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# INTRO TO DIFF EQ MATH 307

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SEVERAL APPROACHES TO NONARCHIMEDEAN GEOMETRY BRIAN CONRAD INTRODUCTION Let k be a non archimedetm eld a eld that is complete with respect to a speci ed nontrivial non archimedean absolute value l There is a classical theory of k analytic manifolds often used in the theory of algebraic groups with k a local eld and it rests upon versions of the inverse and implicit function theorems that can be proved for convergent power series over k by adapting the traditional proofs over R and C Serre7s Harvard lectures S on Lie groups and Lie algebras develop this point of view for example However these kinds of spaces have limited geometric interest because they are totally disconnected For global geometric applications such as uniformization questions as rst arose in Tate7s study of elliptic curves with split multiplicative reduction over a non archimedean eld it is desirable to have a much richer theory one in which there is a meaningful way to say that the closed unit ball is connected More generally we want a satisfactory theory of coherent sheaves and hence a theory of analytic continuation Such a theory was rst introduced by Tate in the early 1960s and then systematically developed building on Tate7s remarkable results by a number of mathematicians Though it was initially a subject of specialized interest in recent years the importance and power of Tate7s theory of rigid analytic spaces and its variants due especially to the work of Raynaud Berkovich and Huber has become ever more apparent To name but a few striking applications the proof of the local Langlands conjecture for GLn by HarrisiTaylor uses etale cohomology on non archimedean analytic spaces in the sense of Berkovich to construct the required Galois representations over local elds the solution by Raynaud and Harbater of Abyhankar7s conjecture concerning fundamental groups of curves in positive characteristic uses the rigid analytic GAGA theorems whose proofs are very similar to Serre7s proofs in the complex analytic case and recent work of Kisin on modularity of Galois representations makes creative use of rigid analytic spaces associated to Galois deformation rings The aim of these lectures is to explain some basic ideas results and examples in Tate7s theory and its re nements In view of time and space constraints we have omitted most proofs in favor of examples to illustrate the main ideas To become a serious user ofthe theory it is best to closely study a more systematic development In particular we recommend BGR for the classical theory due to Tate BLl and BL2 for Raynaud7s approach based on formal schemes and Berl and Ber2 for Berkovich7s theory of k analytic spaces Some recent lecture notes by Bosch B explain both rigid geometry and Raynaud7s theory with complete proofs and a recent Bourbaki survey by Ducros D treats Berkovich7s theory in greater depth There are other points of view as well most notably the work of Huber Date March 11 2007 This work was partially supported by NSF grants DMS0600919 and DMS0602287i 1 2 BRIAN CON RA D but we will pass over these in silence except for a few comments on how Huber7s adic spaces relate to Berkovich spaces Before we begin it is perhaps best to tell a story that illustrates how truly amazing it is that there can be a theory of the sort that Tate created In 1959 Tate showed Grothendieck some ad hoc calculations that he had worked out with p adic theta functions in order to uniformize certain p adic elliptic curves by a multiplicative group similarly to the complex analytic case Tate wondered if his computations could have deeper meaning within a theory of global p adic analytic spaces but Grothendieck was doubtful In fact in an August 18 1959 letter to Serre Grothendieck expressed serious pessimism that such a global theory could possibly exist Tate has written to me about his elliptic curves stu and has asked me if I had any ideas for a global de nition of analytic varieties over complete valuation elds I must admit that I have absolutely not understood why his results might suggest the eccistence of such a de nition and I remain skeptical Nor do I have the impression of having understood his theorem at all it does nothing more than ecchibit via brute formulas a certain isomorphism of analytic groups one could conceive that other equally epplicit formulas might give another one which would be no worse than his until proof to the contrary77 1 AFFINOID ALGEBRAS 11 Tate algebras In this rst lecture we discuss the commutative algebra that forms the foundation for the local theory of rigid analytic spaces much as the theory of polynomial rings over a eld is the basis for classical algebraic geometry The primary reference for this lecture and the next one is BGRD The replacement for polynomial rings over a eld will be Tate algebras Unless we say to the contrary throughout this lecture and all subsequent ones we shall x a non archimedean eld k and we write R to denote its valuation ring and h its residue eld 124le 1 7Rm where m t E R1 ltl lt 1 is the unique maximal ideal of R Exercise 111 The value group of h is lhxl Q RO Prove that R is noetherian if and only if lkxl is a discrete subgroup of R0 in which case B is a discrete valuation ring It is a basic fact that every nite extension h h admits a unique absolute value 1 necessarily non archimedean extending the given one on k and that k is complete with respect to this absolute value Explicitly if x E k then lz l lNkzkx lllkl3kl but it is not obvious that this latter de nition satis es the non archimedean triangle inequality though it is clearly multiplicative The absolute value on k therefore extends uniquely to any algebraic extension of h using that it extends to every nite subextension necessarily compatibly on overlaps due to uniqueness though if k h is not nite then k may not be complete In view of the uniqueness this extended absolute value is invariant under all h automorphisms of h It is an important fact that if h is an algebraic closure of h thenits completion hA is still algebraically closed and its residue eld is an algebraic closure of k In the special case h Qp this completed algebraic closure is usually denoted Cp SEVERAL APPROACHES TO NONARCHIMEDEAN GEOMETRY 3 De nition 112 For n 2 1 the n variable Tate algebra over k is T ml ZaJXJl W a 0 as HJH a 00 where for a multi index J 71 jn we write X to denote and to denote Ziji In other words Tnk is the subring of formal power series in k X1 Xn that converge on B This k algebra is also denoted kX1 Xngt The Gauss norm or sup norm for reasons to become clear shortly on Tn is X J gt 0 H 2a H mfxlaJl Obviously 0 if and only if f 0 Exercise 113 This exercise develops properties of the Gauss norm on Tn This gives Tn a topological structure that goes beyond its mere algebraic structure 1 Prove that the Gauss norm is a k Banach algebra norm on Tn That is f1 M S maxHf1H for all f1f2 E Tn HcfH lcl for all c E k and f E Tn llflng 3 HleHng for all f1f2 E Tn and Tn is complete for the metric Hfl 7 fgl By using kX scaling to reduce to the case of unit vectors show that Hfl for all f1f2 E Tn That is the Gauss norm is multiplicative Let E be an algebraic closure of k endowed with the unique absolute value again denoted l extending the given one on k Using kX scaling to reduce to the case of unit vectors prove that the Gauss norm computes a supremum of magnitudes over the closed unit n ball over E Hfll Slip lf96l mfxl lel Hf1llllf2ll AA 00 3 VV where z 1 xn varies with 7 6 E and S 1 In particular this supremum and maximum are nite Show that the use ofE in the previous part is essential give an example of f E QPXgt such that gt supmezp We want Tnk to be the coordinate ring77 of the closed unit n ball over k but as with algebraic geometry over a eld that may not be algebraically closed we have to expect to work with points whose coordinates are not all in k That is the underlying space for the closed unit n ball over k should admit points with coordinates in nite extensions of k Let7s now see that Tn admits many k algebra maps to nite extensions of k A 4 V Exercise 114 Let k k be a nite extension and choose c c E k with S 1 Prove that there exists a unique continuous k algebra map Tn a k using the Gauss norm on Tn such that X gt gt c for all j Conversely prove that every continuous k algebra map Tn a k arises in this way Hint for converse c E k satis es lcl S 1 if and only if the sequence cmm21 in k is bounded The basic properties of Tn are summarized in the following result that is analogous to properties of polynomial rings over a eld The proofs of these properties are inspired by the local study of complex analytic spaces via Weierstrass Preparation techniques to carry out induction on 4 BRIAN CON RA D Theorem 115 The Tate algebra Tn Tnh satis es the following properties 1 The domain Tn is noetherian regular and a unique factorization domain For every mamimal ideal m of Tn the local ring Tnm has dimension n and residue class eld Tnm that has nite degree ouer h 2 The ring Tn is Jacobson euery prime ideal p of Tn is the intersection of the mamimal ideals containing it In particular if is an ideal of Tn then an element of TnI is nilpotent if and only if it lies in every mamimal ideal of TnI 3 Every ideal in Tn is closed with respect to the Gauss norm As a consequence of this theorem we can reinterpret Exercise 1133 and Exercise 114 in a more geometric manner as follows Consider the set MaxSpecT of maximal ideals of Tn A point in this set will usually be denoted as x though if we want to emphasize its nature as a maximal ideal we may denote it as my To each such point there is associated the residue class eld Tnmm of nite degree over h and this eld is equipped with the unique