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A GLIMPSE OF DEFORMATION THEORY BRIAN OSSERMAN The purpose of these notes as suggested by the title is not to provide any sort of comprehensive introduction to deformation theoryi Rather we attempt to convey the main ideas of the theory with a survey of some applications We do not explore an exhaustive list of possible topics nor do we go into details in many proofsi However in the case of deformations of smooth varieties we do attempt to give a thorough treatment of the theory of rstorder in nitesimal deformations with some hints as to how to generalize to other problems Because the systematic use of rings with nilpotents is one of the major distin guishing characteristics of scheme theory we can view deformation theory as a sub stantial application of scheme theory that is beyond the reach of classical algrebraic geometryi We hope to provide a relatively accessible and motivated introduction to the theory of cohomology of sheaves in the form of the Cech cohomology arising in deformations of smooth varietiesi 1 THE MANY USES OF ARTIN RINGS Recall that A is an Artin ring if it is Noetherian of dimension 0 For the sake of brevity we will in these notes also assume that all Artin rings are local Recall that the maximal ideal of such a ring consists entirely of nilpotentsi Although SpecA for A an Artin ring consists of only a single point the basic idea motivating deformation theory is that such schemes are still big enough77 to provide useful information We have already seen Artin rings arising in two different contexts in tangent spaces and in criteria for smoothnessi In the rst case we had that the tangent space of a scheme X at a point I was equal to the set of maps Spec Mel52 A X with image I where k A special case of the smoothness criterion was that if X is of nite type over a eld 16 then X is smooth over k if and only if for every Artin kalgebra A and every quotient A a A with kernel I a squarezero ideal every map Spec A A X over k can be lifted to a map SpecA H Xi Both of these involve understanding maps SpecA A X for A an Artin ringi Now suppose that X is a moduli space representing some moduli functor or more generally a nonrepresentable moduli functor Then the maps Spec A gt X more generally FSpec A which we abbreviate by correspond to families of objects over Spec Al If we x a map Spec k A X or an element no 6 Fk and restrict our attention to A with residue eld 16 we can consider elements of restricting to no under the natural map Spec k lt gt Spec A and this corresponds to studying families of objects over Spec A which restrict to the xed object no on the reduced point Spec k such families are called in nitesimal deformations of no over Spec A this motivates the terminology One of the most basic ideas of deformation theory is that we can study the tangent space and smoothness ofa moduli space by considering families over Spec Al 1 2 BRIAN OSSERMAN Although completely classifying families of objects over arbitrary schemes T is usually far too complicated in the case of Artin rings the bases are generally small enough that one can write down explicit descriptions 2 PREDEFORMATION FUNCTORS The standard setup introduced by Schlessinger in is as follows we denote the category of Artin rings with a given residue eld k by Artk and consider covariant functors F Art k A Set with the property that Fk is a onepoint set The idea is that such functors should come from deformations over Artin rings of a xed object over k We will call such functors predeformation functors Example 21 In the situation that we have a modulifunctor F we can obtain a predeformation functor F by xing an element 770 E Fk and de ning FA 77 E FA I W 770 In the case that F is representable the above will give a particularly wellbehaved predeformation functor but in general to go between a global moduli problem and an associated deformation functor we will want to do something slightly different which cannot be expressed as well on the level of functors however since plunging into stacks would be a bit much we settle for an example to give the general idea Example 22 Let X0 be a variety over k We de ne a functor F the functor of deformations of X0 as follows FA is the set of schemes X at over Spec A together with a morphism X0 A X such that the diagram Xo gtX Spec k gt Spec A commutes and such that we have an induced isomorphism X0 3 X gtltA k Note that in this case the morphism X0 A X is a homeomorphism on underlying topological spaces so that X differs from X0 only on the level of structure sheaves De nition 23 Let F be a predeformation functor The tangent space TF is de ned to be Fke52 We say that F is formally smooth if for all A a A in Artk with kernel a squarezero ideal and every element 77 E FA there exists an element 77 E FA with n lA 77 Note that if F happens to be obtained from a functor F represented by some X as in the rst example above then TF is in fact the tangent space of X at the chosen point and X is formally smooth at the chosen point if and only if F is formally smooth As suggested above we will focus on the study of tangent spaces with some discussion of obstructions which measure the failure of a predeformation functor to be smooth A different question treated systematically in 4 has to do with with how close a predeformation functor F is to being representable in a suitable sense However we will not pursue this direction here Remark 24 In older references such as 4 the phrase deformation functor77 is frequently reserved for the functor of deformations of schemes with the term func tor of Artin rings77 used more generally However there are many different types of A GLIMPSE OF DEF ORMATION THEORY 3 problems now considered to fall under the rubric of deformation theory so we have adopted the above notation We also mention that as suggested by the terminology we have in mind a notion more speci c than a predeformation functor which we shall call a deformation functor We defer the de nition until such time as we discuss Schlessingerls work in more detail but mention that the conditions on a deformation functor will imply that we can put the natural structure of a kvector space onto its tangent space justifying the terminology 3 A SURVEY OF APPLICATIONS Taking for the moment for granted the idea that deformation theory provides a powerful tool for analyzing moduli problems and in particular for studying tangent spaces and smoothness we survey a number of applications of deformation theory to classical questions Applications to moduli spaces Of course the most direct applications of de formation theory are to the study of moduli spaces If we have a predeformation functor F obtained from a point z of a moduli space X then as mentioned before the notions of tangent space and formal smoothness for X and for F coincide Not ing that if a variety is smooth its dimension is the dimension of its tangent space one could prove in this way that the Grassmannian Cr d is smooth of dimension rd 7 r although of course our proof of representability gave a more direct and stronger statement in showing that it is covered by copies of A2017 However later we will study other ne moduli spaces such as Hilbert spaces where in many cases deformation theory is the only way of obtaining such statements Along these lines one of the archetypical examples is the moduli space My of curves of genus g This parametrizes at families of smooth proper geometrically connected curves of genus 9 Although it is not representable it is nonetheless rather wellbehaved for g 2 2 technically it is a Deligne Mumford stack but from our point of view it suf ces to take for granted that the notions of smoothness and dimension can be de ned independently and we can then state the following theorem proved via deformation theory Theorem 31 For g 2 2 the moduli space My is smooth of dimension 39 7 3 For the sake of giving a corollary which we can actually state precisely we note that My also has a coarse moduli space Mg although the notion of a coarse moduli space has additional hypotheses it is uniquely de ned by the property of corepresenting the moduli functor in question see the notes on Yonedals lemma and representable functors Theorem 32 For g 2 2 the functor associated to My is corepresentable by a variety M Q We then have the following corollary of the earlier theorem Corollary 33 For g 2 2 the coarse moduli space My has dimension 39 7 3 Remark 34 Note that Mg will not be smooth in fact it will have singularities at points corresponding to curves with nontrivial automorphisms This is a typical situation and illustrative of the advantages of working with stacks rather than coarse moduli spaces 4 BRIAN OSSERMAN However deformation theory is an important tool for a variety of arguments so we survey applications beyond moduli spaces Finiteness arguments One very natural application of deformation theory is in proving finiteness results If one wants to prove that a certain set of objects is finite one can break the problem into two steps first show that they are parametrized by a moduli space of nite type and then show that none of the objects have nontrivial firstorder deformations We can then conclude that the moduli space consists of a collection of disjoint points which is necessarily nite While the first step is typically independent deformation theory is often a crucial ingredient in the second step We mention here a typical example a famous conjecture of Shafarevich in a case proved by Parshin using the above approac De nition 35 We say that a morphism X A B is isotrivial if for general distinct points b b E B we have an isomorphism of fibers X1 Xb Theorem 36 Shafarevich Parshin Let B be a smooth proper curve over a eld k and x an integer g 2 2 Then there exist only nitely many nonisotrivial families of curves X gt B which are smooth and proper with bers being curves of genus g This says in essence that for a fixed g 2 2 and a smooth proper curve B of genus g over a eld 16 there are at most finitely many nonconstant families of curves of genus g over B which we can think of as saying that there are only finitely many nonconstant maps from B to My We note that in fact the statement is more general with the same statement holding even when B is not proper However although the proof of the general statement also uses deformation theory it is substantially subtler and was proved by Arakelov Constructing families One important application of deformation theory in volves putting chosen varieties into wellbehaved families of one sort or another Given a variety X0 over 16 we might wish to put X0 into a family X over some base B such that one fiber is isomorphic