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On the Smoothness of the Quot Functor On the Smoothness of the Quot Functor S EBASTIAN W IEGANDT Master of Science Thesis Stockholm, Sweden 2013 On the Smoothness of the Quot Functor S EBASTIAN W IEGANDT Master’s Thesis in Mathematics (30 ECTS credits) Master Programme in Mathematics (120 credits) Royal Institute of Technology year 2013 Supervisor at KTH was Roy Skjelnes Examiner was Roy Skjelnes TRITA-MAT-E 2013:29 ISRN-KTH/MAT/E--13/29--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Abstract For a commutative ring k, we consider free k-modules E, endowing them with k[x1; : : : ; xm]- module structures through a ring homomorphism k[x1; : : : ; xm] ! EndZ(E). These struc- tures are then inspected by encoding the actions of the unknowns xi in matrices X1;:::;Xm. We further introduce the concepts of lifts and formal smoothness for functors, and define the n QuotF=A=k functor acting on the category of k-algebras, taking some k-algebra B to the set of quotients of the form (F ⊗k B)=N, which are locally free as B-modules. Lastly, we find con- 4 2 crete examples of modules showing that the functors Hilbk[x;y;z]=k and QuotL2 k[x;y]=k[x;y]=k are not formally smooth. Contents 1 Introduction 3 1.1 Background . .3 1.2 Overview ........................................4 1.3 Acknowledgments . .5 2 Preliminaries 6 2.1 Categories . .6 2.1.1 Yoneda's Lemma . .6 2.1.2 Representable Functors . .8 3 Free Quotients 10 3.1 Quotients in Rank One . 10 3.1.1 One Variable . 11 3.1.2 Several Variables . 12 3.2 Quotients in Rank Two . 13 3.2.1 One Variable . 14 3.2.2 Two Variables . 14 4 Lifts and Formal Smoothness 16 5 The Quot and Hilb Functors 19 5.1 The Quot Functor and Free Quotients . 19 5.2 Representing Simple Cases . 20 5.3 Smoothness of the Hilb Functor . 23 5.4 Smoothness of the Quot Functor . 27 References 31 1 INTRODUCTION 1 Introduction n The Hilbert scheme HA of n points on a k-algebra A parametrizes ideals I ⊆ A such that A=I is locally free of rank n as a k-module. These parametrizing objects are complicated, but one has that the tangent space of H at the point corresponding to I is given by Hom(I=I2; A=I). This fact has often been used to extract information about the Hilbert scheme. Fogarty, for instance, used this fact to show that the Hilbert scheme with A = k[x; y] is smooth [2]. There is then an open set that has dimension 2n, and Fogarty showed that any point has tangent space of dimension 2n, which implied smoothness. For A = k[x; y; z; w], the Hilbert scheme has an open set of dimension 4n, and for particular choices of I one can compute and find a tangent space of different dimension. This is for example the case with n = 8 and I = (x2; xy; y2; z2; zw; w2; xz − yw); which has tangent space of dimension 25 [3, p. 42]. So, here the Hilbert scheme is not even irreducible. In this work, we will take a different approach using matrix representations. Instead of starting with an ideal I ⊆ k[x1; : : : ; xm] such that the quotient ring is free of rank n as a k- module, we start with an arbitrary free k-module E. Then an k[x1; : : : ; xm]-module structure on E is given by pairwise commuting endomorphisms X1;:::;Xm. Thereafter we ask if these actions can be infinitesimally lifted, which is an equivalent way of describing smoothness of n the moduli space HA. By doing so, we find very small examples where the Hilbert and Quot functors, which are representable by the Hilbert and Quot schemes respectively ([5]), are not smooth. In fact, these are the smallest that can exist. 1.1 Background Given a free k-module E with basis fe1; : : : ; eng, we can endow E with a k[x]-module structure through a ring homomorphism k[x] ! EndZ(E), which is uniquely defined by the endomorphism X it maps the element x to. Consider for example the matrix 0 1 0 0 ··· 0 a0 B1 0 ··· 0 a1 C B C B0 1 ··· 0 a2 C X = B C ; B. .. C @. A 0 0 ··· 1 an−1 encoding the action of x on E. This in fact gives us E as the quotient n n−1 k[x]=(x − (a0 + a1x + ::: + an−1x )); by considering the action of x on f1; x; : : : ; xn−1g. In fact, by introducing m commuting matrices X1;:::;Xm, we can endow E with a k[x1; : : : ; xm]-module structure in the same way. Now, given