absolute value which we also denote as l that extends the given one on h For any f E Tn we write fx to denote the image of f in We can combine Exercise 1133 and Exercise 114 to say that for all f E Tn Hfll Slip lf96l mfxl where now z varies through MaxSpecT there is no intervention of the auxiliary h here In particular the function x gt gt on MaxSpecT is bounded and attains a maximal value It is as if77 MaxSpecT were a compact topological space an idea that becomes a reality within the framework of Berkovich spaces as we shall see later One curious consequence of this formula for the Gauss norm in terms of MaxSpecT and the intrinsic h algebra structure of Tn is that the Gauss norm is intrinsic to the h algebra Tn and does not depend on its coordinates77 X E Tn in particular it is invariant under all h algebra automorphisms of Tn which is not obvious from the initial de nition of the Gauss norm 12 A inoid algebras Much as af ne algebraic schemes over a eld can be obtained from quotients of polynomial rings and these in turn are the local model spaces from which more general algebraic schemes are constructed via gluing the building blocks for rigid analytic spaces will be obtained from quotients of Tate algebras This distinguished class of h algebras is given a special name as follows De nition 121 A h a noid algebra is a h algebra A admitting an isomorphism A 2 TnI as h algebras for some ideal I Q Tn The set MaxSpecA of maximal ideals of A is denoted Encample 122 We have R Q MTn in an evident manner but if h is not algebraically closed eg h Qp then MTn has many more points than just those coming from B This underlies the enormous difference between rigid analytic spaces over h and the more classical notion of a h analytic manifold By Theorem 115 every h af noid algebra A is noetherian and Jacobson with nite Krull dimension and Am is a nite extension of h for every 111 E For a point z E MA we write to denote this associated nite extension of h and we write ad E to denote SEVERAL APPROACHES TO NONARCHIMEDEAN GEOMETRY 5 the image of a E A in By the Jacobson property of A a E A is nilpotent if and only if aw 0 for all z E Obviously a E Agtlt if and only if aw 31 0 for all z E In this respect we can view elements of A as functions on MA valued in varying elds much like we do for coordinate rings of af ne algebraic schemes over a eld and the function x gt gt ax determines a up to nilpotents Exercise 123 Recall that any domain of nite dimension over a eld is itself a eld Using this prove that MA is functorial via pullback That is if f A a A is a map of k af noid algebras then the prime ideal f 1m Q A is a maximal ideal of A for every maximal ideal m of A Geometrically if we choose an isomorphism A 2 TnI and we let f1 fm be gen erators of I then functoriality provides an injection MA gt MTn onto the subset of points x E MTnlf1 fmz 0 x E 0 for all f E I In this sense we want to think of MA as being the underlying set of the space77 of points in the closed unit n ball over k where the fs simultaneously vanish Keep in mind that just as for MTn if k is not algebraically closed then there are generally many points z E MA with 31 k which is to say that MA usually has many points that are not k rational This abundance of non rational points over the base eld is a fundamental distinction between rigid analytic spaces and the more classical concept of a k analytic manifold ln Berkovich7s theory there will nearly always be even more points than these and in particular lots of non rational points even if k is algebraically closed This is analogous to the fact that an algebraic scheme over an algebraically closed eld nearly always has many non rational points In order to give geometric substance to the sets MA we need to endow them with a good function theory and this in turn requires an understanding of the topological structure of A Thus we now turn to this aspect of k af noid algebras A k Btmaeh space is a k vector space V equipped with a function V a R20 such that 0 if and only if 1 0 H71 7J H S maxltllvll7 Hv ll lel lcl M for all 11 E V and c E k and V is complete for the metric H1 71 Likewise a k Banaeh algebra always understood to be commutative is a k algebra 32 equipped with a k Banach space structure that is submultiplicative with respect to the multiplication law on 12 HalagH S HalH HagH for all a1a2 E 42 For example we have seen that Tn with the Gauss norm is a k Banach algebra and in fact any k af noid algebra A admits a k Banach algebra structure To see this we choose an isomorphism A 2 TnI as k algebras and since I is closed in Tn we may use the residue norm from Tn to de ne a k Banach structure on the quotient TnI and hence on A as is explained in the next exercise Exercise 124 Let V be a k Banach space and W a closed subspace For E E VW de ne the residue norm on E to be HEH inf HWH 39umod W39u 6 BRIAN CON RA D the in mum of the norms of all representatives of in V Using that W is closed prove that this is a k Banach space structure on VW what goes wrong if W is not closed in V In the special case that V 32 is a k Banach algebra and W I is a closed ideal prove that the residue norm is a k Banach algebra structure on MI If we choose two different presentations TnI 2 A and TmJ 2 A of a h af noid algebra A as a quotient of a Tate algebra then the resulting residue norms on A are generally not the same In this sense A usually has no canonical k Banach structure in contrast with Tn However it turns out that any two k Banach algebra structures on A even those perhaps not arising from a presentation of A as a quotient of a Tate algebra are bounded by positive multiples of each other and hence the resulting k Banach topology and concepts such as boundedness77 are in fact intrinsic to A In particular for this intrinsic k Banach topology all ideals of A are closed since the residue norm77 construction via an isomorphism A 2 TnI reduces this to the known case of Tate algebras These and further remarkable features of the k Banach algebra structures on k af noid algebras are summarized in the next result Theorem 125 Let A be a k a noid algebra 1 If and H are h Banach algebra norms on A then there eccist C 2 c gt 0 such that Cl H S H l S OH so both norms de ne the same topology and the same concept of boundedness In particular for a E A the property that the sequence anhzl is bounded ie a is power bounded is independent of the choice of h Banach algebra structure Any k algebra map A a A between h a noid algebras is automatically continuous for the intrinsic h Banach topologies or equivalently is a bounded linear map with respect to any choices of h Banach algebra norms Any A algebra A with module nite structure map A a A is necessarily a k a noid algebra 4 Noether normalization theorem Ifd dimA 2 0 then there is a module nite h algebra injection Tdk gt A In particular ifA is a domain then all of its mapimal ideals have height d 5 Maximum Modulus Principle For any f E A we have the equality llfllsup 3 SUP WW max lle lt00 m6MA D A 0 V 6MA In particular the function x gt gt on MA is bounded and attains a mapimal value IfA is reduced ie has no nonzero nilpotent elements then this is a h Banach algebra structure on A The nal part of this theorem provides a canonical k Banach algebra structure on any re duced k af noid algebra recovering the Gauss norm in the special case of Tate algebras This k Banach algebra structure may not be multiplicative but it is clearly power multiplicative Haanup HaHZup for all a E A and n 2 1 In particular for a reduced h af noid algebra A we deduce the important consequence that a E A is power bounded if and only if HaHZup is bounded for n 2 1 which is to say Hallsup lt 1 or in other words that laxl S 1 for SEVERAL APPROACHES TO NONARCHIMEDEAN GEOMETRY 7 all z E In fact it can be shown that this characterization of power boundedness in k af noid algebras is valid without assuming reducedness That is if A is any k af noid algebra then a E A is power bounded if and only if laxl S 1 for all z E Remark 126 For units in k af noid algebras A ie u E A such that 31 0 for all z E MA there is a minimum modulus principle for u E AX inmeMM minmEMM gt 0 Indeed this is a reformulation of the Maximum Modulus Principle for u We conclude this lecture with an exercise that provides a universal mapping property for Tate algebras within the category of k af noid algebras and even k Banach algebras reminiscent of the universal mapping property of polynomial rings Exercise 127 Let 2 be a k Banach algebra and let 20 be the subset of power bounded elements a E 32 such that the sequence anhzl is bounded with respect to the k Banach norm on 12 1 Prove that 20 is a subring of 42 and in fact is a subalgebra over the valuation ring R of k 2 Prove that 20 is functorial in 32 within the category of k Banach algebras using continuous maps In particular any k Banach algebra map Tn Tnk a 32 carries each X to an element 17 E 32 3 Show that the map of sets HomTnM a 30 de ned by b gt gt X1 gtXn is bijective This is the universal mapping property of Tn within the category of k Banach algebras and in particular within the full subcategory of k af noid algebras 4 As an application of the universal property we can recenter the polydisc77 at any k rational point That is for 01 0 E k with 1071 3 1 prove that there is a unique automorphism of Tn satisfying X gt X 7 07 2 GLOBAL RIGID ANALYTIC SPACES 21 Topological preparations In the rst lecture we studied some basic algebraic and topological properties of k af noid algebras and in particular for any such algebra A we introduced the set MA of maximal ideals of A We wish to impose a suitable topology really a mild Grothendieck topology on A with respect to which notions such as connect edness will have a good meaning But before doing that we want to explain how MA has a Hausdorff