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Calculus I

by: Mrs. Preston Lehner

Calculus I MATH 115

Mrs. Preston Lehner
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This 7 page Class Notes was uploaded by Mrs. Preston Lehner on Thursday October 29, 2015. The Class Notes belongs to MATH 115 at University of Michigan taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/231499/math-115-university-of-michigan in Mathematics (M) at University of Michigan.

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Date Created: 10/29/15
Math 615 Lecture of March 14 2007 Open questions tight closure plus closure and localization We want to consider some open questions in tight closure theory and some related problems about when rings split from their module finite extension algebras After we do this we shall prove some speci c results in the characteristic 1 theory It will turn out that to proceed further we will need the structure theory of complete local rings which we will develop next One of the longest standing and most important questions about tight closure is whether tight closure commutes with localization Eg if R is Noetherian of prime characteristic 1 gt 0 1 is an ideal of R and W is a multiplicative system of R is W 113 the same as W 1 V1R It is easy to prove that W 11 Q W 1 V1R This has been open question for more than twenty years It is known to be true in many cases See for example 1 Aberbach M Hochster and C Huneke Localization of tight closure and and modules of nite phantom projective dimension J Reine Angew Math Crelle7s Journal 434 1993 677114 and Hochster and C Huneke Test exponents and localization of tight closure Michigan Math J 48 2000 3057329 for a discussion of the problem We saw in the last Theorem of the Lecture Notes of March 12 that tight closure cap tures77 contracted extension from module finite and even integral extensions We shall add this as 6 to our list of desirable properties for a tight closure theory which becomes the following 0 u 6 N32 if and only if the image E ofu in MN is in 0WN 1 NITI is a submodule ofM andN Q NITI 1fN Q Q Q M then N Q 62 2 If N Q M then NITI 3 HR Q S are domains and 1 Q R is an ideal 1 Q 1S where 1 is taken in R and 1S in S 4 If A is a local domain then under mild conditions on A the class of rings allowed should include local rings of a nitely generated algebra over a complete local ring or over Z and f1 fd is a system ofparameters for A then for 1 S i S d7 1 f1 fiA 2A fi1 Q fh 014 5 If R is regular then 1 1 for every ideal 1 of R 6 For every module finite extension ring R of S and every ideal 1 of R 15 R Q 1 These are all properties of tight closure in prime characteristic 1 gt 0 and also of the theory of tight closure for Noetherian rings containing Q that we described in the Lecture of March 12 In characteristic 1 gt 0 4 holds for homomorphic images of Cohen Macaulay rings and for excellent local rings If R 2 Q7 4 holds if R is excellent We will prove 1 2 that 4 holds in prime characteristic for homomorphic images of Cohen Macaulay rings quite soon We have proved 5 in prime characteristic p gt 0 for polynomial rings over a field but not yet for all regular rings To give the proof for all regular rings we need to prove that the Frobenius endomorphism is at for all such rings and we shall eventually use the structure theory of complete local rings to do this An extremely important open question is whether there exists a closure theory satisfying 1 7 6 for Noetherian rings that need not contain a field The final Theorem of the Lecture of March 12 makes it natural to consider the following variant notion of closure Let R be any integral domain Let R denote integral closure of R in an algebraic closure K of its fraction field IC We refer to this ring as the absolute integral closure of R R is unique up to non unique isomorphism just as the algebraic closure of a field is Any module finite or integral extension domain S of R has fraction field algebraic over IC and so S embeds in K It follows that S embeds in R since the elements of S are integral over R Thus R contains an R subalgebra isomorphic to any other integral extension domain of B it is a maximal extension domain with respect to the property of being integral over R R is the directed union of its finitely generated subrings which are module finite over R R is also charactized as follows it is a domain that is an integral extension of R and every monic polynomial with coef cients in R