· 2013. 9. 19. · created date: 1/13/2012 3:47:51 am

DEFORMATION OF QUOTIENTS ON A PRODUCT

W. D. GILLAM

Abstract. We consider the general problem of deforming a surjective map of modulesf : V → F over a coproduct sheaf of rings R = B ⊗A C when V = B ⊗A E is obtainedvia extension of scalars from a C module E. Assuming C and E are flat over A, we showthat the Atiyah class morphism F → LR/C ⊗L F [1] in the derived category D(R) factorsnaturally through (the shift of) a morphism β : Ker f → LR/C ⊗L F called the reducedAtiyah class. We describe the obstruction to lifting f over a (square zero) extensionB′ → B in terms of β and the class of the extension. As an application, we use thereduced Atiyah class to construct a perfect obstruction theory on the Quot scheme of avector bundle on a smooth curve (and more generally).

Introduction

Fix a base scheme S. Write × for the fibered product of S-schemes and LX for thecotangent complex LX/S of an S-scheme X. Let Y be an S-scheme, E a quasi-coherentsheaf on Y . Let X ↪→ X ′ be a square zero closed immersion of S-schemes with ideal Iand let

0 // N // π∗2E// F // 0(1)

be an exact sequence of sheaves on X × Y with F flat over X. Consider the problem offinding an exact sequence

0 // N ′ // π∗2E// F ′ // 0(2)

on X ′ × Y with F ′ flat over X ′ restricting to (1) on X × Y . Assuming QuotE is repre-sentable, this is equivalent to finding a map X ′ → QuotE lifting the map X → QuotEdetermined by (1). It is “well-known” that there is an obstruction

ω ∈ Ext1X×Y (N, π∗1I ⊗ F )

whose vanishing is necessary and sufficient for the existence of such a sequence and that,when ω = 0, the set of such sequences is a torsor under HomX×Y (N, π

∗1I ⊗ F ). A simple

proof is provided for the reader’s convenience in (1.4).

Following a trick of Nagata, Illusie (IV.3.2)1 constructs the obstruction ω by translatingthis lifting problem into a problem of G := Gm equivariant algebra extensions, which canbe handled using the machinery of the G equivariant cotangent complex. The basic linkbetween the G equivariant cotangent complex and the usual cotangent complex is provided

Date: November 11, 2010.2000 Mathematics Subject Classification. 13D10, 14D20.Key words and phrases. Deformation theory, Atiyah class, Quot scheme.1For a roman numeral N , a reference in the present paper of the form (N.a) always refers to section a

of chapter N in Illusie’s book [Ill].

1

2 W. D. GILLAM

by the equivalence of the two different descriptions of the Atiyah class in (IV.2.3.7.3). Thisdiscussion motivates the main results of this paper, which we now briefly summarize.

For a scheme X and a quasi-coherent sheaf F on X, let X[F ] := SpecX OX ⊕F denotethe trivial square zero extension of X by F . The scheme X[F ] has a natural G = Gmaction obtained by restricting the action of scalars on F to the invertible scalars. (X,F ) 7→X[F ] is a contravariant functor from the category (stack) of quasi-coherent sheaves to thecategory of schemes with G action.

Assume Y is flat over S, so that π1 : X × Y → X is also flat and LX×Y/Y = π∗1LX .From (1) we obtain a commutative diagram

(X × Y )[F ] // (X × Y )[π∗2E] // Y [E]

(X × Y )[F ] // X × Y //

OO

Y

OO(3)

of S-schemes with G action. From the corresponding morphism of distinguished trianglesof G equivariant cotangent complexes, we obtain a commutative diagram

LG(X×Y )[F ]/(X×Y )[π∗2E]// LG(X×Y )[π∗2E]/Y [E] ⊗

LOX×Y [π∗2E]

OX×Y [F ][1]

LG(X×Y )[F ]/(X×Y )

OO

// π∗1LX ⊗LOX×Y OX×Y [F ][1]

OO(4)

in the G equivariant derived category DG((X × Y )[F ]). Pushing forward to X × Y andtaking the weight one subcomplex is an exact functor

k1 : DG((X × Y )[F ]) → D(X × Y ).

Applying k1 to (4) we obtain a commutative diagram in D(X × Y ) that can be written

N [1]β[1] // π∗1LX ⊗L F [1]

F

OO

at(F ) // π∗1LX ⊗L F [1].

(5)

The bottom horizontal arrow is the Atiyah class of F relative to π2 and the D(X × Y )morphism β : N → π∗1LX ⊗L F whose shift appears on the top line is called the reducedAtiyah class of the quotient π∗2E → F . It is studied further in Section 2 where we give adescription of the reduced Atiyah class in terms of principal parts.

Consider the functors

HomD(X)(LX , ), HomD(X×Y )(N, π∗1 ⊗L F ) : D(X) ⇒ Mod(Γ(X,OX))and the (Γ(X,OX) linear) natural transformation

β : HomD(X)(LX , ) → HomD(X×Y )(N,π∗1 ⊗L F )β(M)(g) := (π∗1g ⊗L F )β.

Suppose that the Quot scheme Q = QuotE is representable, X is an open subset of it,and (1) is the (restriction of the) universal sequence. In Theorem 4.2, we show that, for

DEFORMATION OF QUOTIENTS ON A PRODUCT 3

any quasi-coherent sheaf I on X (regarded as a complex I ∈ D(X) supported in degreezero), the map

β(I) : Hom(LX , I) → Hom(N, π∗1I ⊗L F )

is an isomorphism and the map

β(I[1]) : Ext1(LX , I) → Ext1(N,π∗1I ⊗L F )

is injective.

Under appropriate technical hypotheses on Y and F , we show in Theorem 4.6, usingSerre duality, that the functor HomD(X×Y )(N, π

∗1 ⊗L F ) is represented by

E := RH om(Rπ1∗RH om(N,F ),OX) ∈ D(X)

and that the D(X) morphism E → LX corresponding to β is an isomorphism on H0 andsurjective on H−1. If we assume furthermore that N and E are vector bundles and Y/Sis a projective, Gorenstein curve, then we show in (4.8) that E is of perfect amplitude⊆ [−1, 0] and hence E → LX is a perfect obstruction theory (POT) in the sense of [BF].

Similar results have appeared in [CFK2] and [MO], but neither of these referencesexplains how to produce the D(X) morphism E → LX . In any case, the methods usedhere are completely different and should serve to clarify the construction of this POT andits variants, which are used in various places in the literature ([G2], [MO], [MOP], . . . ),as discussed in the examples at the end of the paper.

Acknowledgements. The general idea of the reduced Atiyah class was suggested to meby Johan de Jong. It can be argued that this is already implicit in (IV.3.2). I owe a debtof gratitude to Luc Illusie since most of the results presented here are simple applicationsof the theory developed in his book [Ill].

1. Deformation of quotients

We begin in (1.1)-(1.7) by “recalling” some general results on the deformation theoryof quotients. In (1.8) we specialize this to the situation encountered in the study of Quotschemes. In the remaining subsections we introduce the reduced Atiyah class and explainhow it is related to deformation of quotients.