to X0 and X has some good properties A typical example is the following theorem Theorem 37 Winters Let X0 be a proper geometrically integral curve over a eld h with at worst nodal singularities Then there exists a base scheme B with point b E B and a scheme X over B such that i Mb h and X1 2 X0 ii B is regular of dimension 1 and can be chosen to be a curve over 16 or h has positive characteristic we can set B Spec A where A is a mixed characteristic DVR with residue eld h iii X is a regular surface and its generic ber over B is smooth Such theorems have two major types of applications The first involves the case that X0 is in fact smooth but h has positive char acteristic the theorem in this case preceded Winters7 work Here by choosing B to have mixedcharacteristic ie its generic point is a eld of characteristic 0 we can realize X0 as a specialization of a curve in characteristic 0 Such techniques are often essential to carrying out computations in characteristic p as in the case of the algebraic notion of the fundamental group of a curve of genus g 2 2 where A GLIMPSE OF DEF ORMATION THEORY 5 the only known way to calculate the fundamental group involves using the classi cal topological formula for curves over C applying this to curves in characteristic 0 more generally and realizing the given curve in characteristic p as the specialization of a curve in characteristic 0 as above While the above application involved using knowledge of the generic ber to prove results about X0 one can also work in the other direction typically by choosing X0 to be singulari This is the essence of degeneration arguments A typical situation might be as follows suppose we wish to prove something about curves of genus g and it turns out to be enough to prove the statement for a single curve of genus 9 this is the case for instance with the BrillNoether theoremi One might take smooth curves of genus l and g 7 l and glue them together at a single node to form Xoi In this case the smooth generic ber provided by the above theorem will have genus g and one can hope to prove the desired statement for the generic ber by making sense of it for X0 and understanding the situation for X0 in terms of the two smooth componentsi Since these components each have genus less than 9 assuming 9 gt 1 this potentially sets up an induction with base case 9 ll But to even get started one has to know that the chosen X0 can be put into a family with smooth generic ber and that is where the above theorem comes in We very brie y describe the proof of Winterls theorem as it is typical of a certain class of results The rst step is to show that one can construct compatible systems of families over larger and larger Artin rings for instance over Htlt for each n This is where deformation theory is used One then makes an argument usually fairly straightforward but not completely automatic that one can construct a deformation over the formal scheme77 see lli9 of corresponding to the limit A of the given Artin rings in the case above A Next one applies a theorem of Grothendieck to effective the deformation which means to realize it over the standard scheme Spec Al Here some real conditions arise and there are many examples of deformations which arise in nature and can be constructed over formal schemes but not effectivizedi Finally one applies a theorem of Artin to algebraize the deformation approximating it for instance over a curve of nite type over i Lifting Galois representations One intriguing application of deformation the ory has been Mazurls theory of deformations of Galois representations and its highly successful application to the proof of the Shimura Taniyama Weil conjec ture on modularity of elliptic curves and further progress in recent years Al though there is no geometry involved the theory does fall neatly into the setting of predeformation functorsi It had been known for a long time that the following problem is important Question 38 Given a Galois representation p GQQ A GL2le what are the possible lifts to representations 2 GQQ gt GL2 Z17 The idea is that in order to answer this question one lifts p successively to Zpnz for increasing n and then forms the inverse limit to obtain a lift to p The Artin rings in question are the Zpnz and the possible lifts for each n de ne the predeformation functori ln 3 Mazur considered this problem in the context of Schlessingerls theory showing that the resulting deformation theory is wellbehavedi 6 BRIAN OSSERMAN This formed an integral part of Wiles7 technique and subsequent work on Galois representations 4 SKETCH OF AN ARGUMENT We discuss one more application of deformation theory to a classical question7 with a somewhat more detailed description of how deformation theory can be used to prove the stated result We consider the following special case of the Brill Noether problem Question 41 Given 9 2 07 for which d gt 0 is it the case that every curve of genus g has a nonconstant map to P1 of degree at most For g 07 the only curve is P1 itself7 so the answer is that any d will do For g 17 we cannot have a map of degree 1 to P1 as such a map would have to be an isomorphism But we always have a map of degree 27 so any d 2 2 is okay The general statement is the following Theorem 42 The answer to the above question is my d for which 2d727g 2 0 The proof of this theorem is in two parts an existence statement when 2d 7 2 7 g 2 07 and a nonexistence statement when 2d 7 2 7 g lt 0 We will now sketch a simple proof based on deformation theory for the nonexistence statement We can immediately check the desired statement in the cases 9 0 or g l for g 07 the assertion is only that d 2 17 which is vacuous7 while for g 17 the assertion is that d 2 27 which is necessary because a map of degree 1 between smooth proper curves is necessarily an isomorphism7 and if g gt 0 then C is not isomorphic to P1 We therefore assume 9 2 2 he basic idea is to consider the following sequence of moduli spaces 35 7gt g 7gt lg7 where 0 Mg is the moduli space parametrizing curves C of genus g 0 t is the moduli space of pairs C7 of curves C of genus 9 together with a map f C 7gt P1 of degree d o g is the moduli space of pairs C7 as for 35 except that we consider C7 N C7 f if f and f are related by an automorphism of P1 The argument then works as follows by de nition7 a curve C with a map to P1 of degree d is precisely a point of Mg which has at least a point of mapping to it It follows that in order for every curve of genus g to have a map of degree at most d to P1 we have to have the map 9 7gt My be dominant for some degree d S d in particular7 we have to have dimg 2 dim Mg 39 7 37 with the last equality coming from the earlier theorem The reason for introducing the space 35 is that it turns out that the dimension of t is easiest to compute directly lndeed7 we can do this via deformation theory7 and we nd that with g 2 27 we have dim 3t 2d 29 7 2 Then7 the automorphism group of P1 is 3dimensional indeed7 it consists of all invertible maps of the form 2 gt gt 2112 with simultaneous scaling producing the same map see Exercise 166 or Example H711 of 27 so we have dimg dim t 7 3 2d2g 7 5 We thus have dimg lt dimV1g if and only if 2d 7 2 7 g lt 07 proving the desired nonexistence statement A GLIMPSE OF DEF ORMATION THEORY 7 Warning 43 The above doesn7t quite make sense as written One can write down a correct argument entirely in terms of schemes by replacing Mg with the base for a modular family77 of curves of genus g which is to say an etale cover of Mg The intuition here is that a modular family is like the universal family over Mg but instead of each curve appearing once each is allowed to appear nitely many times The other spaces should then be interpreted in terms of the curves in the modular family and one can make the argument described below in a precise manner 5 DEFORMATIONS OF SMOOTH VARIETIES AND CECH COHOMOLOGY ln Example 22 above we describe the problem of deforming a variety X0 over h In general it is not easy to describe such deformations but in the case that X0 is smooth over 16 we have all the tools to at least describe the tangent space of the problem 7 that is all deformations of X0 over Spec Mel62 The rst two steps of the analysis are the following results Proposition 51 If X0 is a smooth a ne variety over 16 every rstorder defor mation of X0 is isomorphic to the trivial deformation X0 Indeed we will see with the help of a lemma on atness that this is equivalent to Exercise 1187 of 2 which was on Problem Set 12 Part 1 Proposition 52 Let X0 be a smooth variety over 16 Then the sheaf of in nites imal automorphisms of X0 is the tangent sheaf TXO Here recall that the tangent sheaf is de ned for X0 smooth by Ham9 k OX If as above X0 6 denotes the trivial deformation of X0 over Spec Mele2 in nites imal automorphisms of X0 are automorphisms of X0e over Spec Mele2 which restrict to the identity on X0 The sheaf of in nitesimal automorphisms of X0 is the sheaf associating to an open subset U Q X0 the in nitesimal automorphisms of Assuming for the moment these two propositions if we let X0 be any smooth variety over 16 we can analyze the rstorder deformations of X0 as follows suppose X1 is a firstorder deformation of X0 by Proposition 51 if is any af ne open cover of X0 we have that X1 U1 is isomorphic to the trivial deformation for each i Moreover if we choose trivializations goi X1 U1 1 fle for each i then for any i lt j we have a gluing map so Ui le 3 UNIS obtained from goi UN and ng Umfl where UM I Ui UJx We note that because the data ofthe deformation X1 includes the map X0 t gt X1 each so gives the identity when restricted to UN so is an in nitesimal automorphism of UN and hence a section of TXO UM by Proposition 52 We also note that because the so come from the gluing data for schemes we have the cocycle condition so 0 soj sol1c for all i lt j lt k see Exercise H212 of Also we have so for any i lt j Conversely it is clear that given the data of sections so 6 TXO UN for all i lt j which satisfy the cocycle condition for all i lt j lt h and have so ego for all i lt j switching to additive notation as we consider the so as sections of the sheaf TXO we can glue to obtain an X1 which will be a rstorder deformation of X0 Note here that atness over Spec k 62 can be checked locally and since locally we are starting with the trivial deformation our modules are visibly free over Mele2 hence at 8 BRIAN OSSERMAN It remains to analyze the ambiguity of our description or equivalently when two different choices of so give the same deformation up to isomorphism But this is clear our original choice of the goi could be modi ed precisely by an in nitesimal automorphism of Ui which would