Then there exist elements Ti 6 R such that 2 lelTnzni Taking Frobenius powers of this equation we get ZqTgIgTI for all 410 Applying go now gives us sou 24gt 90091 sown e am for an q p8 and so 2 E a What we have proved is that if S Q R is a module nite extension of domains of positive characteristic then aR 5 Q at for all ideals a of 5 If S is weakly Fregular then for all ideals a of S we get aR 5 CL To complete the proof we need a result of Hochster H02 given a module nite extension 5 Q R of excellent domains 5 is a direct summand of R if and only if aR 5 a for all ideals a C 5 see Theorem 510 D Remark 17 We saw that excellent weakly Fregular domains are direct sum mands of module nite extension domains Theorem 16 An integral domain is said to be a splinter if it is a direct summand of every module nite extension domain It is easy to verify that a splinter is a normal domain Characteristic zero If a normal domain 5 contains the eld of rational numbers and R is a module nite extension domain then the trace map of fraction elds QR A can be used to construct a splitting p R a 5 Consequently an integral domain of characteristic zero is a splinter if and only if it is normal Mixed characteristic It is for rings of mixed characteristic that the local homo logical conjectures remain unresolved In this case the canonical element conjecture 6 ANURAG K SINGH the improved new intersection conjecture and the monomial conjecture are equiva lent to the conjecture that every regular local ring is a splinter which is known as the direct summand conjecture For work on these and related homological ques tions we refer the reader to the papers Du EG Hei H01 H03 PS R01 R02 Positive characteristic As we saw excellent weakly Fregular domains of pos itive characteristic are splinters and Hochster and Huneke also proved the converse for Gorenstein rings This was extended by the author to the class of Gorenstein rings in SiQ and more recently to rings whose anticanonical cover is Noetheriani One of the incentives for proving that the splinter property and weak Fregularity agree for rings of positive characteristic is that it is easy to show the localization of a splinter is a splinter It is an open question whether weak Fregularity localizes in general though this is known in the graded case Lysli This explains the nomenclature the term F regular is reserved for rings for which every localization is weakly Fregulari These issues are closely related to the question whether the tight closure at of an ideal a agrees with its plus closure CIT CLRT N R where R4r denotes the integral closure of R in an algebraic closure of its fraction eld An excellent domain R is a splinter if and only if CIT a for all ideals a of R The containment CIT Q at is easily seen and Smith established the equality CIT CI for parameter ideals see De nition 21 of excellent domains Smlli Whether tight closure and plus closure agree is perhaps the most fascinating problem in tight closure theoryi Work on this question eventually led Hochster and Huneke to the celebrated theorem that R4r is a big CohenMacaulay algebra for any excellent local domain R of positive characteristic A related result is that the separable part of R4r is also a big CohenMacaulay algebra Si3li It should be mentioned that amongst various other consequences establishing the equality of Cl and CIT would prove that tight closure localizes a problem that has persisted since the inception of tight closure theoryi Remark 18 A linear algebraic group is Zariski closed subgroup of a general linear group GL7 A linear algebraic group G is linearly reductive if every nite dimensional Gmodule is a direct sum of irreducible Gmodules equivalently if every Gsubmodule has a Gstable complementi Linearly reductive groups in characteristic zero include nite groups algebraic tori ilei products of copies of the multiplicative group of the eld and the classical groups GLAK SLAK Sp2nK OAK and SOAK A linear algebraic group is reductive if its largest closed connected solvable normal subgroup is an algebraic torusi ln characteristic zero linearly reductive groups are precisely those which are reductivei CHARACTERISTIC p METHODS AND TIGHT CLOSURE 7 If a linearly reductive group acts on a nitely generated Kalgebra R say by degree preserving Kalgebra automorphisms then there is an RGlinear map the Reynolds operator p R 8 RC which makes RC a direct summand of R By our earlier discussion it then follows that RC is Noetherian However if the eld K has positive characteristic there need not be a Reynolds operatorireductive groups in positive characteristic usually fail to be linearly reductive Example 46 In the preface of his book Mum Mumford conjectured that reductive groups satisfy a weaker property which should ensure that RC is Noetherian and this led to the notion of geometrically reductive groups A linear algebraic group G is geometrically reductive if for every nite dimensional G module V and Gstable submodule W of codimension one such that G acts trivially on VW there exists n E N such that W S V has a Gstable complement in S V ln NaS Nagata proved that RC is nitely generated if G is geometrically reduc tive and Haboush in Hab settled Mumford7s conjecture by proving that reductive groups are geometrically reductive It is interesting to note that for reductive groups G though aR RC may not be contained in a we always have aR RC Q rada Na3 Lemma 523 2 THE COHEN MACAULAY PROPERTY De nition 21 Let M be an Rmodule Elements 21 2d of R form a regular sequence on M if 1 21 zdM M and 2 2139 is not a zerodivisor on M21 2i1M for every 239 with l S i S d A local ring R m is CohenMacaulay if some