We work in a fixed topos T , as in [Ill, IV]—in all applications T will be the toposof sheaves on a topological space. We write “ring” (etc.) instead of “sheaf of rings” or“ring object of T”. Beginning in (1.6), we will often work with graded rings, modules,etc. These are always graded by N. For a graded ring A and i ∈ N, we write ki for the(exact!) functor from graded A modules (or complexes thereof) to k0A modules given byki(⊕nMn) := Mi.

1.1. Problem. Consider a surjection of rings R′ → R with square zero kernel J and anR′ module V ′. Let V = V ′/JV ′ so we have a surjection p : V ′ → V of R′ modules withkernel JV ′. Here an throughout, we identify R modules with R′ modules annihilated byJ , so V and JV ′ are R modules. We have a natural surjective map of R modules

i : J ⊗R V → JV ′ = Ker p(1.1.1)j ⊗ v 7→ jv′

4 W. D. GILLAM

because jv′ ∈ V ′ is independent of the choice of v′ ∈ V ′ with p(v′) = v since J2 = 0. Weassume that i is an isomorphism so we have an exact sequence

V = ( 0 // J ⊗R Vi // V ′

p // V // 0 )(1.1.2)

of R′ modules.

We will consider a fixed short exact sequence of R modules

0 // N // Vf // F // 0(1.1.3)

throughout the paper. We also fix an R module K and a morphism u : J ⊗R F → K ofR modules. Consider the problem of finding an R′ module extension F of F by K and amorphism of extensions f : V → F of the form:

0 // J ⊗R V //

u(J⊗f)��

V ′

f ′

��

// V //

f��

0

0 // K // F ′ // F // 0

(1.1.4)

We will refer to such an f (or the map f ′ : V ′ → F ′) as a solution to this problem.2

The basic result concerning this problem is the following “classical” theorem, which wewill prove in (1.4) and again in (1.6).

1.2. Theorem. There is an obstruction ω ∈ Ext1R(N,K) whose vanishing is necessaryand sufficient for the existence of a solution to (1.1). When ω = 0, the set of solutions to(1.1) is a torsor under HomR(N,K).

1.3. Remark. An extreme case of the problem (1.1) occurs when u = 0 (equivalentlyu(J ⊗ f) = 0). In this case f always factors through F ′ → F and we see that a solutionto the problem is the same thing as a morphism of extensions

0 // N //

��

V

��

// F // 0

0 // K // F ′ // F // 0.

(1.3.1)

Of course, every such morphism is obtained by pushing out the top row along a uniquemorphism N → K so in this case it is clear that the obstruction vanishes and, in fact, wehave a natural trivialization of the torsor under Hom(N,K) because we have a naturalbase point obtained from the split extension.

1.4. Elementary construction. The obstruction ω ∈ Ext1R(N,K) can be constructedelementarily as follows. Let L be the kernel of the composition V ′ → V → F . We have a

2Of course, solutions are considered up to the evident notion of isomorphism.


commutative diagram

0

��

0

��

0

��0 // J ⊗R V

i // L

��

p // N

��

// 0

0 // J ⊗R Vi //

��

V ′p //

��

V //

f��

0

0 // F

��

F

��

// 0

0 0

(1.4.1)

with exact columns and rows. By pushing out the top row along u(J ⊗ f) we obtain amorphism of extensions

0 // J ⊗R Vu(J⊗f)

��

// L //

��

N // 0

0 // K // M // N // 0

(1.4.2)

where M := K ⊕J⊗V L. Note that M is an R module (i.e. JM = 0):

[jk, jl] = [0, jl] = [0, i(j ⊗ p(l))] = [u(j ⊗ f(p(l))), 0] = [u(0), 0] = 0,

so the bottom row defines a R module extension. Let ω ∈ Ext1R(N,K) be the class of thisextension. I claim that the vanishing of ω is necessary and sufficient for the existence ofa solution to (1.1), and that, in fact, splittings of the bottom row in (1.4.2) (which are apseudo-torsor under HomR(N,K)) correspond bijectively to solutions of (1.1).

Given a solution f : V → F to (1.1) we obtain a commutative diagram

0 // J ⊗R V // N ′

��

// N //

��

0

0 // J ⊗R V //

u(J⊗f)��

// V ′ //

f ′

��

V

f��

// 0

0 // K // F ′ // F // 0

(1.4.3)

with exact rows by pulling back the extension in the middle row along N → V . Notethat N ′ → V ′ is monic and N ′ ⊂ L ⊂ V ′, but N ′ need not be contained in the kernelof f ′. Given n ∈ N , choose a (local) lift n′ ∈ N ′. I claim that n 7→ [−f ′(n′), n′] ∈ Mis well-defined (independent of the choice of lift n′ ∈ N ′), in which case it will clearlyprovide a splitting of the bottom row in (1.4.2). Indeed, if n′′ is another (local) lift, thenn′′ = n′ + ϵ for some ϵ ∈ J ⊗R V and we have

[−f ′(n′ + ϵ), n′ + ϵ] = [−f ′(n′)− u(J ⊗ f)(ϵ), n′ + ϵ] = [−f ′(n′), n′] ∈ M.

6 W. D. GILLAM

Conversely, notice that a splitting of the bottom row in (1.4.2) is the same thing as amap s : L → K making the diagram

J ⊗R Vu(J⊗f)

��

// L

s{{wwwwwwwwww

K

(1.4.4)

commute. By pushing out the sequence defining L along s we obtain a commutativediagram

0 // J ⊗R V //

��

V ′ // V

f��

// 0

0 // L

s

��

// V ′

��

// F // 0

0 // K // F ′ // F // 0

(1.4.5)

with exact rows, where we have set F ′ := K ⊕L V ′. The map from the top row to thebottom row is evidently a solution to (1.1).

It is straightforward to check that the two constructions are inverse.

1.5. Remark. Assume u is surjective, hence u(J ⊗ f) is also surjective. Let S be itskernel. Given any solution f : V → F to (1.1), set N ′ := Ker f ′. We have a commutativediagram

0

��

0

��

0

��0 // S

��

// N ′

��

// N

��

// 0

0 // J ⊗R V

u(J⊗f)��

// V ′

f ′

��

// V

f��

// 0

0 // K //

��

F ′ //

��

F //

��

0

0 0 0

(1.5.1)

with exact rows and columns. Suppose f ′0 : V′ → F ′0 is some fixed solution to (1.1). Then

by following through the above discussion we see that the bijection between solutions to(1.1) and HomR(N,K) (using f

′0 as a basepoint to trivialize the torsor) is realized by

(f : V → F ) 7→ [S // N ′]

��[F ′0

// F ].

(1.5.2)

Here, the top row on the right is a complex quasi-isomorphic to N and the bottom rowon the right is a complex quasi-isomorphic to K. The map N ′ → F ′0 is the composition of


the inclusion N ′ → V ′ and f ′0 : V ′ → F ′0. The right side of (1.5.2) is a map of complexesbecause the composition S → F ′0 is zero, as one sees by factoring S → N ′ → V ′ the otherway in (1.5.1).