then modify each so accordingly for anyj with j gt i We can formalize this as follows we have the group of Cech lCochains for and TXO which is simply the group HR TXO UM We say that a Cech lcochain is a cocycle if naturally enough it satis es the cocycle condition where composition is replaced by addition in TXO Edy16 Finally we have a coboundary map from TXO to the group ofCech lcocycles obtained by taking the section on UN to be the difference of the sections on Uj and Ui We de ne the rst Cech cohomology group on of TXO written H1UiTX0 to be the group of Cech lcocycles modulo the image of the coboundary map We therefore conclude that our two propositions above imply the following theorem Theorem 53 Let X0 be a smooth variety over 16 and any open a ne cover of X0 Then the rstorder deformations of X0 are parametrized by the cohomology group HlUl TXO It remains to proves Propositions 51 and 52 We need a key commutative algebra lemma on atness for the rst Lemma 54 A module M over k662 is flat and only the natural map MeM 5 SM is an isomorphism See Corollary 62 p 123 of Note that the map is always visibly surjective so atness is equivalent to injectivity Proof of Proposition 51 We rst assert that rstorder deformations of X0 are equivalent to extensions X OX of X0 by OXO see Exercise 1187 of 2 with the additional data ofa h vector space structure on OX a xed map i OX a X0 and if f is the kernel of i a xed isomorphism 7r 3 OX0 The main content to be checked comes down to the above lemma on atness but we go through the details below Indeed if X 0 is a rstorder deformation of X0 we are given by hypothesis a map ill OX gt OX0 as well as the structure of a sheaf of he62algebras on OXi Moreover the hypothesis that our map i X0 lt gt X induces an isomorphism X0 3 XspeckMez Speck is equivalent to saying ill induces an isomorphism OXEOX gt OX0 in particular we have ker ill EOXM By atness and the above lemma we have that the map OXEOX 5 EOX is an isomorphism which is to say that the he62algebra structure gives us the isomorphism 7r as well Conversely if we are given the extension X 0y together with ill and Tr we have that ill induces a map X0 a X while we claim that OX has the natural structure of a sheaf of h662algebras by hypothesis it has the structure of a sheaf of h vector spaces so it suf ces to de ne the multiplicationby e map We de ne this by i on 1 so that the kernel is given by We then obtain that X0 2 X X spec cm52 Spec h from the condition in the de nition of extension that ill induces an isomorphism OX Q OX0 It remains to conclude atness of X OX over Spec Mele2 and this follows from the isomorphisms 7r and Oy 3 OX0 by the above lemma We thus conclude that rstorder deformations of X0 do correspond to extensions of X0 by OXO with xed choices of hvector space structure and maps ill and Tr A GLIMPSE OF DEF ORMATION THEORY 9 By Exercise 118 of 2 every such extension is trivial in fact the same argument shows that every extension together with the halgebra structure and maps i and Tr is isomorphic to the trivial one so we conclude the desired result Before proving the second proposition we remark that it is a formalization of an old idea from differential geometry that one should think of a global section of the tangent bundle 7 a vector eld 7 as being an in nitesimal version of an automorphism as it gives an in nitesimal direction for each point to ow along Proof of Proposition 52 The main idea is that in nitesimal automorphisms of X0 correspond to h linear derivations OX0 A OX0 the desired statement will then follow from the universal property of QkOk with regard to derivations However an in nitesimal automorphism of X0 is nothing but a h662linear isomorphism OXOM 3 OXOM restricting to the identity modulo 6 it follows that on af ne opens U it is of the form a 6b gt gt aeb da for any ab E OX0 U where we note 0 6b gt gt 612 because I gt gt b edb for some db and the map is h662linear o d is some h linear map OX0 gt X0 But the multiplication structure tells us that ab gt gt ab edab a edab edb ab edab adb so we see that d must be a h linear derivation as desired Finally we had on af ne opens by de nition that maps 9amp0 k A OX0 which is to say sections of TXO correspond to h linear derivations OX0 A OX0 However since maps of sheaves are de ned on any open cover it is easy to see that this universal property passes to the level of sheaves completing the proof of the proposition D Remark 55 Although the details are beyond the scope of these notes we brie y explain the notions of higher Cech cohomology and of obstruction theory and sketch the fact that obstructions to deforming a smooth variety X0 lie in H2Ui TXO The idea is that it is useful to know when a deformation over Spec A lifts to a deformation over Spec A given a map A a A in Art lf deformations always lift we have by de nition that our predeformation functor is formally smooth One typically approaches this question by factoring the given map into maps with smaller kernel one can always then require that the kernel is a principle square zero ideal which is then necessarily isomorphic to h We call such a map a tiny extension It turns out that there is often a beautiful theory of obstructions to lifting a deformation over a tiny extension Roughly speaking what happens is that there exists a h vector space V called the obstruction space and if we are given any deformation 77A over Spec A and the tiny extension A a A we obtain an element 1 E V called the obstruction to lifting 77A to A with the property that v 0 if and only if 77A can be lifted to a deformation over Spec A We note that if we have an obstruction space V and if we can prove that V 0 it follows automatically that our deformation problem is formally smooth The converse is however false as if V is an obstruction space then any V containing V is likewise an obstruction space We now de ne the higher Cech cohomology groups TXO and sketch the proof of the following Theorem 56 For the deformations of a smooth variety X0 we have that H2Ui TXO is an obstruction space Deformation Theory and Moduli in Algebraic Geometry Deformations b representability and Schlessinger s criterion BRIAN OSSERMAN 1 FUNCTORS OF ARTIN RINGS We have already seen that a scheme X can be reconstructed from its functor of points that is from the data of all morphisms from all schemes to X On a more restricted level given a k valued point z E X we have seen that with mild hypotheses the tangent space to X at x can be recovered from the set of morphisms Spec kH a X with image m The rst gives a very global picture while the second is as local as possible This lecture series will focus on something in between functors of local Artin rings From the point of view of moduli theory we will be studying families over Artin rings It may seem like not much is going on here since geometrically everything is happening over a single point but the in nitesimal thickenings we study turn out to carry a surprisingly rich trove of information essentially equivalent to the complete local rings77 of the moduli space The main focus of the lectures will be two notions of representability for functors of Artin rings and a practical criterion developed by Schlessinger for testing representability 11 Recovering complete local rings We begin by making precise the data one can recover from a scheme X by considering maps SpecA a X where A is an Artin local ring We use the following temporary notation Temporary Notation 111 Given a eld k let Artk denote the category of Artin local rings with residue eld k and with morphisms commuting with the surjection to k Given a locally Noetherian scheme X and a point z E X let Fx Artkm a Set be the covariant functor sending an Artin local ring A to the set of morphisms SpecA a X such that the composition with Spec a SpecA gives the canonical map Spec a X induced by 13 We have seen that when X is over a eld k and k then we have in bijection with the tangent space of X at a The assertion is that the full data of the functor precisely recovers the complete local ring OX Proposition 112 The canonical map Spec a X induces an isomoijohism FSpec XYWm H FXgtm7 and any complete local Noetherian ring R with Spec R gt X inducing such a bijection is canon ically isommjnhic to OX1 Proof The rst statement is equivalent to the assertion that any map f SpecA a X in factors uniquely through Spec We rst observe that if A is any local ring then any map f SpecA a X having the closed point of A mapping to z necessarily factors uniquely through the natural map Spec OXm a X lndeed reduction to the af ne case makes this an easy exercise which we leave informally to the reader It then suf ces to show that any local map of local rings B a A factors uniquely through B as long as B is Noetherian and A is Artin Uniqueness is immediate from the injectivity of the map B a B and we deduce existence from the following observation the maximal ideal 1 2 m3 of B maps into mA and since A is Artin some power mfg 0 so it follows that the map B a A factors through B a Bmg We obtain the desired map B a A by composing with B a B mg It remains to show the second statement asserting in essence that is uniquely determined by the rst statement The point here is that QMml and Rml are both Artin local rings for all n and by hypothesis we are given compatible bijections MorOXm A 3 MorR A for every Artin local ring A In particular the canonical surjections Ox as and R as Rml induce maps as Rml and R as for all n One checks that the hypothesized compatibilities mean that these maps t together to give homomorphisms 5 a R and R gt 5 which are mutually inverse giving the desired assertion D The last statement of the proposition anticipates the concept of prorepresentability which we will begin to investigate in the next lecture One of the most basic but frequently important pieces of data captured by the complete local rings is the dimension of X at m More sharply one can think of the complete local ring as describing the singularity type77 of z in X or as capturing data similar to a neighborhood in the complex analytic topology Both these points of view are reinforced by the Cohen structure theorem of which a special case is Theorem 113 Let X be a locally Noetherian scheme over a eld k having dimension n at a smooth point m Then 9Xz kmm1 This says that the complete local rings at a smooth point of a scheme is determined simply by the dimension and residue eld which one can think of as analogous to the fact that complex manifolds are characterized by having analytic neighborhoods isomorphic to open subsets of C As one would naively picture from looking at small neighborhoods of a point passing to the complete local ring