canonical topology77 that is closer in spirit to the totally disconnected topology that arises in the classical theory of k analytic manifolds This canonical topology is not especially useful but it is psychologically satisfying to know that it exists the subtle issue is that MA usually has many points that are not k rational and it is also not a set of E points either unless k For this reason it requires some thought to de ne the canonical topology The motivation for the de nition comes from the following concrete description of Exercise 211 Let A be a k af noid algebra and E an algebraic closure of k For each x E MA if we choose a k embedding 239 gt P then we get a k algebra map A a E whose image lies in a subextension of nite degree over k Let AE denote the set of k algebra maps A a E with image contained in a sub eld of nite degree over k this set 8 BRIAN CON RA D has contravariant functorial dependence on A Observe that AutEk acts on this set via composition 1 Show that if we change the choice ofi then the resulting map in AE changes by the action of AutEk Hence7 we get a well de ned map of sets MA a AE AutEk into the space of orbits of AutEk on 2 Prove that the map MA a AEAutEk is functorial in A7 and that it is a bijection For any x E AE and f E A we get a well de ned element f E E and hence a number Show that the loci A 0 V x e M l WW 2 61ifltzgti 2 mm n1igmltzgti nm for f1 gm 6 A and 61 77m gt 0 are a basis of open sets for a topology on Give MA the resulting quotient topology Prove that this topology on MA is Hausdorff and totally disconnected7 and that it is functorial in A in the sense that the pullback map MA a MA induced by a k algebra map A a A of k af noid algebras is continuous This is the canonical topology on Show that MTn is the disjoint union of two open sets7 1 and its complement A 4 V Having introduced the canonical topology7 we now prepare to build up the Tate topology that will replace it The basic idea is to artfully restrict both the open sets and the coverings of one open set by others that we permit ourselves to consider In this way7 disconnectedness will be eliminated where it is not desired For example7 the decomposition in Exercise 2114 will be eliminated in Tate7s theory7 and in fact MTn will in an appropriate sense wind up becoming connected To construct Tate7s theory7 we need to introduce several important classes of open subsets of MA Weierstrass domains7 Laurent domains7 and rational domains These are analogues of basic af ne opens as used in algebraic geometry7 but the main difference is that we can consider loci de ned by non strictl inequalities of the type lfll S lfgl on absolute values7 whereas in algebraic geometry with the Zariski topology we can only use conditions of the type f1 31 f2 It should be noted that later in the theory we will permit strict inequalities of the type lfll lt lfgl but at the beginning it is non strict inequalities that are more convenient to use as the building blocks Roughly what is happening is that non strict inequalities de ne loci that will behave as if77 they are compact7 which is in fact what happens within the framework of Berkovich7s theory7 whereas strict inequalities de ne loci that lack a kind of compactness property In order to de ne interesting open domains within MA7 it will be useful to rst introduce a relative version of Tate algebras7 much as we do with polynomial rings over a general commutative ring De nition 212 Let be a k Banach algebra The Tate algebra over 32 in n variables is MKYn Za I emunynpaj o SEVERAL APPROACHES TO NONARCHIMEDEAN GEOMETRY 9 this is also denoted zltXgt if n is understood from context We de ne a norm on this ring as follows J 7 H E aJY H mfxllaJH Exercise 213 Let 2 be a k Banach algebra 1 Check that in the preceding de nition7 the norm is a k Banach algebra structure on zltXgt and that if the k Banach algebra structure on 32 is replaced with an equivalent such norm ie7 one bounded above and below by a positive constant multiple of the given one then the resulting norm on the Tate algebra over 32 is also replaced with an equivalent one In particular7 if 32 is k af noid then all of its k Banach algebra structures de ne equivalent norms on the Tate algebras over 42 2 If 32 A is k af noid7 and say TmI 2 A is an isomorphism7 show that the resulting natural map Tnm a AltY17 Yngt is surjective7 so the Tate algebras over A are also k af noid In the category of k Banach algebras over 32 ie7 the category of k Banach algebras equipped with a continuous map from 42 and morphisms are as zf algebras7 state and prove a universal mapping property similar to that for Tnk in the category of k Banach algebras Using this7 construct a transitivity77 isomorphism of the type ltwltxgtgtltrgt 2 mar Let A be a k af noid algebra The following class of rings will wind up being the coordinate rings of subsets of MA to be called Laurent domains Let A be a k af noid algebra For a a1an E A and g a3 a E Am7 de ne 7 m Mag AltX1XmKYmgtX1 ia1Xniama Y1 717anYm i 1 A 0 V Remark 214 Beware that relative Tate algebras cannot be treated as easily as polynomial rings For example7 consider Altagt AltXgtX 7 a This is generally not the same as A geometrically what happens is that we are forcing a to become power bounded7 which may not be the case in A at the outset We will see that the natural map MAltagt a MA is an injection onto the set of z E MA such that laxl S 1 In view of the universal property of relative Tate algebras7 the A algebra AltQ7Q71gt has the following universal property for any map of k Banach algebras b A a B7 we can ll in a commutative diagram A 9Altacflgt w B in at most one way7 and such a diagram exists if and only if Mai E B0 for all 239 ie7 all Mai are power bounded and Mag 6 Ex with Mag4 6 B0 for all j In more geometric language7 if B is k af noid then we can say that the structure map 15 A a B factors through AltQ7Q71gt if and only if the map of sets M MB a MA factors through the subset x e MAHa1zl 1lanzl 1la 1zl 211a m 21g MA 10 BRIAN CONRAD Exercise 215 By taking B to vary through nite extensions of k use the above universal property to deduce that the map of k af noid algebras A 7 AQ7 Q71gt induces a bijection MAltQ7i 1gt 6 MM l l0196l S 177lan96l S 17la 1l Z 177la n96l Z 1 The signi cance of the conclusion of exercise is that the purely algebraic condition that a map of k af noid algebras i5 A 7 B factors through the canonical map A 7 AQ7 Q71gt is equivalent to the set theoretic condition that M MB 7 MA factors through the locus of points of MA de ned by the pointwise conditions lail S 1 and lan 2 1 Subsets of MA de ned by such conditions are called Laurent domains7 and ifthere are no ays then we call the subset a Weierstrass domain In particular7 by Yoneda7s Lemma7 a Laurent domain in MA functorially determines the k af noid A algebra AQg 1gt that gives rise to it7 so this latter algebra is intrinsic to the image of its MaxSpec in MA7 and hence it enjoys some independence of the choice of the as and a fs An analogue in algebraic geometry is that the localizations Alla and Alla are isomorphic as A algebras if and only if there is the set theoretic equality of the non vanishing loci of a and a in Spec A7 in which case such an isomorphism is unique This characterization of an algebra by means of a set theoretic condition is reminiscent of the situation for af ne open subschemes of an af ne scheme in algebraic geometry if SpecA is an open subscheme of SpecA7 then a map of schemes SpecB 7 SpecA factors through SpecA as schemes if and only if it does so on underlying sets Note that closed subschemes rarely have such a set theoretic characterization unless they are also open7 since we can replace the de ning ideal by its square without changing the underlying set but this nearly always changes the closed subscheme This set theoretic mapping property suggests that we ought to consider a Laurent domain as an open subset77 of MA with associated coordinate ring given by its canonically associated A algebra as above Exercise 216 Let us work out an example of a Laurent domain explaining the reason for the name Laurent domain Pick 0 E k with 0 lt lcl S 17 and consider the Laurent domain in MT1 de ned by the conditions lcl S ltl S 17 where T1 ktgt this is an annulus The associated coordinate ring is klttXYgtX i tc 1tY i 1 klttXYgtX 7 my 7 c Prove that the natural map ktgt 7 ktXgtX 7 t is an isomorphism by considering uni versal mapping properties7 and deduce that the annulus has associated k af noid algebra ktYgttY 7 c Prove that this is a domain Hint show that every element of ktYgttY 7 c can be represented by a unique series of the form 00 221 0717sz ij with 07 7 0 as j 7 00 and 07 7 0 asj 7 00 Show that this de nes an injection into a k algebra of doubly in nite Laurent series in t satisfying certain convergence properties Exercise 217 If you are familiar with tale maps of schemes7 then to put in perspective the role of open subschemes in set theoretic mapping properties7 consider the following problem Let i U 7 X be a locally nitely presented map of schemes with the property that a map of schemes X 7 X factors through i if and only if its image is contained in iU7 in which case such a factorization is unique Prove that i is an open immersion Hint Show that i is SEVERAL APPROACHES TO NONARCHIMEDEAN GEOMETRY 11 etale via the functorial criterion and that it is set theoretically injective and induces purely inseparable residue eld extensions Thus it is an open immersion by EGA lV4 1791l Now we introduce another important class of subsets of MA the rational domains For these subsets the relevant input is a collection of elements a1 an a E A with no common zero Given such data we de ne 11 a A igtAX1XngtaX17a1aXn7an a What is happening in this A algebra is that we are forcing lajl 3 la at all points easier to remember