1.6. Illusie’s construction. Illusie constructs the obstruction ω, following a trick ofNagata, by translating the problem (1.1) into a problem of graded algebra extensions andthen using the machinery of the graded cotangent complex. Let

R′[V ′] := Sym∗R′ V′/Sym>1R′ V

′

be the algebra of dual numbers on V ′ over R′, graded as usual so that V ′ is the degree onecomponent; define graded algebras R[V ] and R[F ] similarly. There is an obvious diagram

R′[V ′] // R[V ] // R[F ](1.6.1)

of graded rings. Regard K as a graded R[F ] module supported in degree one (henceannihilated by F ). A solution to (1.1) is the same thing as an extension of graded algebras

0 // K // G // R[F ] // 0

R′[V ′]

bbDDDDDDDDD

OO(1.6.2)

where the induced map J ⊗R V → V ′ → K is u(J ⊗ f). Indeed, the bijection is givenby taking the degree one part of G; (1.6.2) is uninteresting in all other degrees since K issupported in degree one, so we can always write G = R[F ′] for some R′ module F ′.

Graded algebra extensions (1.6.2) form a graded R[F ] module ExalgrR′[V ′](R[F ],K). By

the fundamental theorem of the graded cotangent complex (IV.2.4.2) and (IV.1.2.2.1)there are isomorphisms

ExalgrR′[V ′](R[F ],K) = Ext1R[F ](L

grR[F ]/R′[V ′],K)gr(1.6.3)

= Ext1R(k1LgrR[F ]/R′[V ′],K).

From (IV.2.2.5) and the triangle associated to R → R[V ] → R[F ] we obtain a naturalisomorphism

k1LgrR[F ]/R[V ] = N [1].(1.6.4)

Using (1.6.4), the degree one part of the distinguished triangle of cotangent complexesassociated to (1.6.1) is a triangle in D(R) that can be written

k1(LgrR[V ]/R′[V ′] ⊗LR[V ] R[F ])

// k1LgrR[F ]/R′[V ′] // N [1].(1.6.5)

Since R′[V ′] → R[V ] is a square zero extension with kernel J ⊕ (J ⊗R V ) (in gradings 0, 1)we have

τ≥−1LgrR[V ]/R′[V ′] = (J ⊕ (J ⊗R V ))[1]

and hence

τ≥−1LgrR[V ]/R′[V ′] ⊗LR[V ] R[F ] = (J ⊕ (J ⊗R F ))[1](1.6.6)

8 W. D. GILLAM

(in gradings 0, 1). Applying Hom( ,K) to (1.6.5) and using (1.6.3) and (1.6.6) we obtainan exact sequence

0 // Hom(N,K) // ExalgrR′[V ′](R[F ],K)// Hom(J ⊗R F,K)

δ // Ext1(N,K)

(1.6.7)

of R modules. A solution to (1.1) is an element of ExalgrR′[V ′](R[F ],K) whose image in

Hom(J ⊗R F,K) is u. Evidently such a solution exists if and only if ω := δu vanishes, inwhich case solutions are a torsor under Hom(N,K).

As a morphism in D(R), ω is the composition

N // k1(LgrR[V ]/R′[V ′] ⊗LR[V ] R[F ])

// J ⊗R F [1]u[1] // K[1],(1.6.8)

where the first map is the one from the triangle (1.6.5) and the second is the truncation.

1.7. Comparison. As the notation suggests, the obstruction ω constructed in (1.4) co-incides with the one constructed in (1.6). To see this, first recall that we have naturalisomorphisms

k1LgrR[F ]/R[V ] = τ≥−1k1LgrR[F ]/R[V ] = N [1],

so the morphism N → J ⊗R F [1] constructed in (1.6) using the degree one part of thetransitivity triangle associated to (1.6.1) coincides with the morphism

τ≥−1k1LgrR[F ]/R[N ][−1] → τ≥−1k

1(LgrR[V ]/R′[V ′] ⊗LR[V ] R[F ]) = J ⊗R F [1]

obtained from the degree one part of the truncated transitivity triangle

τ≥−1LgrR[V ]/R′[V ′] ⊗LR[V ] R[F ]

// τ≥−1LgrR[F ]/R′[V ′] // τ≥−1LgrR[F ]/R[V ].(1.7.1)

The graded algebra maps appearing in (1.6.1) are all surjective, with kernels as indicatedbelow.

R[V ]0⊕N // R[F ]

R′[V ′]

J⊕(J⊗RV )

OO

J⊕L

;;wwwwwwwww

In each case, the direct summand decomposition corresponds to the splitting into thedegree 0, 1 parts, respectively.

Using (1.6.6) and standard facts about the truncated cotangent complex (III.1.3), thetriangle (1.7.1) is naturally identified, after applying [−1], with (the triangle associatedto) the short exact sequence

0 // J ⊕ (J ⊗R F ) //J ⊕ L

(J ⊕ L)2// 0⊕N // 0(1.7.2)

of graded R[F ] modules. Note that J ⊕ (J ⊗R V ) ⊂ R′[V ′] and 0⊕N ⊆ R[V ] are squarezero ideals, while

(J ⊕ L)2 = 0⊕ JL ⊆ J ⊕ L.


The degree one part of the exact sequence (1.7.2) is therefore

0 // J ⊗R F // L/JL // N // 0.(1.7.3)

The extension (1.7.3) is part of the commutative diagram:

0 // J ⊗R V

J⊗f��

i // L

��

p // N // 0

0 // J ⊗R F // L/JL // N // 0

(1.7.4)

As in any map of extensions with isomorphic cokernels, the bottom left square in (1.7.4)is a pushout. Therefore, when we push (1.7.3) out along u : J ⊗R F → K to get theobstruction ω of (1.6), the resulting extension of N by K is the same as the one in thebottom row of (1.4.2) used to define the obstruction ω of (1.4).

1.8. Coproduct case. We now specialize the lifting problem of (1.1) to the sort of liftingproblem encountered in the study of Quot schemes, as in the introduction. The situation isslightly complicated by the fact that the structure sheaf of a fibered product of schemes isnot exactly the tensor product of the structure sheaves of the factors, as will be discussedin (1.10).

Let A be a ring, let B,C be A-algebras, let H := B⊗AC, let S ⊆ H be a multiplicativesubsheaf, and let R = S−1H be the localization of H at S, so H → R is flat and LR/H = 0(II.2.3.2). Let

B = 0 // I // B′ // B // 0

A

``AAAAAAAA

OO(1.8.1)

be an A-algebra extension of B by a B module I.

Let I ′ := I ⊗A C, H ′ := B′ ⊗A C, and assume that

T orA1 (B,C) = 0(1.8.2)

so we have a C-algebra extension

H = 0 // I ′ // H ′ // H // 0

C

``BBBBBBBB

OO(1.8.3)

of H by I ′ obtained from (1.8.1) by applying ⊗A C. Let S′ ⊆ H ′ be the preimage ofS ⊆ H under H ′ → H, let R′ = (S′)−1H ′, and let J = I ′ ⊗H R = S−1I ′. Note that

J ⊗R M = (I ⊗A C)⊗H R⊗R M(1.8.4)= (I ⊗A C)⊗B⊗AC M= I ⊗B M

10 W. D. GILLAM

for every R module M . We have a map H → R of C-algebra extensions lifting H → R asbelow:

0 // I ′ //

��

H ′ //

��

H //

��

0

0 // J // R′ // R // 0

(1.8.5)

Note that H ′ → R′ is flat.Now let E be a C module, let W := B ⊗A E, and let

V := R⊗C E = W ⊗H R = S−1W

be the R module obtained from E by extension of scalars along C → R. Let W ′ = B′⊗AEand assume that

T orA1 (B,E) = 0(1.8.6)

so we have an exact sequence

W = ( 0 // I ′ ⊗H W // W ′ // W // 0 )(1.8.7)

of H ′ modules obtained from (1.8.1) by taking ⊗A E. Extension of scalars of W alongthe flat ring map H ′ → R′ (localization at S′) yields an exact sequence

V = ( 0 // J ⊗R V // V ′ // V // 0 )(1.8.8)

of R′ modules, where V ′ = W ′⊗H′R′ = (S′)−1W ′ and where V and J⊗RV are R modules.