equivalently every system of parameters for R is a regular sequence on R A ring R is CohenMacaulay if the local ring Rm is CohenMacaulay for every maximal ideal m of R If R is an N graded ring nitely generated over a eld R0 K then R is CohenMacaulay if and only if some equivalently every homogeneous system of parameters for R is a regular sequence For a local or N graded ring R m and a graded R module M the depth of M denoted depth M is the length of a maximal sequence of elements of m which form a regular sequence on M Consequently a local ring R is CohenMacaulay if and only if depthR dim R How does the CohenMacaulay property arise in invariant theory We start with an elementary example Example 22 Let K be an in nite eld and R Kzlzg yhyg Consider the action of the multiplicative group G K 0 as follows A E G I 1611712791792 H fO ILAIQaAilylaAilyZ 8 ANURAG K SINGH Note that under this action every monomial is taken to a scalar multiple Let f E R be a polynomial which is xed by the group action If a monomial occurs in f with nonzero coefficient comparing coef cients of this monomial in f and M gives us All vikil 1 for all A 6 G Since G is in nite we must have ij k l It follows that the ring of invariants is precisely RC Klzlyly 11M 12M I2y2l Note that dim RC 3 for example by examining the transcendental degree of the fraction eld The polynomial ring 5 K211212221222 surjects onto RC via the K algebra homomorphism go with go zij gt gt my It is easily seen that 80211222 212221 0 Since dimS 4 the kernel of go must be a height one prime of S and it follows that kergo 211222 7 212221 Remark 23 Given an action of G on a polynomial ring R the rst fundamental problem of invariant theory according to Hermann Weyl We is to nd generators for the ring of invariants RC in other words to nd a polynomial ring 5 with a surjection go S Raf The second fundamental problem is to nd relations amongst these generators ie to nd a free S module 53951 which surjects onto ker so In Example 22 we solved these two fundamental problems for the prescribed group action In general continuing this sequence of fundamental questions one would like to determine the resolution of RC as an S module ie to determine an exact complex gtSb3gt5b2gt5b1 QSLRGQO Hilbert s syzygy theorem implies that a minimal such resolution is nite Since RC is a graded module over the polynomial ring 5 minimal can be taken to mean that the entries of the matrices de ning the maps Sbltl 53951 are homogeneous elements of S which are nonunitsi Knowing the graded free modules occurring in the resolution it is then easy to compute the dimension multiplicity and more generally the Hilbert polynomial of Rel Another fundamental question then arises what is the length of the minimal resolution of RC as an S module ie what is the projective dimension pds RC 7 By the AuslanderBuchsbaum formula pdSRG depth S 7 depth R9 The polynomial ring 5 is CohenMacaulay and depth RC S dimRG so we get a lower bound for the length of a minimal resolution pdSRG 2 dimS 7 dim RC CHARACTERISTIC p METHODS AND TIGHT CLOSURE 9 Equality holds if and only if depth RC dirn RC ie precisely when RC is Cohen Macaulay This leads us to the question When is RC CohenMacaulay Theorem 24 HochsterEagon Let R be a polynomial ring over a eld K and let G be a nite group acting on R by degree preserving Kalgebra auto morphisms If RC is a direct summand of R then RC is CohenMacaulay In particular lGl is invertible in K then RC is CohenMacaulay Proof Let g 21 20 be a homogeneous system of parameters for the ring RC Since G is nite R is an integral extension of RC Consequently g is a system of parameters for R and hence is a regular sequence on R But RC is a direct summand of R so g is a regular sequence on RC as well D Remark 25 There are examples of nite groups G for which RC is not Cohen Macaulay due to Bertin and Fossurn Gri ith Let R Kzl zq be a polyno mial ring over a eld K of characteristic p gt 0 where q p8 Fix a generator 0 for the cyclic group G Zq and consider the Klinear action of G on R where a cyclically permutes the generators 11 zq Then for q 2 4 the ring of invariants RC is a unique factorization domain which is not CohenMacaulay Moreover this is preserved if R is replaced by its completion R at the homogeneous maximal ideal and the action of G on R is the unique continuous action extending the one on R For proofs see Ber and The proof of Theorem 24 works more generally to show that a direct summand S of a CohenMacaulay ring R is CohenMacaulay provided that a system of pa rameters for 5 forms part of a system of parameters for R In general a direct summand of a CohenMacaulay ring need not be CohenMacaulay as we see in the next example Example 26 Let K be an in nite eld and let R be the hypersurface R KryzstI3 y3 23 Then R has a Klinear action of G K 0 where zgt gtz Sgt gt718 A y gty and A for AEG tgt gt71t 2gt gt2 Similar to Example 22 the ring of invariants RC is the Kalgebra generated by the elements sz sy s2 tr ty and t2 It is easy to see that RC is a direct summand of the CohenMacaulay ring R More generally an algebraic torus is linearly reductive However RC is not CohenMacaulay the elements sz ty syitz 10 ANURAG K SINGH form a homogeneous system of parameters for RC and satisfy the relation sztzsy 7 tr sz2ty 7 t2Qsz7 and therefore sy 7 tr is a zerodivisor on RGsz7 tyRG Though the CohenMacaulay property is not preserved by direct summands7 there is a beautiful theorem of Hochster and Roberts which implies that several important rings of invariants are CohenMacaulay Theorem 27 Let G be a linearly reductive group acting linearly on apoly nomial ring R Then RC is CohenMacaulay More