as the imprecise condition laja l S 1 To make this precise we state and prove a universal mapping property 1 Lemma 218 Let A be a h a noid algebra and a1 ama E A with no common zero For any map of k a noid algebras b A a B there is at most one way to ll in the commutative diagram A Alt aquotgt 7 B and such a diagram CCElStS if and only if M MB a MA factors through the subset ofx E MA such that lagml S la l for all j or in other words l ajyl S l a yl for all y E Proof By the universal property of relative Tate algebras to give such a diagram is to give power bounded elements b1bn E B such that a bj Maj for all j This implies that a must be a unit in B because at any y E MB where it vanishes we get that all Maj also vanish so the point M E MA is a common zero of the as and a contrary to hypothesis Hence the bs are uniquely determined if they exist so we get the uniqueness of the diagram if it exists Moreover power boundedness of such bs forces l ajyll a yl S 1 for all y E MB which is to say that M factors through the desired subset of Conversely if this set theoretic condition holds then l ajl S l a l pointwise on MB so a E B has to be a unit because if it is not a unit then there would exist some y E MB at which a vanishes and hence all Maj vanish yielding the point E MA as a common zero of the as and a contrary to hypothesis But with a a unit in B even if a is not a unit in A it makes sense to consider bj aj a E B To construct the desired commutative diagram the problem is to prove that bj is power bounded in B or equivalently that S 1 for all y E This is exactly the assumed system of inequalities l aj S l a yl for all y E MB since l a yl 31 0 for all such y I We call a subset in MA of the form 96 6 MM 1 laj96l S lit901 for a1ana E A with no common zero a rational domain The universal property in the preceding lemma shows that such a subset canonically determines the A algebra Aa1a ana gt 12 BRIAN CONRAD Example 219 We write Q R0 to denote the divisible subgroup that is generated by lkx l which is to say the set of positive real numbers 04 such that 041V 6 lkxl for some integer N gt 0 Note that this is a dense subgroup of R0 If 04 E and 041V lcl with c E kX then for k af noid A and f E A the inequality S 04 for z E MA is equivalent to the inequality lc lfNzl S 1 Thus in the de nitions of Weierstrass Laurent and rational domains it is no more general to permit real scaling factors from in the inequalities For example in the closed unit disc over k Qp the locus ltl S 1f is a Weierstrass domain it is the same as the condition lptzl S 1 and so has associated coordinate ring77 klttXgtX 7 ptz In contrast it is true and unsurprising but perhaps not obvious how to prove that for r Z VlQSl pQ the locus S r in the closed unit disc MQplttgt over Qp is not a Weierstrass domain 22 A inoid subdomains and admissible opens Weierstrass Laurent and rational domains are the most important examples of the following general concept De nition 221 Let A be a k af noid algebra A subset U Q MA is an a noid sub domain if there exists a map 239 A a A of k af noids such that MA a MA lands in U and is universal for this condition in the following sense for any map of k af noid algebras b A a B there is a commutative diagram A A 7 x l B if and only if M carries MB into U in which case such a diagram is unique By Yoneda7s Lemma if U Q MA is an af noid subdomain then the k af noid A algebra A as in the preceding de nition is unique up to unique A algebra isomorphism It is therefore legitimate to denote this A algebra as AU it is functorially determined by U We call AU the coordinate ring of U with respect to A For example the universal property of the domains of Weierstrass Laurent and rational types shows that each is an af noid subdomain and provides an explicit description of AU in such cases By chasing points valued in nite extensions of k it is not hard to show that the natural map MAU a MA is an injection onto U Q By the universal property we likewise see that if V Q U is an inclusion of af noid subdomains of MA then there is a unique A algebra map of coordinate rings pg AU a AV and by uniqueness this is transitive with respect to another inclusion W Q V of af noid subdomains in MA in the sense that pop Lf This is to be considered as analogous to restriction maps for the structure sheaf of a scheme so for f 6 AU we usually write fly to denote p50 6 AV Akin to the case of schemes it can be shown by a method entirely different from the case of schemes that AU is A at for any af noid subdomain U Q Exercise 222 Prove that Weierstrass Laurent and rational domains in MA are all open for the canonical topology If the condition no common zero77 is dropped from the de nition of a rational domain then it still makes sense to consider the underlying set in MA de ned by the simultaneous conditions lagml S la xl Show by example that this locus can fail to be open if there is a common zero SEVERAL APPROACHES TO NONARCHIMEDEAN GEOMETRY 13 Exercise 223 This exercise develops two kinds of completed tensor product operations that arise in rigid analytic geometry The theory of the completed tensor product can be developed in greater generality than we shall do7 but for our limited purposes we adopt a more utilitarian approach 1 Let A and A be k af noid algebras We wish to construct a k af noid completed tensor product77 A kA To do this7 rst choose presentations A 2 TnI and A 2 TnzI Using the natural maps Tn a Tnnz and TM a My onto the rst 71 and last 71 variables7 it makes sense to let J7 J Q Tnnz be the ideals generated by I and 1 respectively Consider the k af noid algebra Tnn J J There are evident k algebra maps L A a Tnn J J and U A a nnzJ J Prove that this pair of maps is universal in the following sense for any k Banach algebra B and any k Banach algebra maps b A a B and 15 A a B7 there is a unique k Banach algebra map h Tnn J a B so that hot b and hot 15 In view of this universal property7 the triple TnnJ J 7 L U is unique up to unique isomorphism7 so we may denote TnnJ J as A kA The product Lat a is usually denoted a a 2 Let j A a A and j A a A be a pair of maps of k af noid algebras De ne the k af noid algebra mam A kA ja 1 7 1 j a la 6 A Formulate and prove a universal property for this in the category of k Banach A algebras analogous to the universal property of tensor products of rings 3 Let A be a k af noid algebra7 and let Kk be an analytic extension eld ie7 a non archimedean eld K equipped with a structure of extension of k respecting the absolute values Beginning with the case of Tate algebras7 de ne a K af noid algebra K kA as a solution to a universal mapping problem for continuous maps over k a K from A to K Banach algebras Exercise 224 Let A be a k af noid algebra 1 If U7 U Q MA are af noid subdomains7 then prove that U U is one as well the k af noid A algebra AUQU is AU AAU Check this via universal mapping properties 2 Let b A a B be a map of k af noid algebras7 and let U Q MA be an af noid subdomain Show that the pullback M 1U Q MB is also an af noid subdo main its coordinate ring is AU AB Check this via universal mapping properties ls there a similar result for Weierstrass7 Laurent7 and rational domains 3 Let U Q MA be an af noid subdomain7 with corresponding coordinate ring AU Using the natural bijection U MAU7 prove that a subset U Q U MAU is an af noid subdomain of MAU if and only if it is an af noid subdomain of The introduction of the concept of af noid subdomains was a genuine advance beyond Tate7s original work7 in which he got by with just Weierstrass7 Laurent7 and rational sub domains ln order to make af noid subdomains easy to handle eg7 are they open7 the crucial result required is the Ger tzeniGmuert theorem that describes them in terms of rational domains 14 BRIAN CONRAD Theorem 225 Let A be a k a noid algebra Every a noid subdomain U Q MA is a nite union of rational domains In particular a noid subdomains are open with respect to the canonical topology It is very hard to determine when a given nite union of rational domains let alone af noid subdomains is an af noid subdomain This is analogous to the dif culty of detecting when a nite union of af ne open subschemes of a scheme is again af ne Since Laurent domains are a basis for the canonical topology7 in order to get a good theory of non archimedean analytic spaces we cannot permit ourselves to work with arbitrary unions of af noid subdomains or else we will encounter the total disconnectedness problem Tate7s idea is to restrict attention to a class of open subsets for the canonical topology and a restricted collection of coverings of these opens by such opens so as to force77 af noid subdomains to appear to be compact The key de nition in the theory is as follows De nition 226 Let A be a k af noid algebra A subset U Q MA is an admissible open subset if it has a set theoretic covering by af noid subdomains Ul Q MA with the following niteness property under af noid pullback for any map of k af noid algebras b A a B such that Mb MB a MA has image contained in U7 there are nitely many Us that cover this image equivalently7 the open covering M 1Ui of MB by af noid subdomains has a nite subcovering A collection of admissible open subsets of MA is an admissible cover of its union V if7 for any k af noid algebra map b A a B with Q V7 the covering of MB has a re nement by a covering consisting of nitely many af noid subdomains This forces V to be admissible open7 by using the af noid subdomain covering VJkjk Kj where VJkheKj is a covering of each by af noid subdomains as in the de nition of admissibility of each Note that the set theoretic covering of U in the de nition of an admissible open subset of MA is necessarily an admissible cover Example 227 Let U17 Un be af noid subdomains of MA for a k af noid algebra A Then U UUJ is an admissible open subset7 with