We can now consider the same problem as in (1.1) for the square zero surjection R′ → R,the extension V in (1.8.8), and an exact sequence of R modules (1.1.3) (a quotient f ofV = R⊗C E).

1.9. Flat deformations. Assume F is flat over B. Then by the flatness criterion, F ′ willbe flat over B′ iff the natural map u(F ) : J ⊗R F → K (IV.3.1) is an isomorphism (notethat J ⊗R F = I ⊗B F by (1.8.4)). Using this isomorphism to make the identificationJ ⊗R F = K, we see that the problem of lifting f : V → F to a surjection f ′ : V ′ → F ′with F ′ flat over B′ is equivalent to the special case of (1.1) where K = J ⊗R F and u isthe identity. In this case, we refer to a solution of (1.1) as a flat deformation of f over B(or “over B′”).

1.10. Dictionary. The setup of (1.8) is related to the situation in the introduction asfollows. We work in the topos of sheaves on X × Y (equivalently, sheaves on X ′ × Y ),and we set A := OS , B := OX , B′ := OX′ , C := OY , R := OX×Y (dropping all notationfor inverse images). The map H → R (resp. H ′ → R′) may be viewed as the comparisonmap between the ringed space fibered product and the locally ringed space (equivalently,scheme-theoretic) fibered product X × Y (resp. X ′ × Y ). The fact that these maps arelocalizations is well-known and amply discussed in [G1, 3.2]. In this situation, R′ = OX′×Yand the E from the introduction is the same as the E from (1.8) (suppressing inverseimages along X × Y → Y ) and V , V ′ in (1.8) are both denoted π∗2E in the introduction.The sequence (1.1.3) is of course the same as the sequence (1) from the introduction.The lifting problem from the introduction is the special case of Problem 1.1 discussed inRemark 1.9 above.


1.11. Assumptions. We will assume from now on that C and E are flat over A. Theseassumptions clearly imply the assumptions (1.8.2) and (1.8.6) made in (1.8). Since A → Cis flat, B → H is also flat (hence so is B → R because H → R is flat) and

LB/A ⊗B H = LH/C(1.11.1)

by base change properties of the cotangent complex (II.2.2). From this, the transitivitytriangle for C → H → R, and the vanishing LR/H = 0, we find

LB/A ⊗B R = LR/C .(1.11.2)

Furthermore, using flatness of A → C and H → R, we have

T orCi (R,E) = T orCi (H,E)⊗H R(1.11.3)

= T orCi (B ⊗A C,E)⊗H R= T orAi (C,E)⊗H R= 0 when i > 0

because we assume E is flat over A.

1.12. Reduced Atiyah class. By taking the degree one part k1 of the map of cotangentcomplex transitivity triangles associated to

R′[V ′] // R[V ] // R[F ]

C[E]

OO

// R[V ] // R[F ]

(1.12.1)

we obtain a commutative diagram

N // k1(LgrR[V ]/R′[V ′] ⊗LR[V ] R[F ])

N // k1(LgrR[V ]/C[E] ⊗LR[V ] R[F ])

OO(1.12.2)

in D(R).

From the commutative diagram of graded rings

R′

$$III

IIII

III

��

C

::uuuuuuuuuu //

��

R

��

R′[V ′]

##HHH

HHHH

HH

C[E] //

;;vvvvvvvvvR[V ]

(1.12.3)

12 W. D. GILLAM

and the naturality of the graded cotangent complex, we obtain a commutative diagram

LR/R′ ⊗LR R[V ] // LgrR[V ]/R′[V ′]

LR/C ⊗LR R[V ]

OO

// LgrR[V ]/C[E]

OO(1.12.4)

in D(R[V ])gr. Since the front square in (1.12.3) is cocartesian,

R[V ] = R[R⊗C E] = R⊗C C[E],and we have the tor-independence (1.11.3), it follows from base change properties of thecotangent complex (II.2.2) that the bottom arrow in (1.12.4) is an isomorphism. Byapplying ⊗LR[V ]R[F ] to (1.12.4) and taking k

1, we obtain a commutative diagram

LR/R′ ⊗LR F // k1(LgrR[V ]/R′[V ′] ⊗

LR[V ] R[F ])

LR/C ⊗LR F

OO

∼= // k1(LgrR[V ]/C[E] ⊗LR[V ] R[F ])

OO(1.12.5)

in D(R), where the bottom arrow is an isomorphism as indicated.

From the maps of cotangent complex transitivity triangles associated to the diagram

A // B′

��

// B

��A // R′ // R

A // C //

OO

R

(1.12.6)


LB/A ⊗B R //

��

LB/B′ ⊗B R //

��

I ⊗B R[1]

(1.8.4)

LR/A //

''OOOOO

OOOOOO

OOLR/R′ // J [1]

LR/C

OO

(1.12.7)

in D(R). The two right horizontal arrows are the truncations τ≥−1 of LB/B′ ⊗B R andLR/R′ , computed using the fact that B′ → B is surjective with kernel I and R′ → R issurjective with kernel J . The composition given by the top row is κ(B)⊗B R, where

κ(B) ∈ Ext1B(LB/A, I) = HomD(B)(LB/A, I[1])

is the Kodaira-Spencer class of B (the image of B under the isomorphism

ExalA(B, I) = Ext1(LB/A, I)

provided by the Fundamental Theorem (III.1.2.3)).


Using the isomorphism given by the bottom map of (1.12.5), we view the bottom mapof (1.12.2) as a morphism

β(f) : N → LR/C ⊗LR F,(1.12.8)

which we will call the reduced Atiyah class of the quotient f . It will be further studied inSection 2. Recall from (1.11.2) that our assumptions (1.11) imply that the natural mapLB/A ⊗B R → LR/C (which appears as a composition in (1.12.7)) is also an isomorphism,thus we can also view β(f) as a morphism

β(f) : N → LB/A ⊗LB F.(1.12.9)

By assembling the diagrams (1.12.2), (1.12.5), and (1.12.7)⊗LRF and using the isomor-phisms noted above, we obtain a commutative diagram

N //

β(f)((QQ

QQQQQQ

QQQQQQ

QQ k1(LgrR[V ]/R′[V ′] ⊗

LR[V ] R[F ])

// J ⊗R F [1]

LB/A ⊗LB Fκ(B)⊗BF

55kkkkkkkkkkkkkkk

(1.12.10)

in D(R). The composition of the top row and u[1] is the obstruction ω (1.6.8). We haveproved the following (c.f. (IV.3.1.8), (IV.3.2.14), and Theorem 2.9 in [HT]):

1.13. Theorem. Assume C and E are flat over A. Then the obstruction ω ∈ Ext1R(N,K)of (1.2) in the coproduct case of (1.8) can be written as the composition

Nβ(f) // LB/A ⊗LB F

κ(B)⊗BF // J ⊗R F [1]u[1] // K[1],

where β(f) is the reduced Atiyah class of f (1.12.8) and κ(B) ∈ Ext1(LB/A, I) is theKodaira-Spencer class of the A-algebra extension B (1.8.1).