can also have effects like separating the branches of an irreducible curve at a node Example 114 Let C be the irreducible nodal curve yz zgizz over a eld k and P 00 Then 90 E kHs tHst the complete local ring at the origin of Spec Ms tlst Of course the complete local ring remembers more than topological information Example 115 Although the projection to the y axis of the cuspidal curve yz 3 is a homeomorphism the complete local rings at the origin are not isomorphic Indeed the complete local ring of the cusp is 7 m3 which has a 2 dimensional Zariski cotangent space mm2 generated by z and y In contrast the axis has complete local ring with 1 dimensional Zariski cotangent space generated by t 12 The functors of interest With this motivation we now discuss the functors we will consider For later convenience we x a eld k and a complete Noetherian ring A with residue eld k and work with the category ArtA k of local Artin A algebras with residue eld k You should think of this as working in a relative setting over Spec A Schlessinger works with the following functors although the terminology is not standard De nition 121 A predeformation functor is a covariant functor F ArtA k a Set such that is the one element set The tangent space TF of F is de ned to be The restriction on re ects the idea that we want to work locally at a xed k valued point For a general predeformation functor we have the unfortunate terminological situation that the tangent space is only a set However we ve seen in Corollary 112 of Martin s lectures that under mild hypotheses the tangent space is in fact a vector space over k Also it is important to keep in mind that the tangent space is a relative one over Spec A 3 From a moduli point of view the idea is to study deformations over Artin rings of a xed object de ned over k We ll see more precisely how these arise in several concrete examples below Remark 122 Some words on A Note that if we start with a scheme X over k and a k valued point m and we set A k then any map Spec A a X with image m makes A into a k algebra so the functor FXm we considered earlier is closely related to a predeformation functor differing only in that the category of Artin rings considered has fewer morphisms since we require our maps to be maps of k algebras One can check that in this setting if we instead took the predeformation functor we would still recover in the same sense as in Proposition 112 For many deformation problems anyone who is happy to work only over elds can set A to k so that we are considering Artin local k algebras with residue eld k Otherwise if k is perfect of characteristic p one typically takes A to be the Witt vectors of k thus allowing one to study deformations in mixed characteristic Indeed this is universal in this case every complete Noetherian local ring with residue eld k is canonically an algebra over the Witt vectors see Proposition 10 of H 5 of In a more involved setting one might be working with families over a given base and A is then frequently a complete ring of the base scheme or something along those lines 2 SOME EXAMPLES AND THE STATEMENT OF SCHLESSINGER S CRITERION We begin by giving some examples of predeformation functors We then describe two forms of representability and give the statement of Schlessinger s necessary and suf cient criterion for either one to hold At rst and thereafter the criterion may appear technical and somewhat lacking in intuition but in practice it is relatively straightforward to check given the appropriate tools 21 Examples We introduce some important examples of predeformation functors If we have a good moduli functor F Sch a Set and an element no 6 FSpec k we can obtain a well behaved predeformation functor F simply by setting FA 7 E ESpec A nlspeck no that is we consider families over Spec A which restrict to the chosen object no over Spec k The following is an example of this Example 211 Deformations of a closed subscheme Let XA be a scheme over Spec A and X its restriction to Speck Let Z Q X be a closed subscheme The predeformation functor DefzX ArtA k a Set is de ned by sending A to the set of closed subschemes YA Q XAlspecA which are at over A and restrict to Z over Spec k However this doesn t work well when the objects parametrizing the moduli functor have automorphisms 7 an early hint that for moduli of objects with automorphisms functors are not the most natural objects to work with In this case we rigidify somewhat as in the following example Example 212 Deformations of an abstract scheme Let X be a scheme over Speck The predeformation functor DefX ArtA k a Set is de ned by sending A to the set of isomorphism classes of pairs XAltp where XA is a scheme at over A and 4p X a XA is a morphism inducing an isomorphism X 3 XAlk Two pairs XA4p and Xf4ltp are considered to be isomorphic if there is an isomorphism XA 3 Xf4 commuting with 4p and Lp Remark 213 For the rst functors we don t need to consider pairs including a map although we are certainly considering YA as a closed subscheme and not an abstract scheme because there is a notion of equality for subschemes and not simply isomorphism However if we passed 4 naively from a global moduli functor of schemes to a predeformation functor in the second case we would obtain a slightly different functor from the functor Def X described above we would not remember the 4p as part of the data so we would send A simply to isomorphism classes of schemes at over A whose restriction to k is isomorphic to X This is subtly different from what we have de ned in that it is possible to have two pairs XAltp XI4 4p which are abstractly isomorphic as schemes but do not admit an isomorphism commuting with 4p and 4p for instance if X has an automorphism not extending to XA and X2 Thus we see that the functor we have de ned is more rigid and it will in general be better behaved See Theorem 818 of 2 for further discussion in the case of DefX An example of a somewhat different avor is the following Example 214 Deformations of a sheaf Let XA be a scheme over SpecA and X its restriction to Speck Let E be a quasicoherent sheaf on X The predeformation functor Defg ArtA k a Set is de ned by sending A to the set of isomorphism classes of pairs EA 4p with EA a quasicoherent sheaf on XAlspecA at over A and 4p EA a E a map of OXA SPSCA modules which induces an isomorphism EA A k 3 5 As with deformations of schemes two pairs are isomorphic if there is an isomorphism EA 3 54 commuting with 4p and 4p In all cases above functoriality is de ned in the obvious way by pullback Remark 215 We conclude with some vague comments on atness which is an opaque fre quently frustrating but extremely powerful condition Each of the above examples imposes a atness condition over A The conceptual reason for this is reasonably clear if we want to consider a family X over a base S we intuitively want to have two properties rst X should surject onto S in some strong sense so that the family is naturally over S and not some smaller subscheme and second the bers of X over S should vary continuously so that we can naturally consider them as a family parametrized by S Flatness accomplishes both these things in a relatively clean manner While it can be under stood in more concrete terms over for instance a discrete valuation ring it is simply equivalent to the condition that X has no associated points over the closed point in general it is much subtler particularly in the case of a non reduced base where it cannot be expressed so geo metrically Since we work extensively with non reduced bases in deformation theory we simply work the concept of atness into all our basic deformation problems without further comment We do however mention that over a local Artin ring atness is very concrete it is equivalent to freeness 22 Prorepresentability and hulls The strongest form of representability we will consider for predeformation functors is not representability in the strict sense but what is called prorep resentability The reason for this is that as we saw in the case of a predeformation functor obtained from maps to a scheme the representing object is typically not necessarily an Artin ring but rather a limit of Artin rings 7 or more speci cally a complete Noetherian local ring There are two equivalent de nitions of prorepresentability De nition 221 Given a covariant functor F ArtA k a Set let KrtA k be the category of complete Noetherian local A algebras with residue eld k and de ne the associated functor F ArtEAJs a Set by FR linn FRm We say that F is prorepresentable if and only if F is representable Note that our de nition follows Schlessinger Grothendieck used a different and somewhat more general de nition Exercise 222 Show that F is prorepresentable if and only if there exists a complete Noether ian local ring R and a collection nn 6 FRm of elements compatible with restriction such 5 that for every A E ArtA k and every 7 E FA there exists a unique map f7 R a A such that if f7 factors through Rm we have that 7 is the image of 7 Warning 223 When we apply the above situation to moduli problems it is frequently the case that for a complete ring R the set FR will not be the same as the set of families over Spec R It will rather correspond to a compatible collection of families over Spec Rml for all 71 but whether these actually come from a family over SpecR is a subtle topic called effectivization This will be addressed in the background lecture on the Grothendieck existence theorem and later on when we examine the issue of which deformations can always be effectivized As is the case with global moduli functors and representability it turns out that prorepre sentability is often too much to hope for The next best situation to hope for is that we have what Schlessinger calls a hull In order to de ne a hull we need to rst de ne the notion of formal smoothness for a morphism of functors De nition 224 Let F F ArtA k a Set be covariant functors with a morphism 4p F a F We say 4p is smooth if for every surjection A as B in ArtA k the map H FB XFB FA is surjective Notice that this is precisely the formal criterion for smoothness for morphisms of nite type between locally Noetherian schemes translated into the abstract setting of functors Notation 225 Let F ArtAk a Set be a covariant functor and F as above Given R a complete Npetherian local ring we denote by hR ArtA k a Set the functor of copoints of R and by 713 its restriction to ArtA De nition 226 Let F be a predeformation functor and F as above We say a pair R 7 with R a complete Noetherian local ring and 7 E FR is a hull for F if the induced map hR a F is smooth and induces a bijection TEE 3 TF on tangent spaces We follow the terminology used by