generally a direct summand of a polynomial ring is CohenMacaulay This has been extended by Hochster and Huneke to all equicharacteristic regular rings using their construction of big CohenMacaulay algebras Theorem 28 Let S be a direct summand of a regular ring containing a eld Then S is CohenMacaulay We have seen that direct summands of weakly Fregular domains are weakly Fregular7 Theorem 16 In view of this7 the following theorem gives us an elementary proof of the Hochster Roberts theorem in the positive characteristic graded case We will return to the characteristic zero case in 4 Theorem 29 Let R7m7K be a complete local domain of characteristic p gt 0 or an Ngraded domain nitely generated over a eld Rlo K Let yl yd be a system of parameters for R or a homogeneous system of parameters in the graded case Then y17 7yizy i1 Q 917mny far all US i S d 1 In particular ifR is weakly F regular then it is CohenMacaulay Proof In the complete case7 R is modulefinite over A Ky1yd and in the graded case over the graded subring A Ky1 yd Suppose there exist 27r17ri E Rwith Zyi1 Tiyi Tili7 taking p8 th powers7 we get zqyg1 ry rgyg for all g p8 Let A be a free Asubmodule of R where t is as large as possible Then RA is a finitely generated torsion Amodule7 hence is killed by some nonzero element c E A Since cR Q A 7 multiplying the equation by c7 we get czqyg1 E 7 yg l CHARACTERISTIC p METHODS AND TIGHT CLOSURE 11 The elements yl yi1 form a regular sequence on A hence on the free Amodule A and so we get czqe y ygA y ygR for all 4108 This implies that 2 E y1 yi as desired D Under mild hypotheses weak Fregularity is preserved on taking direct sum mands and implies the CohenMacaulay property There is another class of rings in characteristic zero with these properties De nition 2 10 Let X be a normal irreducible variety over an algebraically closed eld K of characteristic zero Then X has rational singularities if for some equiv alently every desingularization f Z a X we have Rifi OZ 0 for all i 2 1 If X has rational singularities then all local ring of X are CohenMacaulay We say that R has rational singularities if Spec R has rational singularities There are useful numerical criterion to detect when a graded ring has rational singularities see Fl or Wa2 Theorem 211 Boutot Bo Let R be a nitely generated algebra over a eld of characteristic zero and S be a direct summand ofR IfR has rational singularities then so does 5 The rational singularity property is related to a property which arise in tight closure theory De nition 212 An ideal a 11 zn is said to be a parameter ideal if the images of 11 In form part of a system of parameters in the local ring Rp for every prime ideal p containing a A ring R of positive characteristic is F rational if every parameter ideal of R equals its tight closure Of course a weakly Fregular ring is Frational The notions agree for Gorenstein rings but not in general see A ring R Kzl Inla nitely generated over a eld K of characteristic zero is said to be of dense F rational type if there exists a nitely generated Zalgebra A C K and a nitely generated free A algebra RA AlrlvwwrnlaA such that R 2 RA A K and for all maximal ideals m in a Zariski dense subset of Spec A the ber rings RA 8 Am are Frational rings of characteristic p gt 0 Smith proved that rings of dense Frational type have rational singularities Sm2 and the converse is a theorem of Hara Har as well as Mehta Srinivas Com bining these results we have 12 ANURAG K SINGH Theorem 213 Let R be a ring nitely generated over a eld of characteristic 0 Then R has rational singularities if and only it is of dense F rational type For other striking connections between tight closure theory and singularities in characteristic zero see We conclude this section with some examples of rings of invariants Although we will not prove anything except perhaps in one case and refer the reader to We and DP we hope the following examples give a glimpse of this rich subjectl Example 214 Let X be an n X d matrix of variables over a eld K and consider the polynomial ring R KX iel R is a polynomial ring in nd variables Let G SLAK be the special linear group acting on R as follows M zij MXij ie an element M E G send zij the ij entry of the matrix X to the ij entry of the matrix MXl Since detM 1 it follows that the size n minors of X are xed by the group action It turns out whenever K is in nite RC is the Kalgebra generated by these size n minors The ring RC is the homogeneous coordinate ring of the Grassmann variety of n dimensional subspaces of a d dimensional vector space The relations between the minors are the wellknown Pliicker relations The reader is invited to prove that RC is a unique factorization domain The key point is that since the commutator of the group G SLAK is GG C any homomorphism from G to an abelian group must be trivial In particular there are no nontrivial homomorphisms from G to the multiplicative group of the eld Once we know that RC is a unique factorization domain Murthyls theorem Mur implies that RC is Gorensteinl More generally the ring of invariants of a connected semisimple linear algebraic group acting linearly on a polynomial ring is a CohenMacaulay unique factorization domain hence also Gorensteinl For more on the Gorenstein property of RC see Wall and Example 215 Let X and Y be r X n and n X 8 matrices of variables over an in nite eld K and consider the polynomial ring R KX Y of dimension rnnsl Let G GL7 be the general linear group acting on R where M E G maps the entries of X to corresponding entries of XM 1 and the entries of Y to those of MY Then RC is the K algebra generated by the entries of the product matrix