1.14. Classical truncation. For later use, we will now determine the map induced on H0

by the reduced Atiyah class N → LR/C ⊗LR F (1.12.8). Recall that this map was definedby the commutative diagram

N

β(f)((

// k1(LgrR[V ]/C[E] ⊗LR[V ] R[F ])

LR/C ⊗LR F

∼=OO

(1.14.1)

using the fact that the indicated map is an isomorphism under the assumptions (1.11).The variant (1.12.9) was constructed from this using the fact that the natural map

LB/A ⊗LB R → LR/C(1.14.2)

is an isomorphism under the assumptions (1.11). The assumptions (1.11) are not actuallynecessary in this section because the two maps in question always induce isomorphismson H0 because Kähler differentials are “compatible” with pushouts and localization.

14 W. D. GILLAM

Taking H0 of (1.14.1), we obtain the diagram:

N //

H0(β(f)) ((QQQQQ

QQQQQQ

QQQQQ k

1(ΩR[V ]/C[E] ⊗R[V ] R[F ])

ΩR/C ⊗R F

∼=

OO(1.14.3)

The horizontal arrow is just the “connecting homomorphism” from the long exact se-quence obtained from the triangle associated to the bottom row of (1.12.1) (note N =H−1(LgrR[F ]/R[V ])). Viewing N as a submodule of V = R⊗C E, a typical local section of ntakes the form3 n = r⊗e, and the horizontal arrow in (1.14.3) is given by r⊗e 7→ d(r⊗e)⊗1.Note r ⊗ e ∈ V = k1R[V ], so d(r ⊗ e) is in k1ΩR[V ]/R[F ]. Since 1 ⊗ e is the image ofe ∈ E = k1C[E] under C[E] → R[V ], d(1⊗ e) = 0 in ΩR[V ]/C[E], and we see that

d(r ⊗ e)⊗ 1 = d(r(1⊗ e))⊗ 1(1.14.4)= (1⊗ e) · dr ⊗ 1= dr ⊗ f(1⊗ e),

which is the image of the same expression under the vertical isomorphism in (1.14.3). Wefind that H0(β(f)) is given by

H0(β(f)) : N → ΩR/C ⊗R F(1.14.5)r ⊗ e 7→ dr ⊗ f(1⊗ e).

Notice that the formula in (1.14.5) actually defines a C linear map V → ΩR/C ⊗R F , butthis map is R linear only when restricted to N = Ker f .

2. Reduced Atiyah class

In this section, we give an alternative treatment of the reduced Atiyah class, first in-troduced in (1.12.8), using principal parts instead of the graded cotangent complex. Theresults of this section are not needed elsewhere and are parallel to the analogous discussionin (IV.2.3).

2.1. Principal parts. Let C → R be a ring homomorphism and let V be an R module.Recall that the R module of principal parts of V relative to C → R is given by

P 1R/C(V ) := ((R⊗C R)/I2∆)⊗R V,(2.1.1)

where I∆ is the kernel of the multiplication R⊗C R → R and we use the right R modulestructure on ((R⊗C R)/I2∆) (restriction of scalars along r 7→ 1⊗ r) to define to define thetensor product over R, then we regard the result as an R module using the left R modulestructure. Recall (III.1.2.6.3) the principal parts exact sequence

0 → ΩR/C ⊗C V → P 1R/C(V ) → V → 0(2.1.2)

of V . The cokernel map in (2.1.2) admits a natural C linear section

s : V → P 1R/C(V )(2.1.3)v 7→ 1⊗ 1⊗ v,

3Actually n will be a linear combination of “pure tensors” r ⊗ e, but there is no difficulty in carryingthe sum through the computation (1.14.4), so we will suppress it.


which yields an C module splitting of (2.1.2). The map s is generally not R linear, since

s(r · v)− r · s(v) = 1⊗ 1⊗ r · v − r · (1⊗ 1⊗ v)= 1⊗ r ⊗ v − r ⊗ 1⊗ v= dr ⊗ v.

On the other hand, if V = R⊗C E is obtained by extension of scalars from a C moduleE, then I claim the map t : V → P 1R/C(V ) given by

t : R⊗C E → ((R⊗C R)/I2∆)⊗R (R⊗C E)(2.1.4)r ⊗ e 7→ r ⊗ 1⊗ 1⊗ e

is an R linear section of P 1R/C(V ), hence provides an R module splitting

P 1R/C(V ) = (ΩR/C ⊗ V )⊕ V(2.1.5)

of (2.1.2). Obviously t is a section. It is R linear by the following computation:

t(r′ · r ⊗ e)− r′ · t(r ⊗ e) = t(rr′ ⊗ e)− r′ · (r ⊗ 1⊗ 1⊗ e)= r′r ⊗ 1⊗ 1⊗ e− r′r ⊗ 1⊗ 1⊗ e= 0.

2.2. Classical version. Continuing to assume V = R⊗CE, we consider an exact sequence(1.1.3) of R modules. From the naturality of the principal parts sequence we obtain anexact diagram of R modules as below.

0 0 0

0 // ΩR/C ⊗R F

OO

// P 1R/C(F )

OO

// F

OO

// 0

0 // ΩR/C ⊗R V

ΩR/C⊗fOO

// P 1R/C(V )//

OO

V

f

OO

// 0

0 // ΩR/C ⊗R N

OO

// P 1R/C(N)//

OO

N

OO

// 0

0

OO

(2.2.1)

Using the splitting (2.1.5) and the diagram (2.2.1) we obtain a morphism of doublecomplexes of R modules

ΩR/C ⊗R V ΩR/C ⊗R V

ΩR/C ⊗R N

OO

// P 1R/C(N)

OOα //

ΩR/C ⊗R V

ΩR/C ⊗R N

OO(2.2.2)

16 W. D. GILLAM

(where ΩR/C ⊗R N is placed in degree (0,−1)) by projecting onto the quotient complex.I claim that the map H0(α) coincides with the map (1.14.5):

H0(α) : N → ΩR/C ⊗R F(2.2.3)r ⊗ e 7→ dr ⊗ f(1⊗ e).

Indeed, H0(α)(r⊗e) can be computed by choosing (locally) a lift of r⊗e to P 1R/C(V ), thenapplying ΩR/C ⊗ f to the component of this lift in the summand ΩR/C ⊗R V under thesplitting (2.1.5) (the result will be independent of the lift of r⊗ e ∈ N ⊂ V and will yieldan R linear map). We may as well choose the lift systematically be means of the C linearmap s in (2.1.3). Since the splitting (2.1.5) is defined via t, the ΩR/C ⊗R V component ofs(r ⊗ e) is given by s(r ⊗ e)− t(r ⊗ e). Now we simply compute

H0(α)(r ⊗ e) = (ΩR/C ⊗ f)(s(r ⊗ e)− t(r ⊗ e))= (ΩR/C ⊗ f)(1⊗ 1⊗ r ⊗ e− r ⊗ 1⊗ 1⊗ e)= (ΩR/C ⊗ f)(1⊗ r ⊗ 1⊗ e− r ⊗ 1⊗⊗1⊗ e)= (ΩR/C ⊗ f)(dr ⊗ 1⊗ e)= dr ⊗ f(1⊗ e).