Schlessinger Hulls are also sometimes called minimal versal deformation spaces or miniversal deformation spaces Note that prorepresentability of F is equivalent to the existence of a pair R 7 as above with hR a F an isomorphism Hulls can be thought of as a loose analogue of coarse moduli spaces By Yoneda s lemma a pair R 7 prorepresenting F is unique up to unique isomorphism It turns out that hulls are unique up to non unique isomorphism Proposition 227 Let Rm and R 7 be two hulls for a predeformation functor F Then there exists an isomomhz39sm u R gt R such that 71 The proof is left as an exercise 23 Schlessinger7s criterion We need one more bit of terminology in order to state Sch lessinger s criterion in its strongest form De nition 231 We say a surjective map f A as B in ArtA k is a small thickening if ker fmA 0 and ker f is principal or equivalently if ker f E k Note that this is called a small extension77 by Schlessinger and many others but not uniformly eg 3 uses small extension77 differently It is an easy exercise to check that every surjection in ArtA k can be factored as a series of small thickenings The statement of Schlessinger s criterion frequently refers to the following map which we obtain from the de nition of a functor whenever we are given A a A A a A in ArtA k and a covariant functor F 1 FA gtltA A H FA gtltFA FA Schlessinger s theorem is then the following criterion Theorem 232 Let F be a predeformation functor Then F has a hull if and only if the following conditions H1 H2 are satis ed and F is prorepresentable if and only if in addition is satis ed H1 The map 1 is surjective A gt A is a small thickening H2 The map 1 is bijective whenever A k and A H3 The tangent space TF is nite dimensional over k H4 The map 1 is bijective whenever A A and A gt A is a small thickening Remarks 233 Note that if H2 is satis ed then by Corollary 112 in Martin s lectures the tangent space in fact has a k vector space structure so H3 makes sense The conditions H1 H4 are stated as weakly as possible However they imply substantially stronger conditions For instance if F is prorepresentable it is an easy exercise to verify that 1 is always a bijection Since every surjection can be factored into small thickening H1 implies the same surjectivity condition for arbitrary surjections A a A Similarly if H2 is satis ed then by inducting on the dimension of a vector space V we conclude that the same bijectivity condition is satis ed for A kW with V any k vector space 3 AN EXAMPLE OF SCHLESSINGER S CRITERION We introduce a notion of deformation functor which is in practice always satis ed for natural deformation problems We then examine deformations of abstract schemes and show that the corresponding functors are deformation functors 31 Deformation functors and a rst example Generally speaking we nd that H1 and H2 are always satis ed for every reasonably natural deformation problem we can come up with while H3 tends to require properness conditions H4 is frequently not satis ed and can be best understood in terms of automorphisms of objects being parametrized However because H1 and H2 are so ubiqitous and also because if H2 is satis ed we have that TF is in fact a vector space we make the following de nition De nition 311 A deformation functor is a predeformation functor satisfying H1 and H2 Our rst example of a deformation functor will be DefX the functor of deformations of a scheme over h To give the sharpest statement we de ne De nition 312 Given XA4p E DefXA an automorphism of XA 4p or in nitesimal automorphism of XA is an automorphism of XA over A commuting with 4p Theorem 313 LetX be a scheme over k and DefX the predeformation functorparametrizing flat deformations of X Then i DefX satis es H1 and H2 so is a deformation functor ii DefX satis es ifX is proper over k iii DefX satis es if and only if for every pair XALp over A and A gt A a small thickening every automorphism of XA 1A LpA extends to an automorphism of XA 4p Note that properness is not required for H3 to hold for instance if X is smooth and af ne we saw that the tangent space of Def X is O dimensionall Schlessinger s criterion then implies the following Corollary 314 In the situation of the theorem ifX is proper then DefX has a hull and if further H0XHomQkk Ox 0 then DefX is prorepresentable If X is smooth then Hom9k OX is simply TX so we see for instance the following Example 315 Suppose X is a smooth proper curve of genus 9 Then DefX has a hull and if g 2 2 we have that DefX is prorepresentable In fact the restriction on genus is a bit of a red herring here as it is still possible to have prorepresentability even in the presence of in nitesimal automorphisms as long as the automor phisms always extend 32 Proof of the theorem To prove the theorem we will use some general results on at modules We remark that if we restrict our attention only to local Artin rings the proofs are simpler since every at module is free However for compatibility with future lectures we prove a more general result Lemma 321 Schlessinger 5 Lemma 34 Consider a commutatiue diagram p N a M B A A gtA of compatible ring and module homomorphisms where B A XA A N M XM M and M and M are flat ouer A and Aquot respectively Suppose i A a A with nilpotent kernel ii u induces M Ar A gt M and similarly for u Then N is flat ouer B and p induces N 83 A gt M and similarly for p Next suppose that with the aboue notation we also have a B module L and maps q L gt M and q L gt M such that q induces L 83 A gt M Then the map q X q L gt N is an isomorphism This is left as an exercise We apply the lemmas to prove the following more general proposition Proposition 322 Suppose we are giuen ring homomorphisms A gt A and A gt A where A gt A is surjectiue with nilpotent kernel Let B A XA A Then i Giuen schemes X and Xquot flat ouer A and A and an isomorphism 4p XlA gt XHlA there exists a scheme Y flat ouer B and morphisms Lp X gt Y and 4p X a inducing isomorphisms X gt Y A and X gt Y Au and such that 4p LpH llA 0 WM ii Giuen schemes Y1Y2 flat ouer B the natural map lsomY1 Y2 a lsomY1lAY2lA x150my1 Ay2 A lsomY1lAY2lAr is a bijection 8 Proof To check i we construct Y on the topological space of X If we identify the topological spaces of X and X lA and use 4p to identify both spaces with X lA we denote by i X lA a X the map of underlying topological spaces and we can set OyU OXU XOXA13971U OXi71U It follows immediately from the rst part of Lemma 321 that considered as modules over the base Artin rings we have 9y at over B and restricting to OX and OXu over A and A respectively as desired Finally to check that 9y de nes a scheme structure Y it suf ces to check that ber product commutes with localization which is an easy exercise For ii we see immediately from the second part of Lemma 321 that the given map is surjective on the level of modules and it is clear that a module isomorphism which is an algebra isomorphism after restriction to A and A is necessarily an algebra isomorphism since we now know that our modules over B are all isomorphic to ber products of modules over A and A This gives surjectivity of the map in question But by the same token since all our modules over B are isomorphic to ber products of the restrictions to A and A it is clear that homomorphisms in general and isomorphisms in particular are determined by their restrictions to A and A Proof of theorem It is easy to check that follows from the above proposition lndeed H1 is nearly a special case of from the proposition except that DefX is de ned with the additional data of the rigidifying map 4p X a XA However the fact that our morphisms in DefX are required to commute with 4p means that all the Ap s simply come along for the ride we get an induced map LpY X a Y induced by Lp X a X and 4p X a X which then restricts to Lp and 4p as required We next conclude H2 from ii of the proposition more generally obtaining injectivity of 1 whenever A k In this case the various maps 4p ensure that any isomorphisms we have will restrict to the same isomorphism over A so if we have two pairs over B which are isomorphic over A and A those isomorphisms necessarily agree on A so by the surjectivity in ii come from an isomorphism ofthe original pairs over B giving the necessary injectivity on isomorphism classes for We next prove iii again using the proposition 1 can fail to be injective only if we have Y1 471 and Y2 p2 over B which are isomorphic after restriction to A and A but not isomorphic Suppose we have isomorphisms 11 and 11 on A A by ii of the proposition the only problem comes if 11 and 1 cannot be chosen to give the same isomorphism after restriction to A But the restrictions must differ by an automorphism of Kl 4271114 and if every such automorphism extends to an automorphism of Y14p1 we can modify 11 so that the restrictions to A agree This shows that if every automorphism extends H4 is satis ed But conversely if there is an automorphism 1 over A which doesn t extend over A we can start with X X over A A and choose our isomorphism over A to be either 1 or the identity map and we will obtain non isomorphic schemes over B which are isomorphic over A and A This proves the rst assertion of iii Next we claim that if H0XHomQ k OX 0 then in fact all in nitesimal automor phisms are trivial It then follows see the remarks following Proposition 22 of Martin s lectures that all in nitesimal automorphisms over MI are trivial and more generally that when A as A with squarezero kernel that there is a unique automorphism of any object over A restricting to a given automorphism over A If we take any A E ArtA k writing the map A a k as a sequence of surjections with squarezero kernel we obtain the assertion that every pair over A has only trivial automorphisms Finally we note that ii is an immediate consequence of Proposition 26 of Martin s lectures when X is proper and smooth over k since the tangent space is given by H1X TX and TX 9 is locally free hence coherent More generally one can show that the tangent space is always nitedimensional when X is proper over k See 8 of Martin s lectures and in particular the coherence result in i as well as Theorem 84 ii for an approach using lllusie s cotangent complex See also Remarks 56 and 311 of 2 for an argument using a truncated form of the cotangent complex due to Lichtenbaum and Schlessinger 4 THE PROOF OF SOHLESSINGER S CRITERION Our aim today is to lay out most of the proof of Schlessinger s criterion leaving certain aspects of it to exercises The bulk of the proof is contained in proving the existence of a hull 41 Background results The following proposition sheds some further light on the structure of deformation functors and will come up in the proof of