XY lf Z is an r X 8 matrix of new variables mapping onto the entries of XY the kernel of the induced Kalgebra surjection KZ RC is the ideal generated by the size n 1 minors of the matrix Zl These determinantal rings are the subject of BV The case where r s 2 and n 1 was earlier encountered in Example 22 Let us nally go through a computation in some detail CHARACTERISTIC p METHODS AND TIGHT CLOSURE 13 Example 216 Let X be an n X n matrix of variables over an in nite eld K and consider the polynomial ring R Let G CLAK be the general linear group acting linearly on R where M E G maps entries of the matrix X to corresponding entries of MXM I We shall determine the ring of invariants RC This example is a special case of Pr The matrices X and MXM 1 are conjugate so detX traceX and more generally the coef cients of the characteristic polynomial pt dettI 7 X of X are xed by the group action We claim that RC is the Kalgebra generated by the coef cients of pt Let Y be an n X n matrix of new variables and set S RY 1 detY Given 6 RC consider the element fYXY 1 E 5 When Y is specialized to any matrix in GLAK the specialization of fYXY 1 agrees with Since fYXY 1 7 vanishes for all such specializations it must vanish identically ie fYXY 1 Let L be the algebraic closure of the fraction eld of R When we specialize the offdiagonal entries of X to 0 the resulting matrix has distinct eigenvalues 111I7m and it follows that X has distinct eigenvalues in L Consequently there exists a matrix N E GLnL such that NXN 1 D is diagonal the entries of D being the eigenvalues of X Specializing Y to the matrix N we see that fD Hence is a polynomial in the entries of D ie is a polynomial function of the eigenvalues of X Moreover for a permutation 7r 6 Sn consider the corresponding permutation matrix P Then fPDP 1 fD so fX is a symmetric function of the eigenvalues of The elementary symmetric functions of the eigenvalues are up to sign the coef cients of the characteristic polynomial so we have proved our claim We have now solved the rst fundamental problem for this group action The second fundamental problem is to determine the relations if any between the coef cients of the characteristic 39 of X 1 39 quot 39 the f d39 1 entries 1 of X to 0 the coef cients of the characteristic polynomial of the resulting matrix are the n elementary symmetric functions in 111 mm which are algebraically independent It follows that the coef cients of pt are algebraically independent as well so RC is a polynomial ring in n variables 3 THE BRIANQON SKODA THEOREM De nition 31 Let a be an ideal of a ring R An element 2 E R is in the integral closure of a denoted E if it satis es an equation of the form 2na12n71 a22n72 an 0 14 ANURAG K SINGH where ai E ai for all 1 S i S n If R is Noetherian another characterization of the integral closure is that 2 E E if and only if there exists 5 E R such that CZn E a for in nitely many positive integers n or equivalently for all n gtgt 0 Yet another characterization in the Noetherian case is that 2 E E if and only if z E aV for every homomorphism g0 R a V where V is a rank one discrete valuation ringi It is easy to see that E is an ideal of R with a Q E Q radai Moreover if R has characteristic p gt 0 then at Q Hi Let R Czl i i i Id be the ring of convergent power series in d variables over the complex numbers If f belongs to the maximal ideal of R then using the valuation criterion we can see that 19f 9f f6 lt87 iiif gti This implies that some power fk of f belongs to the ideal generated by the partial derivatives and John Mather asked if there is a bound on this power 16 Wall If f is a homogeneous polynomial of degree n then It 1 suf ces since the Euler identity implies E E nf lt11quot39Id67gtA However k 1 may not be suf cient for an inhomogeneous polynomial f for example take f 12f I5 y5 E R where R Czyi We claim that f does not belong to the ideal 19f 19f a a a This is easily veri ed by comparing power series coef cients or alternatively by gt 21y 5147 212y 5f working modulo the ideal I y Briancon and Skoda answered Mather7s question by proving that fd belongs to the ideal generated by the partial derivatives SB Their proof uses the conver gence of certain integralsi The absence of a purely algebraic proof for such an algebraic statement was highlighted by Hochster in his lectures on Analytic meth ods in commutative algebra and became to quote Lipman and Teissier something of a scandaliperhaps even an insultiand certainly a challenge77 for algebraistsi The rst algebraic proofs were found by Lipman and Teissier and subsequently the result was extended to ideals in arbitrary regular local rings by Lipman and Sat haye Theorem 32 LiT Lisl Let R be a regular ring and a be an ideal generated by n elements Then E Q a CHARACTERISTIC p METHODS AND TIGHT CLOSURE 15 Tight closure theory gives an extremely elementary proof for regular rings of positive characteristic Theorem 33 Hochster Huneke Let R be a Noetherian ring ofprime character istic and a be an ideal generated by n elements Then a7 Q a In particular ifR is weakly F regular then E Q a Proof lt suffices to verify the assertion modulo each minimal prime of R so we assume R is a domain Let a 11 l l l zn and 2 6 Fl By one of the characteri zations of integral closure there is a nonzero element c E R such that k nk c2 6 a for all h gtgt 0 By the pigeonhole principle at Q If i l l zi so restricting h to q p8 we get czq E alql and hence that 2 E a D The reader should have no difficulty in proving that for R and a as above amt Q CLMTIV for all m 2 0 There is a beautiful extension of the Briangon Skoda theorem due to Aberbach and Huneke Theorem 34 Let R