2.3. Derived version. We now generalize the construction of (2.2) by replacing Kählerdifferentials with the cotangent complex. We adopt the setup of (1.8) and make theassumptions (1.11).

Let P := PAB be the standard simplicial resolution of B by free A-algebras4 and let

T := P ⊗A C. Since A → C is flat, the natural map p : T → H is a resolution of Hby termwise free C-algebras. Let T → Q be the localization of T at p−1(S). We have acocartesian diagram of (simplicial) rings

A //

��

C

��P //

��

T

p

��

// Q

��B // H // R = S−1H

(2.3.1)

where the horizontal maps are flat, the top vertical maps are termwise free, and the bot-tom vertical maps are surjective quasi-isomorphisms. The rightmost square is a pushoutbecause surjectivity of p implies p(p−1(S)) = S. In particular, we have

LR/C = LH/C ⊗H R(2.3.2)= ΩT/C ⊗T R= (ΩT/C ⊗T Q)⊗Q R= ΩQ/C ⊗Q R= ΩP/A ⊗P R.

Regard N,V, F ∈ Mod(R) as Q modules via restriction of scalars along Q → R. Weremind the reader that Q → R is surjective, so restriction along Q → R followed by ⊗QR

4any termwise free resolution will do


is the identity on Mod(R). As in (2.2), the exact sequence (1.1.3) (now viewed as anexact sequence of Q modules) yields a commutative diagram

0 0 0

0 // ΩQ/C ⊗Q F

OO

// P 1Q/C(F )

OO

// F

OO

// 0

0 // ΩQ/C ⊗Q V

OO

// P 1Q/C(V )//

OO

V

OO

// 0

0 // ΩQ/C ⊗Q N

OO

// P 1Q/C(N)//

OO

N

OO

// 0

0

OO

0

OO

0

OO

(2.3.3)

of Q modules with exact rows and columns—the “extra” exactness here is due to the factthat ΩQ/C is a flat Q module. Now, we would like to say that the middle row in (2.3.3)splits as in (2.1) because V = R⊗CE is obtained by extension of scalars from a C module,but this is not quite true because we are viewing V here as a Q module. On the otherhand, the parts sequence for M := Q⊗C E does split as in (2.1) and we can see that thenatural map

M = Q⊗C E → R⊗C E = V(2.3.4)

is a quasi-isomorphism of Q modules, as follows. First, T → Q is flat, and from (2.3.1),(2.3.4) is given by extension of scalars of

T ⊗C E → H ⊗C E(2.3.5)

along T → Q, so it suffices to prove that (2.3.5) is a quasi-isomorphism. But from (2.3.1),(2.3.5) is identified with

P ⊗A E → B ⊗A E(2.3.6)

and (2.3.6) is a quasi-isomorphism because P → B is a quasi-isomorphism and we assumeE is A flat (1.11).

From the quasi-isomorphism (2.3.4), we obtain a map of Q module extensions

0 // ΩQ/C ⊗Q M

≃��

// (ΩQ/C ⊗Q M)⊕M

≃��

// M //

≃��

0

0 // ΩQ/C ⊗Q V // P 1Q/C(V ) // V // 0,

(2.3.7)

where the vertical arrows are quasi-isomorphisms (by the Five Lemma and the fact thatΩQ/C is a flat Q module).

We now consider the two term complex

Z := [ (ΩQ/C ⊗Q F )⊕M // P 1Q/C(F ) ](2.3.8)

18 W. D. GILLAM

of Q modules (in degrees 0, 1), where the map M → P 1Q/C(F ) is the composition of themap M → P 1Q/C(V ) appearing in (2.3.7) and the natural map P

1Q/C(V ) → P

1Q/C(F ).

From the exactness of (2.3.3) and the quasi-isomorphisms in (2.3.7) it follows that Z isquasi-isomorphic to N . This is clear once we note that Z is quasi-isomorphic to the doublecomplex below.

(ΩQ/C ⊗Q V )⊕M // P 1Q/C(V )

ΩQ/C ⊗Q N

OO

// P 1Q/C(N)

OO(2.3.9)

There is an obvious morphism of complexes β : Z → ΩQ/C ⊗Q F obtained by projectingto the quotient complex. After extending scalars along Q → R and using the isomorphism(2.3.2), we may view β as a D(R) morphism

β : N → LB/A ⊗LB F,(2.3.10)

which coincides, as the notation suggests, with the reduced Atiyah class of the quotientf : V → F constructed in (1.12.8), as we will see in (2.5).

2.4. Factorization. We now explain the appellation “reduced Atiyah class.” Notice thatthe projection map

[ ΩQ/C ⊗Q F // P 1Q/C(F ) ] // ΩQ/C ⊗Q F(2.4.1)

obviously factors through β by including the domain complex as a subcomplex of Z (2.3.8).

By (IV.2.3.7.3) the image of this projection map in D(R) coincides (up to a shift) withthe Atiyah class

at = atR/C(F ) : F → LR/C ⊗LR F [1](2.4.2)

of F relative to C → R. Note that the aforementioned inclusion of subcomplexes isnaturally isomorphic in D(R) to the map F [−1] → N obtained from (1.1.3). Hence, inD(R), we obtain a commutative diagram

F [−1]

��

at[−1]

%%LLLLL

LLLLL

Nβ // LR/C ⊗L F

(2.4.3)

factoring the (shift of the) Atiyah class of F through the reduced Atiyah class β.

2.5. Comparison. We now prove that the reduced Atiyah class of (2.3) conincides withthe one introduced in (1.12.8). From the transitivity triangle of graded cotangent com-plexes associated to the diagram of graded rings

C[E] // R[V ] // R[F ]

C

OO

// R //

OO

R[F ]

(2.5.1)



N // k1(LgrR[V ]/C[E] ⊗LR[V ] R[F ])

F [−1]

OO

at[−1] // LR/C ⊗LR F

∼=OO

(2.5.2)

where the bottom horizontal arrow is the (shifted) Atiyah class (2.4.2). We saw that theindicated vertical arrow is an isomorphism in (1.12).

2.6. Theorem. The diagram

N //

β ((QQQQQ

QQQQQQ

QQQQQQ k

1(LgrR[V ]/C[E] ⊗LR[V ] R[F ])

F [−1]

OO

// LR/C ⊗LR F

∼=OO

(2.6.1)

obtained from (2.5.2) by inserting the reduced Atiyah class (2.3.10) is commutative.