U7 an admissible covering of U For example7 for a E A the Laurent domains MAltagt la 3 1 and MAlt1agt la 21 constitute an admissible covering of The content here is that the pullback of each Uj under Mb for a k af noid algebra map b A a B is an af noid subdomain of It is dif cult to determine if such a U is an af noid subdomain Now we come to a key example that shows the signi cance of the niteness requirement in the de nition of admissibility Example 228 Let T1 klttgt be the Tate algebra in one variable Within the closed unit ball MT17 the locus V 1 is a Laurent domain The subset U lt 1 is open for the canonical topology7 and more importantly it is an admissible open lndeed7 it is covered by the Weierstrass domains Un S lclln for a xed c E k with 0 lt lcl lt 1 and n 2 17 and these satisfy the admissibility condition due to the Maximum Modulus Principle if b T1 a B is a map to a k af noid algebra such that MB a MT1 lands in U7 then the function gtt E B has absolute value lt 1 pointwise on MB7 and so the Maximum SEVERAL APPROACHES TO NONARCHIMEDEAN GEOMETRY 15 Modulus Principle on MB provides 0 lt 04 lt 1 such that l t S 04 for all y E Hence for no so large that 04 lt loll 0 lt 1 we have Q Uno Thus the required nite subcover property is satis ed The pair of admissible opens V 1 and U lt 1 covers MTl set theoretically and these are disjoint However this is not an admissible covering Indeed by the de nition of admissibility of a covering applied to the identity map of MT1 it would follow that U V has as a re nement a nite covering of MTl by af noid subdomains But by the Maximum Modulus Principle any af noid subdomain of MTl contained in U is contained in some Un S lolln and hence if there were a re nement of UV by a nite collection of af noid subdomains then we would get that MTl is covered by V and by UnO for some large n0 By using a suitable nite extension of k we can certainly nd a point in MTl lying in the locus loll 01l that is disjoint from UnO U V This gives a contradiction so U V is not an admissible covering of MTl Note that here it is essential that our spaces have points that are not necessarily k rational Exercise 229 Choose 0 lt r lt 1 with r Z Prove that S r lt r is an admissible open subset of Mkt and give an admissible covering by Weierstrass domains Prove that this locus does not have an admissible covering by nitely many af noid subdomains Hint use the Maximum Modulus Principle Exercise 2210 Generalize the method of Example 228 to show that for any k af noid algebra A and aa E A the set U x E MA l laxl lt la xl is an admissible open subset give an admissible open af noid covering of U and be careful about points in MA where a vanishes Exercise 2211 Let A be a k af noid algebra Prove that if an admissible open U Q MA is covered set theoretically by some admissible opens U then is an admissible covering of U if and only if it admits an admissible re nement Also show that admissibility for subsets is a local property in the following sense if U Q MA is an admissible open and U is an admissible covering of U by admissible opens then a subset V Q U is admissible open in MA if and only if V U is admissible open in MA for all i 23 The Tate topology The admissible opens and their admissible coverings within MA lead to the de nition of a very mild Grothendieck topology in the sense that it only in volves subsets of the ambient spaces which is not a requirement in the general theory of Grothendieck topologies such as the etale topology on a scheme De nition 231 The Tate topology or G topology on MA has as objects the admissible open subsets and as coverings the admissible open coverings Exercise 232 Let A be a k af noid algebra For a E A let Va be the locus of z E MA for which a 31 0 Prove that this is an admissible open give an admissible Laurent covering and show that the Va7s are a base of opens for a topology on MA this is called the analytic Zaiiski topology Show that the closed sets for this topology are the subsets MAI for ideals I of A these are called analytic sets in MA and that all Zariski opens and Zariski open covers of Zariski opens are admissible Hint for admissibility if B is k af noid with a k Banach algebra norm on B and b1 bn E B some elements that 16 BRIAN CONRAD generate 1 say Z jbj 1 with 67 E B then show that M maxH lH gt 0 and the Laurent domains lbjl 2 1M cover The Tate topology is generally not a topology on MA in the usual sense but is instead a Grothendieck topology It is not crucial for our purposes to delve into the general formalism of Grothendieck topologies The main point that matters for working with the Tate topology is to keep in mind that we do not consider general unions of admissible opens and when doing sheaf theory we only consider admissible coverings of admissible opens It generally does not make sense to evaluate a sheaf for the Tate topology on a general open set for the canonical topology and even for evaluation on admissible opens we cannot expect a sheaf for the Tate topology to satisfy the sheaf axioms for a non admissible covering of an admissible open by admissible opens eg the pair U V in the closed unit disc in Example 228 When we de ne disconnectedness later it will be expressed in terms of an admissible covering by a pair of disjoint non empty admissible opens Example 228 shows that this rules out many classical sources of disconnectedness ofthe canonical topology The fundamental result that gets the theory off the ground is the existence of a structure sheaf77 with respect to the Tate topology This is Tate7s Acylicity Theorem Theorem 233 Let A be a k a noid algebra The assignment U gt gt AU of the coordinate ring to every a noid subdomain of MA uniquely ecctends to a sheaf 64 with respect to the Tate topology on In particular if is a nite collection of a noid subdomains with U UUZ also an a noid subdomain of MA then the evident sequence 0 a AU a HALZ a HALWU is eccact Tate proved this theorem by heavy use of Cech theoretic methods to reduce to the special case of a Laurent covering of MA by the pair MAltagtMAlt1agt for a E A In this special case he could carry out a direct calculation The next exercise gives the simplest instance of this calculation Exercise 234 Choose c E k such that 0 lt lcl lt 1 ln MT1 let U S and V 2 lcl so U V By calculating with convergent Laurent series int using Exercise 216 to describe AV 2 Mt YgttY 7 c as a kltrgt algebra of certain Laurent series Znez cnt show that if f 6 AU and g 6 AV satisfy flU V glUnV in AU V then there is a unique h 6 T1 such that hlU f and hlV 9 De nition 235 Let A be a h af noid algebra The a noid space SpA is the pair MA 4 consisting of the set MA endowed with its Tate topology and sheaf of k algebras 624 with respect to the Tate topology If A Tn Tnh then this is denoted B BE Usually we write x rather than 624 with X SpA Exercise 236 Prove that for z E X SpA the stalk Xm XUJ mEU SEVERAL APPROACHES TO NONARCHIMEDEAN GEOMETRY 17 limit over admissible opens7 or equivalently af noid subdomains7 containing p is a local ring Describe 1319 as an intermediate ring strictly between the algebraic local ring htt and the completion ln general7 6 is a noetherian ring that is faithfully at over the algebraic local ring AW and in fact it has the same completion7 but this requires more work to prove This is the key to developing a good dimension theory for rigid analytic spaces In order to make global de nitions7 we have to rst de ne the category in which we will be working A G topologized space is a set X equipped with a set 11 of subsets U Q X the open subsets for the G topology and a set of set theoretic coverings CovU of each U E 11 by collections of members of 11 such that certain natural properties from ordinary topology are satis ed we will not give the list here see BGR7 Ch 9 for a precise discussion except to note that it includes basic locality and transitivity conditions eg7 11 is stable under nite intersections7 if E CovU and V Q U then V E 11 if and only if V Ul E 11 for all i7 U E CovU7 Q E 11 etc and we omit the requirement that 11 is stable under arbitrary or even nite unions One very important construction for G topologized subspaces is the analogue of the open subspace topology To be precise7 if X is a G topologized space with associated collection 11 of open subsets7 then for any U E 11 we endow U with a structure of G topologized space by using fly V 6 ill V Q U as the collection of open subsets of U and using the same collection of coverings that if7 for each V 6 fly its associated collection CovUV of set theoretic coverings is CovV This construction does satisfy all of the axioms to be a G topologized space7 and it is called the open subspace structure on U A sheaf of sets7 groups7 etc on a G topologized space X is a contravariant assignment U gt gt 9U of a set or group7 etc to each U E 11 such that the usual sheaf axioms are satis ed for coverings in CovU for all U E 11 In this respect7 we are restricting both the concept of openness and the concept of open covering from ordinary topology Note that if U E 11 and 37 is a sheaf on X then the functor 97h V gt gt 9V on fly is easily seen to be a sheaf on U with respect to its open subspace structure We leave it to the readers imagination or see BGR7 Ch 9 to formulate how one glues G topologized spaces or shea es presheaves this latter issue requires some care Example 237 Consider a pair X7 x consisting of a G topologized space X and a sheaf of k algebras x on this space If U E 11 then with the open subspace structure on U we get another such pair U7 y where y is the sheaf of k algebras le This is called an open subspace of X7 x For a second such pair X 7 x a morphism X 7 63 a XX is a pair f X a X and fit x a frX where f is continuous in the sense that pullback under f respects the class of opens and their coverings and fit is a map