be an F rational ring of positive characteristic or a nitely generated algebra over a eld of characteristic zero which has rational singularities If a is an ngenerated ideal of R then amt Q am for all m 2 0i Lipman used the notion of adjoint ideals to obtain improved Briangon Skoda theorems in Lil Improvements involving coefficient ideals may be found in AHZ and for applications to Rees rings see AHL AHTll Rees and Sally studied the BrianconSkoda theorem from another viewpoint in RS and Swanson7s related work on joint reductions appears in SwL Sw2ll ln Wall7s lectures on Matherls work where it all began it is amusing to find the sentence Wal page 185 Once the seed of algebra is sown it grows fast 4 REDUCTION MODULO p The first use of reduction modulo p methods we usually encounter is in the proof that cyclotomic polynomials are irreducible Dedekind 1857 typically done in a graduate course in abstract algebral The basic idea in Dedekind s proof as in most reduction modulo p proofs is to start with a statement in characteristic zero reduce modulo p and then exploit the Frobenius The technique has proved extremely 16 ANURAG K SINGH useful in commutative algebra and yielded results for the equicharacteristic cases of the homological conjectures We shall use reduction modulo p methods here to prove the BrianconSkoda theorem for regular rings of characteristic zero as well as the Hochster Roberts theoremi There are beautiful results relating the characteristic 0 and characteristic p prop erties of algebraic setsi Starting with a polynomial fzli i i rd 6 Zn i i i zd the solution set of f 0 in Cd is a topological space The Weil conjecturesinow theorems of Grothendieck and Deligneirelate the Betti numbers of this topological space to the number of roots of E Zpzli i i rd in the nite elds lea where p8 is a prime poweri Closer to the applications we have in mind here is the following elementary result Proposition 41 Consider a family of polynomials f1ixfn E Zhl 1 1 Then these polynomials have a common root over C if and only for all but nitely many prime integers p their images have a common root over the algebraic closure of Zp Proof If 11 i i i ad 6 Cd is a common root of the given polynomials consider the subring A Za1 i i i ad of C Let m be any maximal ideal of Al Then Am is a eld which is nitely generated as a Zalgebra and hence is a nite eld see AM Exercise 76 Let p be the characteristic of Ami Using 7 to denote images modulo m the point 61 i i i Ed 6 Amd is a common root of the images of the fi in Zpzli 1dli It remains to verify that A has maximal ideals containing in nitely many prime integersi By the Noether s normalization lemma AQ Qa1i i i ad is an integral exten sion of a polynomial subring Qy1 i i i ytli Each ai satis es an equation of integral dependence over Qy1 i i i ytli Each of these d equations involves nitely many coefficients from Q so after localizing at a suitable integer r we have an integral extension Zlylyu 7yt71Tl Q AllTl For every prime integer p not dividing r there is a maximal ideal of Zlyi i i yt lr which contains pi Since the extension is integral there exists a maximal ideal of Alr lying over every maximal ideal of Zy1 i i i yt lrli Conversely if the polynomials do not have a common root in Cd then Hilbert s Nullstellensatz implies that f1 i i i fn generate the unit ideal in Czl i i i 10 ie that Czlxizdf1 i i i fn 0 But Czlxizd Qzlxizd E C flit m Japan Q so Qzl i i i Lilfl i i i 0 since C is faithfully flat over This implies that f1 i i fn generate the unit ideal in Qzli i i 10 as well and so after multiplying CHARACTERISTIC 1 METHODS AND TIGHT CLOSURE 17 by a common denominator7 we have an equation of the form f191fngnm where g17 7gn E Zzl7 7 Idl7 and m is a nonzero integer For every prime integer p not dividing m7 the images of f17 7 fn generate the unit ideal in Zpzl7 7 Idl7 and hence cannot have a common root over the algebraic closure of Zp B One of the ingredients we will need in subsequent applications of reduction mod ulo p methods is the generic freeness lemma of HochsterRoberts A special case may be found in Mal7 Chapter 8 and the general result is HR7 Lemma 81 Lemma 42 Generic freeness Let A be a Noetherian domain7 B a nitely gener ated Aalgebra7 and C a nitely generated Balgebra Let N be a nitely generated Cmodule7 M a nitely generated Bsubmodule of N7 and L a nitely generated Asubmodule of N Then there exists a nonzero element a E A such that the local i L M a We next prove the BrianconSkoda theorem for regular rings of characteristic ization is a free Allmodule zero Theorem 43 Let a be an ngenerated ideal of a regular ring which contains a eld of characteristic zero Then angai Proof We rst consider the case where the regular ring R is nitely generated over a eld K7 and return to the general case later in this section Consider R as a homomorphic image of a polynomial ring T Kzl7 7 Idl7 say R Tg17 7gm Let f17 7 fn be elements of T which map to generators of the ideal a C R If the statement of the theorem is false7 there exists 2 E T whose image in R belongs to the set 07 a Hence there exist elements ai E fl7 i 7 and h17 7hm E g177gm such that zla12l 1azh191hmgm 7 Let A be a nitely generated Z subalgebra of K containing the coef cients of 27 fi7 gi7 ai7 and hi as polynomials in 117 7 1017 and also the coef cients of polynomials needed to express each ai as an element of 161 7 In the next few steps7 we shall replace A by its localization at nitely many elements The conditions we require of A are preserved under further localization7 and we do not change our notation for A It is worth emphasizing that when we 18 ANURAG K SINGH adjoin the inverses of