Proof. The lower triangle commutes by (2.4.3); to see that the upper triangle commutes,we continue with the setup of (2.2). We first replace (2.5.1) with a quasi-isomorphicdiagram of graded simplicial rings

C[E] // Q[M ] // L

C

OO

// Q //

OO

K

OO(2.6.2)

lying over it, where Q → K and Q[M ] → L are termwise free. The map of transitivitytriangles associated to (2.5.1) is then the image in D(R[F ]) of the map of exact sequences

0 // ΩQ[M ]/C[E] ⊗Q[M ] L // ΩL/C[E] // ΩL/Q[M ] // 0

0 // ΩQ/C ⊗Q L // ΩK/C ⊗K L //

OO

ΩK/Q ⊗K L

O O

// 0

(2.6.3)

of L modules (after applying ⊗LR[F ]). The left vertical arrow above is an isomorphismbecause the left square of (2.6.2) is a pushout. Set L1 := k1L. The composition N →LR/C ⊗LR F of the top horizontal and (inverse) right vertical arrow in (2.6.1) is the imagein D(R) of the quotient complex projection

[ ΩQ/C ⊗Q k1L // k1ΩL/C[E] ] → ΩQ/C ⊗Q L1(2.6.4)

after extending scalars along L0 := k0L → k0(R[F ]) = R. Note that

k1ΩL/Q[M ] ⊗L0 R = k1LR[F ]/R[V ] = N [1].

From the graded ring maps C → C[E] → L and the fact that ΩC[E]/C = k1ΩC[E]/C = E,we obtain an exact sequence

0 // E ⊗C L1 // k1ΩL/C[E] // k1ΩL/C // 0(2.6.5)

20 W. D. GILLAM

of L0 modules. From (II.1.2.6.7) and the proof of (IV.2.3.7.3) we obtain a morphism

0 // ΩQ/C ⊗Q L1 // P 1Q/C(L1) //

≃��

L1

d≃��

// 0

0 // ΩQ/C ⊗Q L1 // k1(ΩL/C) // ΩL/Q // 0

(2.6.6)

of exact sequences of L0 modules, where the indicated maps are quasi-isomorphisms be-cause Q → L is a quasi-isomorphism in grading zero and Q is supported in grading zero(IV.2.2.5). We now have a sequence of quasi-isomorphisms, compatible with the projec-tions to ΩQ/C ⊗Q L1 ≃ ΩQ/C ⊗Q F , as follows:

[ ΩQ/C ⊗Q L1 // k1ΩL/C[E] ](2.6.7)

≃ [ (ΩQ/C ⊗Q L1)⊕ (E ⊗C L0) // k1ΩL/C ]

≃ [ (ΩQ/C ⊗Q L1)⊕ (E ⊗C L0) // P 1Q/C(L1) ]

≃ [ (ΩQ/C ⊗Q F )⊕M // P 1Q/C(F ) ].

The first is obtained using the sequence (2.6.5), the second is obtained using (2.6.6),and the others are obtained from the natural augmentation maps from (2.6.2) to (2.5.1).Extending scalars along L0 → R, we see from (2.6.7) that the domain of (2.6.4) becomesquasi-isomorphic, over ΩQ/C ⊗Q F , to the complex Z (2.3.8) used to define β, hence theproof is complete.

2.7. Functoriality. The reduced Atiyah class is functorial in various ways that we leaveto the reader to make precise. We only point out that, given a morphism of R moduleextensions of the form

0 // N //

��

Vf // F //

��

0

0 // N ′ // Vg // F ′ // 0,

(2.7.1)

we have a commutative diagram in D(R):

N

��

β(f) // LR/C ⊗LR F

��N ′

β(g) // LR/C ⊗LR F ′

(2.7.2)

It is a good exercise to prove this using both constructions of the reduced Atiyah class.


3. Perfect quotients

3.1. Setup. We continue with the setup (1.8) and the assumptions (1.11). We also assumethat the extension B (1.8.1) is trivialized,

B = 0 // I // B[I] // B // 0

A,

aaCCCCCCCC

OO(3.1.1)

and that F is flat over B.

The C-algebra extensions H and R in (1.8.5) are also naturally trivialized and we have:

H ′ = B[I]⊗A C = H[I ′] R′ = (S′)−1(B[I]⊗A C) = R[J ]W = B ⊗A E V = R⊗C EW ′ = B[I]⊗A E V ′ = R[J ]⊗C ES′ = S × (I ⊗A C)

Given an A-algebra section s : B → B[I] of B[I] → B, we also obtain C-algebra sectionss⊗A C : H → H ′ = H[I ′] and S−1(s⊗A C) : R → R′ = R[J ], both of which we will alsoabusively denote by s. We will write B[I]⊗s (resp. H[I ′]⊗s, R[J ]⊗s) to emphasize thatthe tensor product is taken regarding B[I] (resp. H[I ′], R[J ]) as a B (resp. H, R) modulevia restriction of scalars along s. We have a natural isomorphism of H ′ modules

B[I]⊗s (B ⊗A E) → B[I]⊗A E = W ′(3.1.2)(b+ i)⊗ b′ ⊗ e 7→ (b+ i)s(b′)⊗ e.

The localization of (3.1.2) at S′ (that is, (3.1.2) ⊗H′R′) yields an isomorphism

R[J ]⊗s (R⊗C E) → R[J ]⊗C E = V ′(3.1.3)(r + j)⊗ r′ ⊗ e 7→ (r + j)s(r′)⊗ e

of R[J ] modules.

We have a diagram of R[J ] modules

0

��

0

��

0

��0 // J ⊗R N //

��

R[J ]⊗s N //

��

N //

��

0

0 // J ⊗R V //

J⊗f��

R[J ]⊗s V //

R[J ]⊗sf��

V

f

��

// 0

0 // J ⊗R F

��

// R[J ]⊗s F //

��

F //

��

0

0 0 0

(3.1.4)

22 W. D. GILLAM

with exact columns and rows. Note that the left column is exact because it is identifiedwith

0 // I ⊗B N // I ⊗B V // I ⊗B F // 0(3.1.5)

by (1.8.4) and (3.1.5) is exact since we assume F is B flat. Using the isomorphism (3.1.3),we can regard the bottom two rows of (3.1.4) as a solution

0 // J ⊗R V //

J⊗f��

V ′ //

��

V

f

��

// 0

0 // J ⊗R F // R[J ]⊗s F // F // 0

(3.1.6)

to Problem 1.1. In fact this solution, which we will denote s∗f , is a flat deformation off over B[I] in the sense of (1.9). Notice that the diagram (1.5.1) built from the solutions∗f as in Remark 1.5 is nothing but the diagram (3.1.4).

3.2. Definition. The surjection f : V → F is a perfect quotient if and only if the map

{A−algebra sections of B[I] → B } → { flat deformations of f over B[I] }s 7→ s∗f (3.1.6)

is bijective for every B module I.

The geometric significance of perfect quotients will be made clear in Theorem 4.2.Roughly speaking, f is a perfect quotient if and only if B is a “universal deformationspace” for quotients of the C module E.

Given an A-algebra section s of B[I] → B, let ds : B → I be the corresponding Alinear derivation and let gs : ΩB/A → I be the corresponding map of B modules. Lets0 : B → B[I] be the zero section b 7→ b.

3.3. Lemma. The map N → J ⊗R F = I ⊗B F defined by the map of complexes

[J ⊗R N // R[J ]⊗s N ]

��[R[J ]⊗s0 F // F ].

(3.3.1)

as in Remark 1.5 coincides with the composition

NH0(β(f)) // ΩB/A ⊗B F

gs⊗F // I ⊗B F,(3.3.2)

where β(f) is the reduced Atiyah class of f (1.12.9). Both maps are given by

N → J ⊗R F(3.3.3)r ⊗ e 7→ ds(r)⊗ f(1⊗ e).