of sheaves of k algebras with fr de ned in the usual way fr carries sheaves to sheaves because f is continuous Composition of morphisms is de ned exactly as in the theory of ringed spaces In case the stalks of the structure sheaves are local rings7 we can also de ne the more restrictive notion of a morphism of locally ringed G topologized spaces As a fundamental example7 for any h af noid algebra A we have constructed such a space SpA7 and if U Q SpA is an af noid subdomain then the corresponding open subspace 18 BRIAN CONRAD U Aly is naturally identi ed with SpAU due to Exercise 2243 More importantly if b A a B is a map of k af noid algebras then we get a morphism of locally ringed G topologized spaces Sp SpB a SpA as follows on underlying spaces we use the map f M MB a MA and the map 624 a fB that is de ned on af noid subdomains U Q SpA via the k algebra map AU a AU AB Bray and is uniquely extended to general admissible opens via the sheaf axioms One checks readily that this assignment A w SpA is thereby a contravariant functor and one shows by copying the proof in the case of af ne schemes that this is a fully faithful functor In this way the opposite category of the category of k af noid algebras is identi ed with a full subcategory of the category of locally ringed G topologized spaces with k algebra structure sheaf and maps that respect this k structure An unfortunate fact of life is that the concept of stalk at a point of X is not as useful as in ordinary topology For example even for the G topologized spaces X MA it can and does happen that there exist abelian sheaves 97 on X and nonzero s E 9X such that 51 E 971 vanishes for all z E X The reason that this does not violate the sheaf axiom is that the vanishing in the stalk merely provides Um E 11 containing z E X so that sly E 37Um vanishes but perhaps Umhex is not in CovXl Hence one cannot conclude s 0 For the purposes of coherent sheaf theory in rigid analytic geometry this pathology will not intervene However it is a very serious issue when working with more general abelian sheaves and so it arises in any attempt to set up a good theory of etale cohomology on such spaces The work of Berkovich and Huber enlarges the underlying sets of af noid spaces and their global counterparts to have enough points77 so as to permit stalk arguments to work as in classical sheaf theory This is one of the technical merits of these other approaches to non archimedean geometry 24 Globalization Having introduced the notion of G topologized spaces and sheaves on them we can now make the key global de nition De nition 241 A rigid analytic space over k is a pair X x consisting of a locally ringed G topologized space whose structure sheaf is a sheaf of k algebras such that there is a covering E CovX with each open subspace Ui lei isomorphic to an af noid space SpA for a k af noid algebra A Morphisms are taken in the sense of locally ringed G topologized spaces with k algebra structure sheaf and maps respecting this k structure as in Example 237 Example 242 Let X x be a rigid analytic space and let U Q X be an admissible open subset The pair U y is a rigid analytic space lndeed let be an admissible open covering of X such that each Ui 612 is an af noid space By the axioms for a G topologized space Ul U is an admissible open covering of U Thus if we can nd an admissible open covering of each U U by af noid spaces Vi j E J then the entire collection is an admissible covering of U by the G topology axioms and the de nition of the open subspace structure so U y thereby has an admissible covering by af noid spaces as required Hence we can rename U as X to reduce to the case when X SpA is an af noid space But then by de nition of the G topology on MA any admissible open U Q SpA has an admissible covering by af noid domains SEVERAL APPROACHES TO NONARCHIMEDEAN GEOMETRY 19 For any rigid analytic space X and x E X the stalk 6 can be computed using any af noid open around x so its residue eld is a nite extension of k For any f E XX we therefore get a value fx 6 and hence a number 2 0 We can imitate the de nitions of Weierstrass Laurent and rational domains by imposing non strict inequalities on the lfxl7s and these are admissible opens in X since they meet each admissible af noid open in an admissible open subset One can also construct rigid analytic spaces by gluing procedures that are similar to the case of gluing ringed spaces Rather than delve into general details we illustrate with some examples Example 243 We rst construct rigid analytic af ne n space over k This rigid space AZ is de ned by choosing c E k with 0 lt lcl lt 1 and gluing a rising chain of closed balls centered at the origin with polyradius lcl j for j 2 1 More precisely let Dj B be the closed unit n ball over k with coordinates 51 w We de ne the map Dj a DHl to correspond to the k af noid algebra map 5 gt gt cg which makes sense since lcl S 1 This identi es Dj with an af noid subdomain S in D741 and in particular for each 1 S x39 S n the analytic functions gmci on Dj are compatible with change in j Thus on the gluing of the Ds we get unique global sections of the structure sheaf such that 513 gmci for all j This gluing is denoted AZ and the locus 5 S lcl jl nl S lcl j is the open subspace Dj with 0le 5 It is instructive to see that this deserves to be called an af ne n space by showing that it has the right universal property Namely for any rigid space X and any morphism f X a A we get pullback functions f i E XX and hence a natural map of sets HomXAZquotm a XX given by f gt gt fwgl f n We claim that this is bijective so it provides a universal property af ne n space is the universal rigid space equipped with an ordered n tuple of global functions This gives a viewpoint that is independent of the auxiliary choice of c To prove the bijectivity by naturality and gluing for morphisms it suf ces to treat the case when X is af noid But then by the Maximum Modulus Principle the k af noid ring A of global functions on X is the rising union of its subsets A of elements with sup norm at most lcl j forj 2 1 since 0 lt lcl lt 1 Since a map f X a A 1 lands in Dj if and only if each f i E A has sup norm at most lcl j why we reduce ourselves to a problem for the Ds separately Upon working with g cj ilpj the problem for each Dj viewed as a closed unit n ball becomes exactly the universal property of the n variable Tate algebra Example 244 By imitating the gluing procedures used to make projective spaces as a union of af ne spaces we can construct rigid analytic projective spaces These can also be constructed by gluing closed unit polydiscs These satisfy the usual universal property in terms of line bundles by the same method of proof as in algebraic geometry once the theory of coherent sheaves to be discussed later is fully developed Example 245 In Exercise 223 we saw how to de ne the completed tensor product A A A for a pair of maps of k af noid algebras A a A and A a A Via its universal property one readily checks much like in the case of af ne schemes with tensor products that SpA AuA equipped with its evident morphisms to SpA and SpA agreeing upon composition to SpA is a ber product SpA gtltspA SpA in the category of af noid spaces and then 20 BRIAN CONRAD by gluing maps that it is such a ber product in the category of rigid analytic spaces One can then copy the same gluing method as for schemes to globalize this construction to obtain the existence of bers products X gtltXu X for any pair of maps X a X and X a X of rigid analytic spaces Example 246 In complex analytic geometry7 a very useful tool is the procedure of analyti cation for both algebraic C schemes and coherent sheaves on them The resulting functors 3 w 3 and 97 w 973 from algebraic C schemes to complex analytic spaces and on their categories of sheaves of modules satisfy a number of nice properties that we will not list here The one aspect we note is that there is a natural map My 3quot a 3 of locally ringed spaces with C algebra structure sheaves such that My carries 3 bijectively onto C it induces isomorphisms on completed local rings7 and it is nal among maps from complex analytic spaces to 3 For any X module 97 on X7 one de nes 973 The GAGA theorems of Serre as extended by Grothendieck from the projective to the proper case concern three aspects the equivalence of categories of coherent sheaves on 3 and 3 when 3 is proper7 the comparison isomorphisms of cohomology for 97 and 973 when 3 is proper and 97 is coherent7 and the full faithfulness of the functor 3 w 3 when 3 is proper A similar procedure works in the rigid analytic setting as follows First of all7 algebraic af ne n space A equipped with its standard ordered n tuple of global functions can be shown to be a nal object in the category of locally ringed G topologized spaces equipped with k algebra structure sheaf ln particular7 we get a unique morphism man n Ak a Ak compatible with the standard ordered n tuple of global functions on each space By the universal property of the source and target7 this is nal among all rigid analytic spaces over k equipped with a morphism to AZ In general we de ne an analyti catz39on of a locally nite type k scheme 3 to be a map My 3quot a 3 that is nal among all maps from rigid spaces over k to 3 as locally ringed G topologized spaces with k algebra structure sheaf7 and maps that respect this k structure The preceding shows that for 3 A an analyti cation exists rigid analytic af ne n space Then one uses arguments with coherent ideal sheaves discussed in the next lecture to pass from this case to all af ne algebraic k schemes7 and nally to the general case by gluing arguments We omit the details7 except to remark that analyti cation is naturally compatible with the formation of ber products and that My carries 3 bijectively onto the set of closed points in 