Schlessinger s criterion Proposition 411 Let F be a deformation functor and A gt A a small thickening with kernel I Then for every n E FA the set of if E FA restricting to 7 when non empty has a transitive action of TF k I This action commutes with morphisms of deformation functors F gt F Moreover the condition is equivalent to the condition that for all small thickenings and all n E FA lifting to A the above action is free In order to check the hull condition we will also need the following concept De nition 412 A surjection p A a A in ArtA k is essential if for any morphism q A a A such that p o q is surjective we have that q is surjective Lemma 413 pr A gt A is a small thickening then p is not essential if and only ifp has a section which is to say there exists a homomorphism s A gt A such thatp o s id Both of these results are left as exercises 42 The existence of a hull The following proposition is the hard direction of Schlessinger s theorem Proposition 421 Suppose that F is a predeformation functor satisfying H1 H2 and Then F has a hull Proof of proposition The proof breaks up into two parts constructing the hull and verifying that it is indeed a hull We rst carry out the construction Let n be the maximal ideal of A and if r dim TF set S At1 t and let m be the maximal ideal of S We will construct a hull R as a quotient of S by a certain ideal J which we construct inductively We set J2 m2 nS so that SJg kT and let R2 SJg Choosing an isomorphism SJg kTF is equivalent to choosing an identi cation of t1 t with a basis for TF which we now do The dual basis of t then give in particular an r tuple of elements of TF which by H2 used inductively to describe FkTF as the r fold product of TF with itself gives us a 52 E FR2 Following through the de nitions one checks that 52 induces the bijection TR2 3 TF dual to the isomorphism we xed by identifying the ti as a basis of T We then construct pairs Jifi inductively for i gt 2 with 5139 E FSJi Given Ji71 we choose Ji to be the minimal ideal among all J with mJFl Q J Q Ji71 and such that 71 can be lifted to some element of FSJ We want to see that the collection of such ideals is non empty and closed under intersection Non emptiness is trivial since JFl has the desired properties Similarly the rst condition is trivially closed under arbitrary intersection so it is enough to check the second For this we note that the ideals J correspond to vector subspaces of the nitedimensional vector space Ji71mJi71 so it is in fact enough to check that the condition is closed under nite and hence under pairwise intersections But suppose that J and 10 J satisfy the two conditions Again working in the vector space Ji71mJi71 we see that we can replace J with some larger J satisfying the same conditions with J O J J O J and further J J Ji71 Then we have SJ gtltk5 Ji71SvHg SJ O J so by H1 we nd we can lift 71 to SJ J and J O J J O J satis es our conditions as desired and we can de ne Ji as the minimal such ideal and choose 5139 to be any lift of 513971 to SJi We now take J to be the intersection of all the J1 and set R SJ Because mi Q J for all i we also have R liiniRJi and we can set E Note that our construction for i 2 then means that TR E TF so it remains to check that hR a F is smooth In order to check smoothness we suppose we have p A a A a small thickening and 77 E FA such that Fp7 7 E We also suppose we have u R a A such that 7 By the de nition of smoothness we need to show that there exists a morphism u R gt A lifting u and with Fu 7 We rst claim that there exists a u with 1901 u Since A is an Artin ring u factors through R for some i and although this map may not lift we show that it is possible to produce a lift to a map R a A recovering the given map R H A which then induces a lift R a A This may be rephrased equivalently as follows we need to nd a morphism v R a R XA A commuting with the projections to Bi Now p1 R XA A gt R is a small thickening because A a A is If p1 has a section we can de ne I simply using the section On the other hand if p1 does not have a section then p1 is essential If we choose any map S a A such that the composed map to A agrees with S a R a R H A we obtain a map w S a R XA A making the following square commutative S 4gt R x A A 171 Ri1 Ri Because p1 is essential we see that w is surjective and in order to produce the desired map I it is enough to show that kerw Q Ji11 But applying H1 to R XA A we have that E lifts to R XA A so by the de nition of Ji11 we nd Ji1 Q ker w as desired We have therefore proven that the desired 1 exists It remains to check that we can modify 1 without changing 1901 so that Fu 7 Thus we wish to modify 1 within Rp 1n and Fu will vary within Fp 1n We know from Proposition 411 that TF X I acts transitively and functorially on both Fp 1n and Rp 1n using the given isomorphism TF E TR for the latter so there is some 739 E TF X I sending Fu to 7 and by functoriality the same 739 sends u to some 1 with the desired properties This completes the proof that R E is smooth over F and hence a hull D 5 WRAPPING UP SCHLESSINGER S CRITERION We begin by completing the proof of Schlessinger s theorem and then discuss further exam ples 51 The rest of the proof It is now straightforward to complete the proof of Schlessinger s criterion Proof of Schlessmger s criterion We now know that H1 H3 imply that F has a hull Con versely suppose that RE is a hull for F Since TR E TF and is nite dimensional because R is Noetherian we see immediately that F satis es Now suppose we have 19 A a A and p A a A in ArtA k where p is a surjection and suppose we have objects 77 E FA and 7 E FA restricting to the same object 7 E Since we saw in the exercises that a smooth morphism is surjective there exists a u R a A such that u 77 and additionally by smoothness applied to p we have u R a A with u 7 and p o u p o u If 11 we set C u x u in FA XA A we then see that C projects to n and n which shows that H1 is satis ed Now we suppose that A k and A Me and want to show that the C we constructed is unique But suppose to E FA XA A also restricts to n and 7 Using the same 1 we can use smoothness applied to A xk Me a A to nd a v R a kH such that u x U C w But because TF E TR we have a u so to C as desired Now it remains to see that if in addition H4 is satis ed our hull in fact prorepresents F and that conversely if F is prorepresentable H4 is satis ed For the rst statement we suppose Rf is a hull for F and H4 is satis ed and we argue that hRA 3 FA for all A E ArtAk by induction on the length of A Let p A a A a small thickening with kernel I and suppose hRA 3 For each n E FA by Proposition 411 and H4 we have that hRp 1n and Fp 1n are compatibly torsors over TF X I the surjectivity of hRp 1n a Fp 1n then implies bijectivity proving the induction step and allowing us to conclude that R C prorepresents F Finally for the converse we note that by the fact that our ring ber products are ber products in the categories of rings we consider if F is prorepresentable then we have 1 bijective for all A a A A a A and in particular H4 is satis ed D 52 Deformations of quotient sheaves We have already checked H1 and H2 for de formations of schemes and in the exercises deformations of sheaves Of the three original predeformation functors we de ned that leaves only deformations of closed subschemes which we now address In fact we address a more general situation that of deformations of quotient sheaves Example 521 Deformations of a quotient sheaf Let XA be a scheme over Spec A with qua sicoherent sheaf EA Write X and E for the restrictions to Spec k Let E as f be a quasicoherent quotient sheaf The predeformation functor Defy ArtA k a Set is de ned by sending A to the set of quotient sheaves 4 of E which are at over A and restrict to f over Spec k Note that as in the case of closed subschemes there is a notion of equality for quotient sheaves since quotients correspond to kernel subsheaves Thus in this case we do not have to concern ourselves with isomorphisms In fact without any further hypotheses we have that H1 and H2 are satis ed by Defjffi Theorem 522 With notation as above Defgg is a deformation functor and indeed always satis es as well If further 5 is coherent over k and X is proper then is satis ed so Defgg is prorep resentable Setting EA OXA we immediately obtain the following corollary Corollary 523 Let XA be a scheme over A with restriction X to k and let Z Q X be a closed subscheme Then DefZX is a deformation functor and satis es If further X is proper over k then is satis ed so DefZX is prorepresentable Proof of theorem sketch The proof that Deff is a deformation functor proceeds in much the same way as the proofs for DefX and Defg lndeed given A a A A a A and quotients fAz fAu restricting to a given 4 on A one can de ne f3 on B A XA A as fA gtltA fAz and even though we need not have 6MB EAlA X5A A EMAz we nonetheless have an induced quotient map 6MB as f3 which one can check has the desired properties This proves H1 and H2 and H4 follow easily because our objects have no non trivial automorphisms so one can check directly that the above construction is an inverse to Finally to check H3 we mention that the tangent space is given by H0XHomgf where g ker a f This is left as an amusing exercise This space is nitedimensional when X is proper and Homf g is coherent which is the case under our hypotheses D Having reduced deformations of subschemes to deformations of quotient sheaves we go further with another reduction allowing us to easily treat another deformation problem Example 524 Deformations of a morphism Let XA and YA be schemes over Spec A Write X and Y for the restrictions to Spec k Let f X a Y be a morphism over k The predeformation functor Deff ArtA k a Set is de ned by sending A to the set ofmorphisms fA XAlA a YAlA over A Rather generally we can use graphs to reduce deformations of morphisms to deformations of subschemes We have Corollary 525 Suppose that XA and YA are locally of nite type over A and that XA is flat and YA is separated over A Then Deff is a deformation functor and satis es If further X and Y are proper over k is also satis ed so Deff is prorepresentable Proof We claim that this is a special case of deformations of closed subschemes of XA XA YA by considering the graph of 1 We note that the atness of XA implies the atness of the graph of a deformation of 1 while the separatedness of YA ensures we obtain a closed subscheme The main point to check is to check that conversely a at deformation of a graph is still a graph which is to say that under our niteness hypotheses if we have a morphism from a at scheme the condition that it is an isomorphism may be checked on bers This is Corollary 1795 of 8 