nitely many elements to the ring A we retain the property that it is a nitely generated Z subalgebra of Kl Consider the Aalgebra RA Azlulzdlglplqgm Let denote the fraction eld of Al The inclusion a K is a at homo morphism of Amodules so upon tensoring with RA we get a at homomorphism RA A QM A RA A K E R The ring R is regular so it follows that RA AQA is regular as well Since is a eld of characteristic zero it follows that RA A is a smooth QAalgebrai After inverting an element of A we may assume that A 8 RA is smooth Given a homomorphism from A to a eld H we use the notation RN RA A H For every such H the ring RN is smooth over H by base change In particular RN is regular After localizing A at one element we may assume that RAfl i i fnRA and RAz f1 i fnRA are free Amodulesl This ensures that 161 fnRA A H and 2161 i i fnRA A H are ideals of the ring Rm Since the image of 2 in R does not belong to the ideal 161 i i fnR after inverting one more element of A we may assume that 2f1i lfnRA f1 s 7 nRA is a nonzero free Amodulei Let m be a maximal ideal of Al The eld H Am is nitely generated as a Z algebra hence is a nite eld We use 2m to denote the image of 2 in Rm The freeness hypotheses give us the isomorphism 27 f1v7fngtRA A H g nyiwwfnRm flvwwfn zA f17 7fnRH 7 and also ensure that this module is nonzero ie that 2N f1i i 167le The image of equation in RN implies that Zn 6 f17wfnnRHA But RN is a regular ring of positive characteristic so this contradicts the charac teristic p BrianconSkoda theorem we proved earlier as Theorem 33 D For the general case we will need a rather deep result on regular homomorphisms which we state next See the appendix particularly the subsection Fibers and geometric regularity for relevant de nitions Theorem 44 General Neron Desingularization P0 Let go R a S be a regular homomorphism which factors through a nitely presented Ralgebra R CHARACTERISTIC p METHODS AND TIGHT CLOSURE 19 Then there exists an R algebra T such that the composite homomorphism R a T is smooth and the following diagram commutes R a R a S T T Consider the special case where R is the localization of a polynomial ring over a eld and go R a R is the map to the completion Since R is excellent the map go is regular and so General Neron Desingularization applies here This special case is a theorem of Artin and Rotthaus Theorem 45 Let R Kzlluzdm ie R is the localization of a polynomial ring at its homogeneous maximal ideal m Let R denote the madic completion of R Then given any nitely generated Rsubalgebra R of R the inclusion R a R factors as R a R a T a R where R a T is smooth Proof of Theorem 43 in the general case Suppose the assertion of the theorem is false there exists 2 E a7 with 2 CL Let p be a prime ideal containing the ideal a z 2 Localizing at p and completing we then have a counterexample in a power series ring KHzlp 1dll where K is a eld of characteristic zerol Let R Kzlluzd 1m dl Let R be an R subalgebra of R KHILHWIdH which contains the element 2 a set of generators for the ideal a and the elements of R occurring in one equation demonstrating that 2 E a7 in terms of the chosen generating set for CL By the ArtinRotthaus theorem the maps R a R factors as R a R a T a R where R a T is smooth In particular T is regular and the images of 2 and a in T also yield a counterexample say 20 E aol Note that T is a regular ring of the form T S IB where B is a nitely generated algebra over the eld Kl Since S IB is regular 5 contains an element a of the de ning ideal of the singular locus of El The element 20 the generators for a0 and the elements occurring in one equation implying 20 E involve nitely many elements hence nitely many denominators from S and we take I to be the product of these denominatorsl We then obtain a counterexample in the ring Bab which is a regular ring nitely generated over the eld K but we have already proved the BrianconSkoda theorem in this case B 20 ANURAG K SINGH It should be mentioned that this approach is not the only way to deduce char acteristic zero results from positive characteristic theorems Schoutens Sc uses modeltheoretic methods to obtain the BrianconSkoda theorem for power series rings CHXl i i Xdll using the characteristic p version Theorem 33 We now return to the Hochster Roberts theorem in characteristic zeroi One subtle point is that the property that a ring 5 is a direct summand of R may not be preserved when we reduce modulo primes p as we see in the following example Example 46 Consider the special case of Example 214 where n 2 and d 3 ire K is an in nite eld and G SL2K acts on the polynomial ring R Kuvwzyz where M E G maps the entries of the matrix Xltuvwgt Iyz to those of MXi The ring of invariants for this action is RC KA1A2 A3 where A1vziwy A2wziuz A3uy7vz are the size two minors of the matrix Xi These minors are algebraically independent over K and hence RC is a polynomial ringi If K has characteristic zero then G is linearly reductive and hence RC is a direct summand of R We shall see that RC is not a direct summand of R if K is any eld of characteristic p gt 0 Let a A1 A2 A3R and consider the local cohomology module H ltRgt 113m Raam Inn Examam 2 1175 where the isomorphism follows from the fact that the sequences of ideals atheN and alpglheN are co nali The projective resolution of Ra is P oaRZLRSMRau Since the Frobenius F R a R is at the complex F8 R ER P is acyclic and hence is a resolution of Ralqli This implies that Ext 3Ralql R 0 for all 4 p8 and hence that H R 0 Using the description of H R as the cohomology of the Cech complex 0 A R A RA1 GERAQ SEEM A RAgAg EB RAgAl RA1A2 A RAlAgAg 07 HE R 0 implies that 1 7 7 1 7 2 7 3 A1A2A3 A2A3 