3 inducing an isomorphism on completed local rings Exercise 247 Let X be a rigid analytic space and a collection of admissible opens that cover X set theoretically Show that this is an admissible covering if and only if for every morphism f SpA a X from an af noid space7 the set theoretic covering f 1Ul by admissible opens has a nite af noid open re nement This is an easy exercise in unwinding de nitions7 with morphism77 de ned as in Example 237 Keep in mind that by de nition X has an admissible covering by open af noid subspaces7 and that the functor A w SpA is a fully faithful functor in the sense discussed near the end of Example 237 SEVERAL APPROACHES TO NONARCHIMEDEAN GEOMETRY 21 Exercise 248 A rigid analytic space X is said to be quasi compact if it has an admissible covering consisting of nitely many af noid opens A morphism f X a X of rigid analytic spaces is quasi compact if there is an admissible covering of X by af noid opens Ul such that each admissible open f 1UZ in X endowed with its open subspace structure is quasi compact for all 239 1 Prove that if X SpA is af noid and f X a X is quasi compact7 then for every af noid subdomain V SpB Q SpA the preimage f 1V is quasi compact ln particular7 X is quasi compact Hint Use the Gerritzen Grauert Theorem Theo rem 225 to reduce to the case when V is a rational domain 2 Prove that if f X a X is quasi compact in the sense of the above de nition then for every quasi compact admissible open U Q X the preimage admissible open f 1U Q X is quasi compact Hint Show that it suf ces to treat the case when U is af noid Assume that f X a X is a local isomorphism in the sense that there is an admissible open covering of X such that f maps U isomorphically onto an admissible open Ul Q X If f is bijective and quasi compact then prove that it is an isomorphism Also give an example of such an f that is bijective but not an isomorphism A 0 V We conclude with an exercise that demonstrates the power of Tate7s theory by rescuing connectedness Exercise 249 A rigid analytic space X is disconnected if there exists an admissible open covering U7 V of X with U7 V 31 Q and U V 0 Otherwise we say that X is connected Under this de nition7 X Q is connected 1 Using that the k algebra of global functions on SpA is A7 prove that SpA is connected if and only if A has no nontrivial idempotents This includes A 0 as an uninteresting special case Equivalently7 SpA is connected if and only if SpecA is connected Also show that if A A1 gtlt A2 is a product of nonzero k af noid algebras then SpA1 and SpA2 are af noid subdomains of SpA hint impose inequalities on pointwise absolute values of idempotents with SpA1SpA2 an admissible covering of SpA 2 Let X be a rigid analytic space For any x 6 X7 let Um be the set of points x E X that can be linked to x by a connected chain of nitely many connected admissible af noid opens That is7 there exist connected admissible af noid opens U17 Un in X such that z E U17 x E U 7 and Ul Ui1 31 Q for all 1 S 239 lt n Prove that the Um7s are admissible open in X and that for any 172 E X either Um1 Um2 or Um1 Um2 Q Prove that the collection of Um7s without repetition is an admissible cover of X 3 Building on the previous part7 prove that X is connected if and only if XX has no nontrivial idempotents just like for locally ringed spaces Moreover7 in the context of the previous part7 show that the Um7s are connected and that any connected admissible open in X is contained in some Um For this reason7 we call the Um7s the connected components of X 22 BRIAN CONRAD 3 COHERENT SHEAVES AND RAYNAUD7S THEORY 31 Coherent sheaves To go further in the theory eg to de ne closed immersions separatedness etc we need to discuss coherent sheaves Kiehl extended Tate7s methods to prove the following basic result Theorem 311 Let X SpA be an a noid space over h and let M be a nite A module The assignment U gt gt AU A Mior a noid subdornains U Q X uniquely ecctends to an X rnodule M In particular M 2 MX and the natural map Hom XM 7 a HomAM97X induced by the global sections functor is bijectiye for any X rnodule 95 Kiehl also proved the following globalization Theorem 312 Let X be a rigid space and an admissible a noid cover Let 97 be an X rnodule The following properties are equivalent 1 For every admissible a noid open V Q X 91V 2 M for a nite XV rnodule V 2 For every i 2 M for a nite XU rnodule M An X module 97 that satis es these equivalent conditions is called coherent As for locally noetherian schemes coherence is inherited by kernels cokernels tensor products and extensions There is a naive approach to trying to de ne quasi coherence but it is not satisfactory as we now explain Motivated by the case of locally noetherian schemes one may consider to de ne a quasi coherent sheaf on X to be an X module that locally on the space ie on the constituents of an admissible covering can be expressed as a direct limit of coherent sheaves This is the de nition suggested in FvR Exer 467 It can be shown that this property is preserved under the formation of kernels cokernels extensions tensor products and direct limits and that it suf ces to work with coherent subsheayes in the local direct limit process used in the de nition However it is generally not true that on an arbitrary admissible af noid open in the space such a sheaf is a direct limit of coherent sheaves thereby answering in the negative the open problem77 mentioned in FvP Exer 467l More speci cally Gabber has given an example of a sheaf of modules 97 on the closed unit disk B1 such that 97 is locally a direct limit of coherent sheaves but with nonzero degree 1 sheaf cohomology so 97 cannot be expressed as a direct limit of coherent sheaves over the entire af noid space because the formation of sheaf cohomology commutes with the formation of direct limits on an af noid rigid space this is left as an exercise for readers who are familiar with the Cech to derived functor cohomology spectral sequence which has to be carried over to the rigid analytic case There may be a better de nition of quasi coherence that enjoys the stability properties under basic operations as in algebraic geometry and is equivalent over an af noid space to some module theoretic data perhaps with topological structure over the coordinate ring but I do not know what such a de nition should be Exercise 313 If X SpA is af noid and M is a nite A module prove that ME 2 M A 6 for all z E X Using that 6 is a local noetherian ring with the same completion SEVERAL APPROACHES TO NONARCHIMEDEAN GEOMETRY 23 as AW deduce that if m E M vanishes in 17 for all z E X then m 0 More globally7 deduce that a global section of a coherent sheaf on any rigid analytic space vanishes if and only if it vanishes in all stalks on the space This property is special to coherent sheaves it fails for general abelian sheaves Example 314 Let X SpB a SpA Y be a map of k af noid spaces and let I be the kernel of the natural map B AB a B induced by multiplication Then 12 is a nite module over B ABI B This is generally not the same as the module of relative algebraic Kahler differentials which is typically huge7 since the usual tensor product B A B may not be noetherian This nite B module gives rise to a coherent sheaf Q qy on X There is an y linear derivation x a QkY that can be globalized in accordance with local formulas similar to the situation in algebraic geometry Example 315 A map of rigid spaces f X a X is a closed immersiori if there exists an admissible af noid covering of X such that U f 1Ul is af noid and the map of af noids U a Ul corresponds to a surjection on coordinate rings In that case it can be proved that for every admissible af noid open U Q X the preimage f 1U is af noid and f 1U a U corresponds to a surjection on coordinate rings7 and moreover that y a f1X is surjective with coherent kernel J In fact7 this coherent ideal sheaf J determines the map f X gt X up to unique X isomorphism7 and conversely every coherent ideal sheaf I Q x arises in this way by gluing SpAI7s for admissible af noid open SpA Q X with I JSpA Q XSpA A Using closed immersions and quasi compactness7 we can carry over some notions from algebraic geometry involving diagonal maps De nition 316 A map f X a Y of rigid spaces is separated if the diagonal map Af X a X gtltyX is a closed immersion In case Y Spk7 we say that X is separated lf Af is merely quasi compact equivalently7 the overlap of any two af noid opens in X over a common af noid open in Y is quasi compact then f is quasi separated A map of rigid spaces f X a X is riite if there exists an admissible af noid covering Ui of X such that each U f 1Ul is af noid and the map of coordinate rings XUi a X U is module nite In this case it can then be shown that f 1U is af noid with coordinate ring nite over that of U for any af noid open U Q X Remark 317 There is no rigid analytic notion of af noid morphism77 akin to the concept of af ne morphism in algebraic geometry The problem is that there is no good analogue of Serre7s cohomological criterion for af neness see Liu for counterexamples ie7 quasi compact and separated non af noid spaces whose coherent sheaves all have vanishing higher cohomology lnterestingly7 Liu also gives an example of a quasi compact and separated non af noid space admitting a nite surjection from an af noid space7 in contrast with a theorem of Chevalley in the case of schemes a separated scheme admitting a nite surjection from an af ne scheme is necessarily af ne Exercise 318 Copy the proof from the case of schemes to show that if f X a Y is a map of rigid spaces and Y is separated then for any admissible af noid opens U Q Y and V Q X7 the overlap V f 1U is af noid Hint Show that the graph map 17f X a X gtlt Y is 24 BRIAN CONRAD a closed immersion and consider U gtlt V Q X gtlt Y Taking f to be the identity map7 deduce that an