factors into monic linear polynomials over R Given an ideal I Q R the following two conditions on f E R are equivalent 1 f E IRJr R 2 For some module finite extension S of R f 6 IS R The set of such elements which is IRJr R is denoted 1 and is called the plus closure of I The definition can be extended to modules N Q M by defining NITI to be the kernel of the map M a R R By the last Theorem of the Lecture Notes of March 12 which is property 6 above in characteristic p gt 0 we have that I g 1 g 1 in prime characteristic p gt 0 Whether 1 1 in general under mild conditions for Noetherian rings of prime characterisitc p gt 0 is another very important open question It is not known to be true even in finitely generated algebras of Krull dimension 2 over a field However there are some substantial positive results It is known that under the mild conditions on the local domain R eg when R is excellent if I is generated by part of a system of parameters for R then 1 1 See E Smith Tight closure of parameter ideals lnventiones Math 115 1994 410 Moreover H Brenner Brenner Tight closure and plus closure in dimension two Amer J Math 128 2006 5317539 proved that if R is the homogeneous coordinate ring of a smooth projective curve over the algebraic closure of ZpZ for some prime integer p gt 0 then 1 1 for homogeneous ideals primary to the homogeneous maximal ideal In G Dietz Closure operations in positive 3 characteristic and big Cohen Macaulay algebras Thesis Univ of Michigan 2005 the condition that the ideal be homogeneous is removed in fact there is a corresponding result for modules N Q M when M N has finite length Brenner7s methods involve the theory of semi stable vector bundles over a smooth curve in fact one needs the notion of a strongly semi stable vector bundle where strongly means that the bundle remains semi stable after pullback by the Frobenius map One reason for the great interest in whether plus closure commutes with tight closure is that it is known that plus closure commutes with localization Hence if 1 1 in general under mild conditions on the ring one gets the result that tight closure commutes with localization The notion of plus closure is of almost no help in understanding tight closure when the ring contains the rationals The reason for this is the result established on pp 475 of the Lecture Notes of March 12 which we restate formally here Theorem Let R be a normal Noetherian domain with fraction eld IC and let S be a module nite ezrtension domain with fraction eld A Let h 1C IfQ Q R or more generally ifh has an inverse in R then EngK gives an R module retraction S a R D It follows that if Q Q R and R is a normal domain then 1 I for every ideal I of R Many normal rings in some sense most normal rings that are essentially of finite type over Q are not Cohen Macaulay and so contain parameter ideals that are not tightly closed This shows that plus closure is not a greatly useful notion in Noetherian domains that contain Q Weakly Fregular rings and Fregular rings We define a Noetherian ring R of prime characteristic 1 gt 0 to be weahly F regular if every ideal is equal to its tight closure ie every ideal is tightly closed We define R to be F regular if all of its localizations are weakly F regular It is not known whether weakly F regular implies F regular even for domains finitely generated over a field This would follow if tight closure were known to commute with localization We have already proved that polynomial rings over a field of positive characteristic are weakly F regular and we shall prove that every regular ring of positive characteristic is F regular This is one reason for the terminology The F suggest the involvement of the Frobenius endomorphism We shall soon show that a weakly F regular ring is normal and if it is a homomorphic image of a Cohen Macaulay ring is itself Cohen Macaulay Theorem A direct summand A of a weakly F regular domain is weakly F regular and a direct summand of an F regular domain is F regular 4 Proof Assume that R is weakly F regular If f E 2 then f 6 IR A IR A I Since the direct summand condition is preserved by localization on A it follows that a direct summand of an F regular domain is F regular D Examples of F regular rings Fix a field K of characteristic p gt 0 Normal rings nitely generated over K by monomials are direct summand of regular rings and so are F regular If X is an r gtlt s matrix of indeterminates over K with l S t S r S s then it is known that is F regular and