A3A1 A1A2 for Ti 6 R and t E N Clearing denominators gives us A1A2A3quot1 T1A WA TSAg 6 A3 A3 Agm CHARACTERISTIC p METHODS AND TIGHT CLOSURE 21 Since A1A2A3quot1 A37 A37 A RG7 it follows that RC is not a direct summand of Rf We now return to the Hochster Roberts theorem in the form Theorem 47 Let R Kzl7 i i i 7177 be a polynomial Ting over a eld K7 and let S be an Ngmded subring of R which is a direct summahd of R Then S is CohenMacaulay Proof of Theorem 27 Let y17i i i 7 yd be a homogeneous system of parameters for the ring 5 Suppose there exist si 6 S with Slylquot395mym0 and sm y17ui7ym715 then7 since 5 is a direct summand of R7 we also have sm y17 i i i 7ym71Ri The ring 5 is module nite over its polynomial subring B Klyh i i 7ydl7 say 5 Bui where ui are nitely many homogeneous elements of 5 For all 1 S i7j7t S N7 there exist elements cijt E B such that uiuj 251 cijtuti Let A be the ring obtained by adjoining to Z the following elements of K o the coef cients that occur when each yi and each ui is written as a polyno mial in 117 i i f 7177 0 coef cients occurring when each cijt is written as a polynomial in M7 i i i 7 W7 0 for each si7 the coef cients of the polynomials in M7 i i i 7yd needed in one equation expressing si as a Blinear combination of u17i i i 7 uNi Note that A is a nitely generated Z algebra of Ki Set N BA Aly17iu7ydl7 SA EBAui7 and RA Azl7i i i 7177 i1 Then SA is a subring of RA and is module nite over its subring BA Since sm 917 i i i 7 ym71R7 after replacing A by a localization at one element7 we may assume that 917 7ym71RA is a nonzero free Amodulei After localizing A at an element7 we may also assume that each of RA vaylvwwymilRAl ylyu yymilRAl is a free Amodulei SABA7 and RASA Let m be a maximal ideal of Al Then H Am is a nitely generated Z algebra7 hence a nite eld We use the notation MN to denote M A H7 and 7 to denote the image of an element modulo mi Note that RN Mil i i 7En and 22 ANURAG K SINGH BN H 1Hld are polynomial rings over the eld H and that the freeness hypotheses ensure that BN Q SN Q Rm The image of equation in 5H gives us 31 39 39 39 mym 07 which is a relation on the homogeneous system of parameters Q1 i i 3 of Sm By Theorem 29 it follows that 5m E y17 7m71gt5r gt Since SN Q RN and RN is a polynomial ring we have gm 6 317 73m71 Raf 317 l l 7371171 Rm but this contradicts the condition that Srnyle llyym71RA N Eng1w 7gm71Rm A H f 917 l 7ym1RA y17 7ym71RN is a nonzero module D 5 APPENDIX Bill The structure theory of complete local rings Every ring R admits a ring homomorphism go Z a R where l is the unit element of Ri The kernel of this homomorphism is an ideal of Z generated be a unique nonnegative integer char R the characteristic of Rf A local ring R m K is equicharacteristic or of equal characteristic if char R char For a local ring R m K the possible values of char and char R are 0 char char R 0 in which case Q Q Ri 0 char char R p gt 0 in which case ZpZ Q Ri 0 char p gt 0 and char R 0 As an example of this take R to be the ring of p adic integersl char p gt 0 and char R p8 for some e 2 2 In this case R is not a reduced ringl If R m is a local ring containing a eld we say that a eld L Q R is a coe cient eld for R if the composition L lt gt R a R m is an isomorphisml Proofs of the following theorems due to Cohen maybe found in Mal Chapter 11 and Mal Appendix Theorem 51 Let R m K be a complete local ring containing a eld Then 1 The ring R contains a coe cient eld 2 IfL Q R is a coe cient eld then R is a homomorphic image of a formal power series ring over L ie R E LT1Hl TnHa CHARACTERISTIC p METHODS AND TIGHT CLOSURE 23 3 If L Q R is a coe icient eld and x17lu7xd is a system of parameters for R then the subring A Lx17l 7Idll of R is isomorphic to a formal power series ring LT17l 7 TA and R is a nitely generated Amodule 4 The ring R is regular is and only it isomorphic to a formal power series ring LT17 7 lel over some coe icient eld L In the case where R7 m7 K does not contain a eld7 for the sake of simplicity we make the assumption that R is an integral domain This ensures that the only possibility is char p gt 0 and char R 07 and this case is usually referred to as mixed characteristic The role of a coefficient eld is now replaced by that of a discrete valuation ring V7 pV7 which serves as a coe icient ring A regular local ring R7m7K of mixed characteristic is unrami ed if p m27 and is rami ed if p E In Theorem 52 Let R7m7K be a complete local domain with char R 0 and char p gt 0 Then there exists a discrete valuation ring V7 pV which is a subring of R such V E R induces an isomorphism VpV Rm 1 The ring R is a homomorphic image of aformal power series ring over V7 ie7 R E VT17 l l l 7 TnHa 2 pr7 x27 7 x01 is a system ofparameters for the ring R consider the subring A Vx27 7xd Then A is isomorphic to a formal power series ring VT27 7 Tag and R is a nitely generated Amodule 3 The ring R is an unrami ed regular local ring and only if it is isomorphic to a formal power series ring VT27 7 lel 4 The ring R is a rami ed regular local ring and only it is isomorphic to VT17 7lelp 7 where f is an element in the square of the maximal ideal of VT17A l 7 TA 52 Excellent rings The class of excellent rings was introduced by Grothendieck to circumvent some pathological behavior that can occur in the larger class of Noe therian rings The precise de nition is somewhat technical in nature7 but is satis ed by most Noetherian rings that are likely to be encountered by mathematicians working in algebraic geometry7 number theory7 or several variable complex analysis For a detailed treatment and proofs of results summarized here7 the reader should consult EGA7 