Remark 526 In fact in the non proper case one can de ne corresponding deformation prob lems in which one imposes hypotheses of proper support and this makes it possible to prove prorepresentability for Deff in the case that only XA is proper Remark 527 We conclude with a remark that these prorepresentability results require fewer hypotheses than global representability results for the corresponding functors 7 the Quot scheme Hilbert scheme and Hom scheme lndeed Grothendieck was frustrated that the global functors were not always representable in the proper case but required projectivity hypotheses However the fact that the local functors are always prorepresentable under the suitable properness hypotheses points to a slight generalization of scheme the notion of algebraic space developed by D Knutson and M Artin Artin showed that while Hilbert schemes for proper schemes may not be represented by schemes they are at least represented by algebraic spaces and in practice this is often just as good as having representability by a scheme 6 DIMENSIONS OF HULLs A key technical result underyling Mori s bend and break argument for the existence of rational curves on Fano varieties involved an argument to use deformation theory to give a lower bound for the dimension of certain moduli spaces of maps This bound is given in terms of tangent and obstruction spaces 61 Obstruction theories and the statement We begin with some general comments on obstructions spaces for predeformation functors Artin s de nition of obstruction theory is some what complicated and not quite what we need so we give a simpler de nition De nition 611 A morphism A a A in ArtA k with kernel I is a thickening if ImA 0 so that I inherits a k vector space structure Note that this is a specialization to ArtA k of the ring maps in Martin s de nition of a deformation situation since the map to k is always surjective 13 De nition 612 Given a predeformation functor F an obstruction theory for F taking values in a k vector space V consists of the data for each thickening A a A with kernel I and each 7 E FA of an element 0b7 A E V 816 I such that i 0b7A 0 if and only if there exists 7 E FA with 7 1A 7 ii given an intermediate thickening A a B with kernel J Q I we have that 0b7 B is the image of 0b7 A under the natural map V k I a V 816 IJ Mori proved the following theorem in the special case of deformations of morphisms However the argument generalizes almost immediately Theorem 613 Let F be a deformation functor satis es H3 so that TF is nite dimensional and F has a hull R Suppose also that F has an obstruction theory taking ualues in a uector space V Then we haue dimA dikaF 7 dimk V g dim R g dim A dimk TF and if further A is regular and the rst inequality is an equality we have that R is a local complete intersection ring 62 The proof The version of this theorem in the usual sources only discusses the case that F is prorepresentable but the following lemma reduces the general case to this one Lemma 621 Suppose that F1F2 are functors ArtAk gt Set and we have a morphism 7r F1 gt F2 which is smooth and an obstruction theory for F2 taking ualues in V Then 7139 induces an obstruction theory for F1 taking ualues in V Proof Given 7 E F1A and a small extension A a A we can de ne the obstruction 0b to be simply obr7 The smoothness of 7139 then implies that 7 can be lifted to A if and only if 7r7 can be lifted to A so the main conditions for an obstruction theory are satis ed and it remains only to check functoriality which follows from functoriality of the obstruction theory given for F2 together with the required functoriality of 7139 Proof of theorem It is clear that the lemma reduces the theorem down to the case that F 713 since if R is a hull for F we have TF E TR and 713 is smooth over F We therefore suppose that in fact F is prorepresentable so that F 7113 In this case we work explicitly if we write at dim TF Schlessinger s construction of R is as a quotient of S At1 tdll by some ideal J so to prove the theorem it is enough to show that the number of generators of J is bounded above by dim V By the Artin Rees lemma J mg C Jms for some n We now set A At1 tdllmsJ mg and A Rml At1 tdllJ 1 mg so that we get a small extension OHIHA AHO with I J 1 mmRJ 1 mfg JmRJ From the natural map R a A we obtain an object EA 6 FA with an obstruction ob AA E V k I to extending EA to A We can write ob AA 2731 u ij where the u form a basis for V and the i are the images in I of elements z E J We then consider the ring B A m1 zdjmv this surjects onto A with kernel I and we again have an obstruction 010514 B to extending EA to B We note that again by the functoriality of obstructions we will necessarily have 0105A7 B 0 Thus EA may be lifted to B and because F 7113 this means we can lift the map R a A to a map R a B We wish to show that this implies lt2 J 2 w mm m2 14 which is equivalent to the stronger assertion that we have a lifting R a B which commutes with the natural quotient maps from At1 td to R and to B Now if we are given any lifting we have 5 Allt177tdll HR l l 4P l SAt1tdll HBHA and we can ll in the dashed arrow 4p to make the diagram commute by choosing appropriate values for 4pt i 1 d By hypothesis 4p commutes with the maps to A so must be the identity modulo J mg In particular we conclude that 4p induces the identity map on Insmg so is an isomorphism and in particular maps m5 bijectively to itself Since 4p factors the given map S a B we have 4p 1J C J m2 so that J C 4pJ Wong 4pJ m2 But by commutativity of the maps to R and B we see 4pJ Q mSJ mg and putting these together gives Since we had originally J O m2 Q mSJ we nally conclude that J is contained in hence equal to msJ By Nakayama s lemma we conclude that J is generated by mjj as desired D We have the following silly corollary which can of course be proved directly Corollary 622 UPC 6 KRUX k is smooth over A then R At1t7 for some 7 Remark 623 In his guest lecture Jason discussed a somewhat more informative proof of the above theorem The proof is too involved to give all the details but we recap the main points Given a ring R E EMA k let T dim TR and S At1 257 and consider R as a quotient of S with kernel ideal Let V HomRI12kHom5 lk where fl is the free module on S generated by the 125 and the map sends a functional dt gt gt 0 to f gt gt 21 ngici Then V is an s dimensional vector space where s is the minimal number of generators of 112 and the rst point is that ER has a canonical obstruction theory taking values in V The next point is that for any obstruction theory for ER taking values in some V there is a unique linear map V a V inducing the obstruction theory from the canonical one and moreover this map is linear Since V is s dimensional this implies that the number of generators for I is bounded from above by the dimension of any obstruction theory which recovers the theorem above 63 Examples One can often express tangent and obstruction space dimensions in terms of cohomology groups and this is a powerful tool for computing dimensions We mention two examples where the computation can be made particularly simply Example 631 The context that Mori applied his theorem was to spaces of morphisms from a curve to a variety For simplicity we assume that X Y are smooth over k with X a proper curve and f X a Y is given Then deformations of 1 have tangent space given by H0X fTy and an obstruction theory with values in H1 X fTy so we nd that to obtain a lower bound on the dimension at f of the space of morphisms X a Y we only need to compute the Euler characteristic xfTy which can be computed using Riemann Roch from the dimension Y and the degree on X of fTy Example 632 Another case in which one can often reduce the computation to Riemann Roch is for abstract deformations of a smooth proper surface X Here we know the tangent space is given by H1X TX and obstructions lie in H2X TX and we want to compute hl X TX7h2X TX to obtain a lower bound This is reduced to a Riemann Roch computation when we know H0X TX which is the dimension of the space of in nitesimal automorphisms 15 of X and in characteristic 0 is equal to the dimension of the automorphism group of X For instance if X has nitely many automorphisms which is the case in particular if it is of general type we have H0X TX O and then h1X TX 7 h2X TX ixTX 7 EFFECTIVIZATION AND ALGEBRAIZATION We have now concluded our discussion of functors of Artin rings but certain important questions remain Two such questions are the following Question Effectivity Suppose F is a deformation functor obtained from a global moduli problem and R a complete local Noetherian ring and we have an object 7 E When does 7 actually correspond to an object of the original moduli problem over Spec R Question Algebraization In the above situation when does an object over SpecR arise as the base change of an algebraic object 7 that is an object over a ring of nite type over the base Neither property is always satis ed although both are satis ed in many important cases At rst glance it may appear that effectivity is a minor technical issue and algebraization is more substantive However it turns out that effectivity is more frequently a sticking point We note that an important special case of both questions is when R n is actually a hull for F these questions then ask for the existence of a universal object over Spec R and over an algebraic subring respectively 71 Effectivity Grothendieck proved the rst major effectivity result the Grothendieck exis tence theorem for coherent sheaves which was discussed in the background lectures We recall a rough form of its statement Theorem 711 Let f X gt SpecA be aproper morphism with A a complete local Noetherian ring Let An Am and Xn X A An Suppose fn is a compatible collection of coherent sheaves on X Then there exists a coherent sheaff on X restricting to fn on each X One can apply Grothendieck s theorem immediately to moduli of coherent sheaves on a proper scheme concluding that formal objects ie objects of can always be effectivized How ever this does not extend to other important examples most notably moduli of schemes Example 712 A smooth projective surface S is a K3 surface if ws OS and H1S OS 0 It turns out that if we x a K3 surface S and consider deformations of S the hull is 20 dimensional but roughly speaking only 19 of these dimensions can be effectivized This is a consequence of the fact that the space of K3 surfaces considered as complex manifolds is 20 dimensional but the space of algebraic K3 surfaces is a countable union of 19 dimensional closed subspaces It is possible to deform formally in the direction of an analytic non algebraic surface but such a deformation cannot be effectivized