that the ring generated by the r gtlt r minors of X over K is F regular this is the homogeneous coordinate ring of the Grassmann variety See M Hochster and C Huneke Tight closure of parameter ideals and splitting in module nite extensions J of Algeraic Geometry 3 1994 5997670 Theorem 714 We have already observed that these rings are direct summands of polynomial rings when K has characteristic 0 but this is not true in any obvious way when the characteristic is positive Splitting from module nite extension rings It is natural to attempt to characterize the Noetherian domains R such that R is a direct summand as an R module of every module finite extension ring S We define a Noetherian ring R with this property to be a splinter We then have the following result which was actually proved in the preceding lecture although it was not made explicit there Theorem Let R be a Noetherian domain a IfR is a splinter then every ideal ofR is contracted from every integral ezrtension b IfR is a splinter then R is normal c R is a splinter if and only if it is a direct summand of every module nite domain ezrtension d IfQ Q R then R is a splinter if and only ifR is normal Proof For part a suppose f f1 fh E R and f 221 fisi with the si in S Then we have the same situation when S is replaced by Rs1 sh Hence it su ices to show that every ideal of R is contracted from every module finite extension S But then we have an R linear retraction o S a R and the result is part a of the Lemma at the top of p 2 of the Lecture of February 16 Part b has already been established in the fourth paragraph on p 4 of the Lecture of March 12 For part c we have already observed that S has a minimal prime p disjoint from R 7 0 and it su ices to split the injection R gt Sp Finally for part d the existence of the required splitting when S is a domain is proved at the bottom of p 4 and top of p 5 of the Lecture Notes of March 12 using field trace and restated on p 3 here D Math 711 Lecture of November 12 2007 Before proceeding further with our treatment of test elements we note the following consequence of the theory of approximately Gorenstein rings We shall need similar split ting results in the proof of the generalization stated in the first Theorem on p 2 of the Corollary near the bottom of p 5 of the Lecture Notes from November 9 Theorem Let R be a weakly F regular ring Then R is a direct summand of every module nite ezrtension ring S Moreover if R is a complete local ring as well R is a direct summand ofR In particular these results hold when R is regular Proof Both weak F regularity and the issue of whether R a S splits are local on the maximal ideals of R Therefore we may assume that R m K is local Since R is approximately Gorenstein there is a descending chain It of m primary irreducible ideals cofinal with the powers of m By the splitting criterion in the Theorem at bottom of p 4 and top of p 5 of the Lecture Notes from October 24 R is a direct summand of S or R in case B is complete local if and only if It is contracted for all t In fact ItS R Q If by the Theorem near the bottom of p 1 of the Lecture Notes of October 12 and by hypothesis If It D It is an open question whether a locally excellent Noetherian domain R of prime char acteristic p gt 0 is weakly F regular if and only if R is a direct summand of every module finite extension ring The issue is local on the maximal ideals of R and reduces to the excellent local case By the main result of E Smith Tight Closure of Parameter Ideals lnventiones Math 115 1994 41760 the tight closure of an ideal generated by part of a system of parameters is the same as its plus closure From this result it is easy to see that implies that R is F rational In the Gorenstein case F rational is equivalent to F regular so that the equivalence of the two conditions holds in the locally excellent Gorenstein case We shall prove Smith7s result that tight closure is the same as plus clo sure for parameter ideals The argument depends on the use of local cohomology and also utiliuzes general Neron desingularization We shall also need the main result of Hochster and C Huneke In nite integral extensions and big Cohen Macaulay algebras Annals of Math bf 135 1992 53789 that if R is an excellent local domain then R is a big Cohen Macaulay algebra over R We shall prove this using a recent idea method of Huneke and Lyubeznik cf 0 Huneke and G Lyubeznik Absolute integral closure in positive characteristic Advances in Math 210 2007 