overlap of nitely many admissible af noid opens in a separated rigid space is again af noid Exercise 319 Let X be a rigid space over k Prove that X is quasi separated if and only if it has an admissible covering by af noid opens Ul such that each overlap Ul U7 is quasi compact7 in which case U V is quasi compact for any quasi compact admissible opens U and V in X Prove that if X is a quasi separated rigid space then for any nite collection of quasi compact admissible opens in X7 the union U UUZ is an admissible open in X for which the Us are an admissible covering Give an example of a quasi compact rigid space that is not quasi separated7 and one such for which some nite union of admissible af noid opens is not an admissible open subset Remark 3110 By a gluing procedure7 on the category of quasi separated rigid spaces over k one can de ne change of base eld functors X w XK from rigid spaces over k to ones over K for any analytic extension eld Kk7 compatibly with ber products This is very useful7 such as with k Q17 and K Cp7 or more generally K E for any k It is a general fact that if E k is in nite then E is not complete7 so EA contains elements that are transcendental over k in such cases a notable such example is k Qp The idea underlying the de nition of this functor is to rst de ne it in the af noid case via the operation A w K kA from k af noid algebras to K af noid algebras compatibly with completed tensor products7 and to then globalize by gluing in the separated case since an overlap of af noids is af noid The quasi separated case is obtained by another repetition of this process7 using that an overlap of admissible af noid opens in a quasi separated space may fail to be af noid but is at least quasi compact and separated We omit the details7 except to remark that this is merely a construction77 and it is not really a ber product or characterized by an abstract universal property as in the case of schemes if Kk has in nite degree because in such cases rigid spaces over K cannot be mapped to rigid spaces over k in any reasonable way MaxSpec is not functorial with respect to pullback of prime ideals between k af noid algebras and K af noid algebrasl This lack of functoriality is a real nuisance7 but in the approaches of Berkovich and Huber there are many more points in the underlying spaces and one can view change of base eld functors as actual ber products and more speci cally one can consider analytic spaces over k and K as part of a common category For purposes of analogy7 consider the classical concepts of a variety over 6 and over C using only closed points77 from the scheme perspective no universal domain and try to formulate the idea of the map VC a V for a Q variety V the trascendental points in V0 have nowhere to go in V since the variety V does not have generic points This is due to the use of MaxSpec in classical algebraic geometry7 and exactly the same problem arises in rigid geometry ie7 rigid spaces lack enough points Exercise 3111 Since elements of kX17 7Xn have only nitely many nonzero coef cients whereas elements of kltX17 7X can have in nitely many nonzero coef cients7 change of the base eld in rigid geometry exhibits some features that may be surprising from the viewpoint of algebraic geometry For example7 it can happen that an af noid space X SpA over k remains reduced after any nite extension on k but not after SEVERAL APPROACHES TO NONARCHIMEDEAN GEOMETRY 25 some in nite degree analytic extension on k This never happens for algebraic schemes over a eld lndeed7 consider the following example Let k be a non archimedean eld of characteristic p gt 0 such that k W is in nite 1 Show that an example of such a k is with the y adic absolute value7 where F is a eld of characteristic p such that F FF is in nite Find such an F 2 Show that there exists an in nite sequence can in k tending to 0 such that lanl S 1 for all n and the all 7 generate an in nite degree extension of k 3 Choose a sequence can as just constructed7 and let f Zaanp 6 T1 and A T1 YlYp 7 f which is Tl nite7 hence af noid Prove that A 2 T1YYp 7 f and that A kk is a domain for any k k of nite degree7 but A kk1r1 is not reduced Also show that 771 is complete Assume p 31 2 and let B kXY7 tt2 7 Yp 7 Prove that B k k is a normal domain for any nite degree extension k k but that 3ka1 is reduced and not normal It can be shown that B kK is reduced for any analytic extension eld A 4 V Exercise 3112 This exercise addresses some subtle features of the general concept of an admissible open subset7 even within an af noid space Let X be a quasi separated rigid space over k and let Kk be an analytic extension eld Let 239 U 7 X be the natural inclusion from an admissible open U Q X We get an induced map of rigid spaces Z39K UK 7 XK over K ls this an isomorphism onto an admissible open To appreciate where the dif culties lie7 consider some special cases as follows 1 Assume that X SpA is af noid and that U is an af noid subdomain Prove that UK 7 XK is an isomorphism onto an af noid subdomain Do this by rst showing that UK has admissible open image in XK via the Gerritzen7Grauert theorem Theorem 2257 and then work with the scalar extension of the coordinate ring AU of U Exercise 2482 will be useful here The reason that this special case requires serious input Theorem 225 is that the universal property for U Q X7 even when formulated in purely algebraic terms via k af noid algebras7 only involves maps with k af noid spaces7 and not also K af noid spaces if K k is in nite 2 If X is af noid and U is a quasi compact admissible open7 use the previous part to show that Z39K is an isomorphism onto a quasi compact admissible open in XX 3 Prove an af rmative answer ifz39 is a quasi compact map The key dif culty in the general case appears to be to determine if Z39K has admissible open image in XK 32 Cohomology properness and atness No discussion of coherent sheaves would be complete without addressing their cohomology7 especially in the proper case We rst make some general observations concerning how to de ne sheaf cohomology on rigid spaces Despite the problematic nature of stalks at points on rigid spaces when working with general abelian sheaves7 one can adapt some methods of Grothendieck to prove that the category of abelian sheaves and the category of X modules on a rigid space X each have enough injectives7 so we may and do de ne sheaf cohomology via derived functors in both cases The concept of asque sheaf as traditionally used on ringed spaces is a bit problematic in the general rigid analytic case7 but nonetheless the following result can be proved and it is left as an exercise just for those whose taste is inclined toward such questionsl 26 BRIAN CONRAD Exercise 321 Let X be a quasi separated rigid space Prove that for an X module 95 the natural map from its sheaf cohomology in the sense of X modules to its sheaf coho mology in the sense of abelian sheaves is an isomorphism More speci cally7 prove that an injective X module has vanishing higher sheaf cohomology in the sense of abelian sheaves7 and that restriction to an admissible open preserves the property of being an injective sheaf of modules resp an injective abelian sheaf Thus7 on quasi separated spaces the theory of sheaf cohomology via derived functors presents no ambiguities Kiehl showed that there is a good cohomology theory for coherent sheaves on rigid spaces His work7 coupled with some auxiliary arguments7 gives the following result Theorem 322 Let X be a rigid space and 97 a coherent sheaf on X 1 Acyclicity theorem for coherent sheaves IfX is a noid andil is a nite covering ofX by admissible a noid opens then the C39ech cohomology H 1195 vanishes for alli gt 0 Moreover the sheaf cohomology de ned via derived functors H X95 vanishes for alli gt 0 2 IfX is a quasi compact and separated rigid space andil is a nite admissible a noid open covering ofX then the natural map H 1195 a HiX95 is an isomorphism for all i We next turn to the de nition of properness In view ofthe nature of the Tate topology7 the condition of universal closedness that is used for schemes is not the right one to use in rigid geometry lnstead7 we adapt a formulation similar to one that works in the complex analytic case De nition 323 A map f X a Y of rigid spaces is proper if it is separated and quasi compact and there exists an admissible af noid open covering of Y and a pair of nite necessarily admissible af noid open coverings Viibeg and Wjeji same index set JZ of j7sl of f 1Ul such that two conditions hold Vi Q for all j and for allj E JZ there is an n 2 1 and a closed immersion gt Ul gtlt B over Ul such that Vi Q Ul gtlt lt1l 7 S r for some 0 lt r lt 1 with r E Equivalently7 by the Maximum Modulus Principle7 we can replace 3 r with lt 1 The condition on the inclusion Vi Q over Ul in De nition 323 is called relative compactness of Vi in over Ui It is a replacement for saying that Vi has Ui proper closure in in ordinary topology Algebraically7 if the coordinate rings of Ui Vii and are Ai Big7 and ng then the condition is that the k af noid Ai algebra ng can be expressed as a quotient of a relative n variable Tate algebra over Ai such that the images bi 7b in ng of the standard variables X17 Xn from the Tate algebra have images b17 bn 6 Bi with sup norms less than 1 That is7 all b spwij have sup norm less than 1 for all j E Ji Exercise 324 In the case of a submersion of complex manifolds7 use the implicit function theorem to describe the condition of properness of the underlying map of topological spaces in terms similar to the de nition of properness in the rigid analytic case Example 325 Let X PZ be rigid analytic projective n space This can be constructed by gluing n 1 af ne n spaces U07 7 Un exactly as in algebraic geometry so as to establish

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