we have a k[x1; : : : ; xm]-module structure on E, suppose we have a k[x1; : : : ; xm]- module homomorphism ': k[x1; : : : ; xm] E 3 1.2 Overview 1 INTRODUCTION onto E. By the first isomorphism theorem for modules, we then have that E = k[x1; : : : ; xm]= ker(') is a k[x1; : : : ; xm]-quotient of rank n as a free k-module. Generalizing this to arbitrary k-algebras n A, the functor QuotF=A=k is then defined as taking a k-algebra B to the set of F ⊗k B-module quotients F ⊗k B E which are locally free of rank n as B-modules, with F some fixed n A-module. In particular, if F = A we get the functor HilbA=k as a special case. The concept of formal smoothness for these functors arise naturally from the corresponding concept of formal smoothness for a ring homomorphism f : A ! B, which is in this sense an n infinitesimal lifting property. We will then say that the functor QuotF=A=k is smooth if it fulfills n a certain infinitesimal lifting property itself, namely that any element E in QuotF=A=k(B=N), n for some N ⊆ B with certain properties, admits a lift to some E in QuotF=A=k(B), that reduces to E modulo N. 2 Example 1.1. As a descriptive example of this formal smoothness criterion, let D2 = k["]=(" ) and consider the free D2-module 2 2 3 E = D2[x; y]=(x − "y; y − "x; xy) 2 Hilbk[x;y]=k(D2): 3 With D3 = k["]=(" ), this admits a lift to 2 2 2 3 E = D3[x; y]=(x − "y; y − "x; xy − " ) 2 Hilbk[x;y]=k(D3); and we can identify B = k["]=("3) and N = ("2) in the above discussion. We see that reducing E modulo "2 indeed gives us back E, and note that such a lift was bound to exist, since the Hilbert scheme on k[x; y] is smooth. The aim of this text is to expand on this construction of free module structures, with the ultimate goal of finding elements of certain Hilb and Quot functors that do not admit lifts in the preceding sense. With this in mind, we will develop additional ways of interpreting the smoothness condition given, primarily concerning ourselves with previously mentioned matrix representations of module structures. A brief aside will also be given on the representability of the Hilb and Quot functors in some simple cases. 1.2 Overview In Section 2, we introduce some category theoretic language, which we will use in the later stages of this paper. In particular, the concepts of categories and functors are introduced, followed by an aside on the Yoneda lemma and representable functors. In Section 3, we consider free k-modules E, and endow them with k[x1; : : : ; xm]-module structures through a ring homomorphism k[x1; : : : ; xm] ! EndZ(E), taking xi to an endomor- phism Xi : E ! E. Given a basis fe1; : : : ; eng for E, these can be considered as n × n matrices acting on E, encoding the actions of the unknown variables. For example the actions of x and y on k[x; y]=(x2; y2; xy) can be encoded in the matrices 00 0 01 00 0 01 X = @1 0 0A ;Y = @0 0 0A ; 0 0 0 1 0 0 4 1 INTRODUCTION 1.3 Acknowledgments by considering the actions of x and y on the elements 1, x and y. Further, given r elements e1; : : : ; er 2 E, we induce a k[x1; : : : ; xm]-module homomorphism r M ': k[x1; : : : ; xm] ! E by mapping (f1; : : : ; fr) to f1(X1;:::;Xm)e1 +:::+fr(X1;:::;Xm)er. If this mapping is surjec- Lr tive, we get E as a quotient ( k[x1; : : : ; xm])= ker('), which is free of rank n as a k-module. In Section 4, we introduce the concepts of lifts and formal smoothness, in this sense an infinitesimal lifting property concerning the ability of lifting certain B=N-modules for some k-algebra B and nilpotent N ⊆ B to the larger algebra B. Lastly, we will in Section 5 revisit many of the examples given in Section 3, translated into n a category theoretic language. We will introduce the Quot functor QuotF=A=k : k-alg ! Set for a commutative ring k, a k-algebra A and an A-module F. This functor maps a k-algebra algebra B to quotients of the form (F ⊗k B)=N, which are locally free as B-modules. We will inspect some specific cases, followed by a discussion on the formal smoothness of this functor. This discussion concludes with showing the non-smoothness of two specific cases of the Quot functor, using concrete examples of non-liftable elements.
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