4987504 Our next immediate goal is to prove a strengthened version of the Corollary on p 5 of the Lecture Notes from November 9 First note that if A is a regular local ring we can choose a Z valued valuation ord that is nonnegative on A and positive on m For example if a 31 0 we can let ord a be the largest integer k such that a 6 m We thus have an inclusion A Q V where V is a Noetherian discrete valuation ring Now assume that A is 1 complete and complete V as well That is we have a local injection A gt V We also have an injection A gt V For every module finite extension domain R of A where we think of R as a subring of 14 we may form VjR within V VjR is a complete local domain of dimension one that contains V lts normalization which we may form within V is a complete local normal domain of dimension one and is therefore a discrete valuation ring VR The generator of the maximal ideal of V is a unit times a power of the generator of the maximal ideal of VR Hence ord extends to a valuation on R with values in the abelian group generated 1 by E where h is the order of the generator of the maximal ideal of V in VB Since A is the union of all of these rings R ord extends to a Q valued valuation on R that is nonnegative on R and postive on the maximal ideal of R If R is any complete local domain we can represent R as a module finite extension of a complete regular local ring A Hence we can choose a complete discrete valuation ring VR and a local injection R a VB and extend the corresponding Z valued valuation to a Q valued valuation that is nonnegative on R and positive on the maximal ideal of R Theorem valuation test for tight closure Let R m K be a complete local do main of prime characteristic p gt 0 and let ord be a Q valued valuation on R that is nonnegative on R and positive on the maximal ideal of R Let N Q M be arbitary R modules and u E M Then the following two conditions are equivalent 1 u E NITI 2 There exists a sequence vn of elements of R 7 0 such that ord vn a 0 as n a 00 and v7 u is in the image ofR R N in R R M for all n We need several preliminary results in order to prove this The following generalization of colon capturing can be further generalized in several ways We only give a version su icient for our needs here Theorem Let R m K be a reduced ezrcellent local ring of prime characteristic p gt 0 Let 1 zk E m be part of a system of parameters modulo every minimal prime of B Let a1 ak b1 bk 6 N and assume that ai lt bi for alli Then z1z2k 2R x111ka filial zZiTGiV Proo Let di bi 7 ai It is easy to see that each di multiplies ital ak into 2 1 k b b b b 117 z kk g 117 7 lgk7 since di ai bi for every i But if I is tightly closed so is any ideal of the form I R y This is equivalent to the statement that if 0 is tightly closed in RI then it is also tightly closed in the smaller module yRI E RI 2R Since 1117 7 g 217 z zkly ZR t 2 we also have d d b b If Mcka Q If ickk 2R 11116 Thus it su ices to prove the opposite inclusion By induction on the number of ai that are not 0 we reduce at once to the case where only one of the ai is not 0 because quite generally 2R yz 12R y R 2 By symmetry we may assume that only ak 31 0 We write xk ac ak a bk b and dk d Let J 111 261 Suppose Joan E J xbR Let c E R0 be a test element Then cacqauq E qul icqu for all 1 gtgt 0 and for such 1 we can write cacqauq jq icqbrq where jq E qul and rq E R Then xq cuq 7 rqich E Jquot and by the form of colon capturing already established we have that cuq 7 rqich E qul and hence c2uq 7 crqach E qul Consequently 02uq E qul quR J deq for all 1 gtgt 0 and so u E J acdR as required D Theorem Let A m K be a complete regular local ring of characteristic p and let ord be a Q ualued ualuation nonnegatiue on A and positive on the maximal ideal of A Let u E A 7 0 be an element such that ord u is strictly smaller than the order of any element ofm it su ces to check the generators of Then the map A 7 A such that l gt gt u splits ie there is A linear map 6 A 7 A such that 61 1 Proof Let 1 xn be minimal generators of m Since A is complete and Gorenstein it su ices by the Theorem at the top of p 3 of the Lecture Notes from October 24 to check that for all t Z ic iH xf1A Suppose that n t t 7 115 xluwnui slacz 21 Let S Au s1 si In S we have u E ic iH xfoS 5 3152 and so u 6 x1 WS by the preceding Theorem on colon capturing But then u is in the integral closure of 1 xnS and this contradicts ord u lt min ord D


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