particularly 77 and Mal7 Chapter 13 The expository article MaZ provides a nice introduction to the theory of excellent rings of characteristic zerol Various examples of nonexcellent rings were constructed by Nagata7 and are included as an appendix in his book Na2ll De nition 53 A ring R is said to be excellent if 1 R is Noetherian7 24 ANURAG K SINGH 2 R is universally catenary 3 for every prime ideal p E Spec R the formal bers of the local ring RP are geometrically regular and 4 for every nitely generated R algebra S the regular locus of the ring 5 ie the set p E SpecS Sp is a regular local ring is an open subset of Spec Si Of course we need to de ne some of the terms occurring above Universally catenary rings A ring R is catenary if for all prime ideals p Q q of R all saturated chains of prime ideals joining p and q have the same length A ring R is universally catenary if every nitely generated algebra over R is a catenary ringi Several attempts had been made to prove that all Noetherian rings were catenary until Nagata constructed the rst examples of noncatenary Noetherian rings in 1956 He constructed a local integral domain R m of dimension 3 which is not catenary R has saturated chains of prime ideals joining p 0 and q m of lengths 2 and 3 as illustrated in the diagram below q m n 0 Theorem 54 Dimension formula Let R Q S be integral domains such that R is universally catenary and S is a nitely generated Ralgebra Let q E Spec 5 and p q NR 6 SpecR Then height q trideg Nmam height p trideg 35 where H03 denotes the eld RPpRP and Hq Sqqu and by trideg 35 we mean the transcendence degree offraction eld of 5 over the fraction eld of R Ratliff showed that the above dimension formula in a sense characterizes uni versally catenary rings see RaL Ra2li Fibers and geometric regularity For a ring homomorphism g0 R A S by the ber of go at a prime p E Spec R we mean the ring 5 ER H03 where H03 denotes the eld RPpRPi Note that the inverse image of p under the induced CHARACTERISTIC p METHODS AND TIGHT CLOSURE 25 map Spec 5 Spec R is homeomorphic to Spec S R H03 which explains the use rather the misuse of the word ber For a local ring R m the formal bers of R are the bers of the homomorphism R a R where R denotes the completion of R at its maximal ideal m If a is an ideal of R then m RaR and so a formal ber of the ring Ra is also a formal ber of the ring R A K algebra R is geometrically regular if R K L is a regular ring for every nite extension L of the eld K This is equivalent to the condition that R K L is regular for every nite purely inseparable extension L of K A ring homomorphism go R a S is regular if it is at and for all p E Spec R the ber H03 ER 5 is geometrically regular A ring homomorphism go R a S is smooth if it is regular and S is nitely pre sented over the image of R In this case for every R algebra T the homomorphism R R T S R T is also smooth The ubiquity of excellent rings The result below explains why the Noetherian rings we encounter are almost always excellent rings Theorem 55 If a ring R is obtained by adjoining nitely many variables to a eld or a complete discrete valuation ring taking a homomorphic image and localizing at some multiplicative set then R is an excellent ring More precisely 1 Every complete local ring particular every eld is excellent The ring of convergent power series over R or C is excellent A Dedekind domain whose eld of fractions has characteristic zero eg Z is excellent 2 A nitely generated algebra over an excellent ring is excellent in particular a homomorphic image of an excellent ring is excellent 3 A localization of an excellent ring is excellent There is an interesting class of rings of characteristic p gt 0 which are excellent by the following theorem of Kunz Theorem 56 Kul Ku2 For a ring R of characteristic p gt 0 let F R a R be the Frobenius homomorphism fR is module nite over FR then R is excellent Some properties of excellent rings Theorem 57 If R is an excellent ring then the normal locus as well as the CohenMacaulay locus ie p E SpecR RF is normal and p E SpecR RF is CohenMacaulay are open subsets of Spec R 26 ANURAG K SINGH Nagata de ned a ring R to be a pseudogeometric ring if for every prime p E Spec R and for every nite extension eld K of the eld of fractions of Rp the integral closure of Rp in K is a nitely generated Rp module Examples of Noe therian rings which did not satisfy this property were rst constructed by Akizuki in Ak In honor of the Japanese school of commutative algebra Grothendieck re named pseudogeometric rings as anneaux universellement japonais or universally Japanese rings EGA 77 At some point they were again renamed and are now called Nagata rings Theorem 58 An excellent ring is a Nagata ring The excellence property also ensures that certain properties of a local ring R are inherited by its m adic completion R and this is the essence of the next theorem Theorem 59 Let R be an excellent local ring with maximal ideal m 1 fR is reduced ie has no nonzero nilpotent elements then R its madic completion is also a reduced ring 2 If R is a domain then by 1 above R is a reduced ring In this case there is a bijection between the minimal primes ofR and maximal ideals of R where R denotes the integral closure of R in its eld of fractions In particular R is a domain and only R is local 3 fR is a normal ring then R is also a normal ring We conclude with the following theorem of Hochster Theorem 510 H02 Let S be a reduced excellent local ring with maximal ideal m Then for all n gt 0 there is an mprimary ideal an E m such that the ring San is injective as a module over itself 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