Nonetheless all is not lost The typical patch is to consider moduli of polarized varieties 7 that is varieties together with a choice of ample line bundle This is similar to but weaker than restricting our consideration to subvarieties of projective space but it turns out to rigidify the problem suf ciently to allow effectivization to work As one might expect the proof involves a reduction to Grothendieck s existence theorem for coherent sheaves although we note that the necessity of working with morphisms of sheaves means that the proof requires a stronger statement of the existence theorem than the version given above asserting that one actually obtains an appropriate categorical equivalence 16 72 Algebraization Having more or less dealt with the question of effectivity we now turn to algebraization We should not expect any completely general results but Artin showed that if we restrict our attention to deformations 135 which induce smooth maps liR a F in which case we will say R E is smooth over F it is possible to prove general results under surprisingly mild hypotheses We need one preliminary de nition De nition 721 Let F SchS a Set be a contravariant functor We say F is locally of nite presentation over S if for all ltering projective systems of af ne schemes Z 6 Schg we have LIME FOLD ZA Note that the arrows here are in the opposite direction from the effectivity condition here we consider direct limits of rings The reason for this de nition is that it is shown in Proposition 8142 of 7 that if F hX for some X E Schs the above de nition is in fact equivalent to the usual de nition of the morphism X a S being locally of nite presentation Artin proved his algebraization theorem using his earlier results on approximations and we are therefore required to restrict our base scheme S to be locally of nite type over a eld or an excellent Dedekind domain Given F Schs a Set and no 6 FSpec k we denote by F7O the predeformation functor obtained by setting F7O A n E FA nlk no Artin s theorem is then the following Theorem 722 Suppose F Schg a Set is locally of nite presentation and no 6 FSpec k for a eld k where Speck gt S is giuen and of nite type having image 3 E S Let R be a complete local Noetherian USES algebra with residue eld k and E E FSpec R inducing no on k and smooth ouer FWD Then there is an S scheme X of nite type with a closed point m E X hauing residue eld k and an element n E FX such that there exists an isomorphism 5Xm gt R with n inducing En E FRm 1 for each it Note that effectivity is built into the hypotheses since we start with an element of FSpec R and not merely an inverse system of elements over the Artin quotients of R Also note that it is not necessarily the case that n induces E E FR unless as is often the case the element 5 is uniquely determined by the truncations En In this case we have the following uniqueness theorem which says that the formal deformation theory is determining X uniquely at least etale locally Theorem 723 Under the hypotheses of the above theorem suppose further that E is uniquely determined by the Then X an is unique up to etale base change in the sense that any two such triples are related by a common etale morphism These theorems are Theorems 16 and 17 of 1 respectively 8 GROUPOID PERSPECTIVE As we ve seen in Max s and Martin s lectures when working with moduli problems and related deformations it is frequently bene cial to work in the context of categories bered in groupoids We ve even seen this indirectly in the rst couple days when de ning some of the basic examples of predeformation functors For instance we noted that Def X can t be de ned formally from a global functors of at families of schemes since the functor doesn t keep track of the data of pullback morphisms On the other hand when working with categories bered in groupoids one can formally restrict from global to local moduli problems and still obtain good behavior However it turns out that thinking in terms of categories bered in groupoids can also shed additional light on Schlessinger s criterion and leads to other good behavior as well 17 81 Deformation stacks We rst observe that H1 and H2 can be subsumed by a single somewhat stronger condition in the groupoid setting7 which we make precise7 following some background de nitions De nition 811 Let C be a category A category 5 co bered in groupoids over C is a category 5 together with a functor usually omitted from the notation to C which makes 5 into a category bered in groupoids over Copp De nition 812 Let D be a category such that every morphism is an isomorphism We say D is the trivial groupoid if every pair of objects has a unique morphism between them We frequently refer to the trivial groupoid7 since any trivial groupoid is equivalent to the category with one object and only the identity morphism For categories co bered in groupoids over ArtA7 k7 the condition corresponding to H1 and H2 has gone by various names Rim called a category satisfying this condition a homoge neous groupoid 7 while Artin referred to the condition itself as 51 We propose the following terminology De nition 813 A category 5 co bered in groupoids over ArtA7 k is a deformation stack if 5k is the trivial groupoid7 and if for every pair of morphisms A a A7 A a A in ArtA7 k7 with A a A a surjection7 the following conditions are satis ed i for any pair of objects 711712 6 SAXAAH the natural map MorA XAA 71772 H MOFA WllA JIZlA gtltMorA71lA72lA MOFA 71lA 772lA is a bijection ii given a pair of objects 77 6 SA and n 6 SAN and an isomorphism 4p n lA gt 7 lA7 there exists an object C E SAXAAH restricting to n and n on A and A 7 and recovering 4p on A To any deformation stack 57 we have the associated functor to sets F5 obtained by sending A to the set of isomorphism classes of SA We observe the following Proposition 814 Let S be a deformation stack Then F5 is a deformation functor Proof That F5 is a predeformation functor follows from the hypothesis that 5k is the trivial groupoid We thus need to see that H1 and H2 follow from the additional conditions But H2 follows immediately from condition ii of a deformation stack7 while H1 and more generally7 that 1 is a bijection whenever A k then follows from condition i7 since any two objects have a unique isomorphism over k so any isomorphisms over A and A necessarily come from an isomorphism over A xk A 82 Ubiquity and utility of deformation stacks On a purely formal level7 it seems that the conditions for a deformation stack are stronger than a deformation functor However7 in practice it seems that any time one has an argument showing that a given functor satis es Schlessinger s H1 and H27 then the same argument will show that actually the deformation stack condition is satis ed as well In particular7 we see that this is the case for deformations of abstract schemes7 and deformations of quasicoherent sheaves on a scheme Proposition 322 immediately implies that DefX given the natural groupoid structure is a deformation stack7 and one can similarly deduce that Defg is a deformation stack from Lemma 321 In another direction7 in Lemma 144 of 4 Martin gives an argument showing that any deformation problem obtained from a point of an Artin stack will satisfy the conditions of a deformation stack We can therefore think of the deformation stack condition as explaining where H1 and H2 come from7 and why it is natural for these conditions to be satis ed Additionally7 Martin s lemma gives an indication of why it is natural to assume that A a A is surjective there is 18 some natural asymmetry arising from the use of a smooth cover of the stack by a scheme and the need to apply the formal criterion of smoothness on one side Indeed the deformation stack axiom is automatically satis ed for a deformation problem coming from a point of a scheme simply by applying the universal property of ber products In order to generalize to the case of Artin stacks one then wants to reduce to the case of schemes via the smooth cover and this is where the formal criterion for smoothness arises However deformation stacks have additional nice behavior building on some of the structure we have discussed for deformation functors For instance with a deformation functor F we had that for any thickening A a A with kernel I and any n E FA the vector space TF 816 I acts transitively on if E FA n lA 7 but we do not in general obtain a torsor However with deformation stacks we can describe a torsor and we can do similarly with automorphisms Proposition 821 Suppose S is a deformation stack and A gt A is a thickening with kernel I Then for any n 6 SA we have that n w i 7 E FA 7 WA 5 A is a pseudotorsor for T5 k I Similarly if we write 5 for the trivial deformation over He obtained by base change under h gt He then for any 7 6 SA and any 4p 6 Audi 14 we have that it 6 AutW 3 MA w is a pseudotorsor for AutC5 k I Here we can put a k vector space structure on Aut5 using the same construction that we used to put a vector space structure on T5 TF5 An amusing fact is that the additional law we obtain in this way agrees with the composition law on Aut5 which is given to us a priori in particular the deformation stack axioms automatically imply that this automorphism group is abelian As pointed out informally by Schlessinger and formalized by Rim we also obtain a geometric interpretation for H4 in the deformation stack context Proposition 822 Let S be a deformation stack Then F5 satis es if and only if for every svrjection A gt A in ArtA k and every 7 6 SA the natural map Aut7 gt Audi 14 is a svrjection Proof This is not hard to see and essentially the same as the proof of the same statement for DefX we sketch one direction If the map on automorphism groups is surjective and we have 711712 6 SAXAA which are isomorphic after restriction to the rst and second factors the two isomorphisms differ on A by an element of Autn1lA which by hypothesis can be lifted to an element of Autp2n1 If we modify the isomorphism on the second factor by this automorphism we obtain isomorphisms on both factors which agree on A so by the deformation stack axioms we obtain an isomorphism m a 712 on all of A XA A When the deformation stack is obtained from an object no of a global stack the proposition is saying that H4 is equivalent to the associated lsom stack being smooth at the identity section in Autn0 83 Deformation stacks as stacks We conclude with a brief remark on the analogy be tween deformation stacks and stacks in the usual sense From a descent theory perspective the ber product rings appearing in Schlessinger s criterion are slightly mysterious but the basic observation is the following

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