Proof. Continuing our abuse of notation, we also write ds : R → J and gs : ΩR/C → Jfor the maps corresponding to the C-algebra section s : R → R[J ]. By definition of the


reduced Atiyah class (1.12.9), the calculation in (1.14), and naturality of the formation ofgs from s, we have a commutative diagram as below.

NH0(β(f)) //

(1.14.5) ''OOOOO

OOOOOO

OOOΩB/A ⊗B F

∼=��

gs⊗F // I ⊗B F

(1.8.4)

ΩR/C ⊗R Fgs⊗F // J ⊗R F

(3.3.4)

The maps s, gs, and ds are related by the formulas

s(x) = x+ ds(x)(3.3.5)

gs(dx) = ds(x),

so the fact that (3.3.2) is given by (3.3.3) is clear from the formula (1.14.5).

To calculate the map defined by (3.3.1), we will make use of the natural C linear sectiont : n 7→ 1⊗ n of R[J ]⊗s N → N . (This map is not R linear, but all that matters is thatit is a “set-theoretic” section.) The isomorphism R[J ] ⊗s V ∼= R[J ] ⊗s0 V obtained byidentifying both sides with V ′ via the natural isomorphism (3.1.3) can be explicitly written

R[J ]⊗s (R⊗C E) → R[J ]⊗s0 (R⊗C E)(3.3.6)(r + j)⊗ r′ ⊗ e 7→ (r + j)s(r′)⊗ 1⊗ e.

As in (1.14), we view N = Ker(f : V → F ) as a submodule of V = R⊗C E and we writen = r ⊗ e by slight abuse of notation. The map defined by (3.3.1) can be computed on nwith the aid of the section t by following t(n) = 1⊗ r ⊗ e through the sequence of maps

R[J ]⊗s (R⊗C E)(3.3.6)

∼=// R[J ]⊗s0 (R⊗C E)

R[J ]⊗s0f // R[J ]⊗s0 F

and noting that the result lies in the submodule J ⊗R F ⊆ R[J ]⊗s0 F . Carrying this out,1⊗ r ⊗ e 7→ s(r)⊗ 1⊗ e

= r ⊗ 1⊗ e+ ds(r)⊗ 1⊗ e= 1⊗ r ⊗ e+ ds(r)⊗ 1⊗ e7→ 1⊗ f(r ⊗ e) + ds(r)⊗ f(1⊗ e)= ds(r)⊗ f(1⊗ e),

we see that the map induced by (3.3.1) is also given by (3.3.3), so the proof is complete.

4. Applications to the Quot scheme

4.1. Setup. We work throughout in the category Sch of schemes over a fixed base schemeS, which we refer to simply as “schemes”. All relative constructions done without explicitreference to a morphism are assumed to be relative to the terminal object S, so, forexample, LX = LX/S , X×Y = X×S Y , etc. Let Y be a scheme, E a quasi-coherent sheafon Y . We make the assumptions on E and Y corresponding to the assumptions (1.11)under the dictionary (1.10), namely that Y and E are flat over S. We let Q denote theQuot functor

Q : Schop → Sets(4.1.1)X 7→ { quotients of π∗2E on X × Y flat over X }.

24 W. D. GILLAM

The presheaf Q is representable under various hypotheses: For example, if S is noetherian,Y/S is projective, and E is coherent, then Q is representable and the components of Q/Sare projective [Gro]. The Quot functor (4.1.1) is representable by an algebraic space inmuch more generality. The Quot functor Q may be defined in the same way when Y/Sis a representable morphism of algebraic stacks. Assuming the basic machinery of Serreduality, etc. for algebraic spaces/stacks, the results of this section carry over to that settingas well, though we often view X simply as a scheme for the sake of concreteness.

Since we assume Y/S is flat, we have LX×Y/Y = π∗1LX for any scheme X.5 For an exactsequence

0 // N // π∗2Ef // F // 0(4.1.2)

on X × Y with F flat over X, we let β(f) : N → π∗1LX ⊗ F (or just β if f is clear fromcontext) denote the reduced Atiyah class of f (1.12.9). Recall the natural transformation

β : HomD(X)(LX , ) → HomD(X×Y )(N, π∗1 ⊗ F )(4.1.3)β(M)(g) := (π∗1g ⊗ F )β.

from the introduction. By abuse of notation, we write f : X → Q for the morphismcorresponding to the quotient f .

Since all the technical results are already in place, the proof of the next theorem willamount to little more than unwinding various definitions. I view this theorem as the mainresult of this paper.

4.2. Theorem. The following are equivalent:

(1) f is a perfect quotient.(2) β(I) is an isomorphism for every quasi-coherent sheaf I on X.

If f : X → Q is formally étale, then these two equivalent conditions hold, and, furthermore,

β(I[1]) : Ext1(LX , I) → Ext1(N,π∗1I ⊗ F )

is injective for every quasi-coherent sheaf I on X.

Proof. For a quasi-coherent sheaf I, let ιI : X ↪→ X[I] be the trivial square zero closedimmersion with ideal I. By Lemma 3.3 we have a commutative diagram

Hom(ΩX , I)β(I) // Hom(N, π∗1I ⊗ F )

{ retracts of ιI } // { flat deformations of f over X[I] }

(4.2.1)

where the bottom arrow is given by pullback. By definition, the latter is an isomorphismfor every I iff f is a perfect quotient, so the equivalence of the two conditions is clear.

5We recorded this as formula (1.11.2) using the dictionary (1.10).


If f : X → Q is formally étale, then by definition of “formally étale” s 7→ fs gives abijection between retracts of ιI and completions of the solid diagram

XιI //

f

��

X[I]

}}Q

in (presheaves on) Sch. On the other hand, by definition of Q, such completions are thesame thing as flat deformations of f over X[I]. This proves the second statement.

For the final statement: The Fundamental Theorem (III.1.2.3) identifies Ext1(LX , I)with (isomorphism classes of) square zero thickenings X ↪→ X ′ of X with ideal sheaf I.According to (1.13), the map

β(I[1]) : Ext1(LX , I) → Ext1(N,π∗1I ⊗ F )takes the class κ(X ′) : LX → I[1] of the thickening X ↪→ X ′ to the obstruction

ω = (π∗1κ(X′)⊗ F )β

to finding a flat deformation of f over X ′. But by definition of Q, finding such a flatdeformation is the same thing as finding a commutative diagram of solid arrows

X _

��

X

f��

X ′ //

>>

Q

and by defintion of formally étale every such diagram can be completed to a commutativediagram as indicated by the dotted arrow. Such a dotted arrow is the same thing as aretract of X ↪→ X ′, which is the same thing as an identification of X ′ with the trivialthickening: X ′ = X[I]. If β(I[1]) failed to be injective we would therefore find somenontrivial thickening X ′ with a retract, which is absurd.

4.3. Lemma. Let Z be a scheme. Suppose F ∈ D(Z) is a perfect complex6 whose rankr is invertible in Γ(Z,OZ). Set F∨ := RH om(F,OZ). For any N,E ∈ D(Z), there is anatural isomorphism

HomD(Z)(N,E ⊗L F ) → HomD(Z)(N ⊗L F∨, E).

Proof. Since F is perfect, we have F ⊗L F∨ = RH om(F, F ), and we have a tracemorphism

tr : RH om(F, F ) → OZand a “scalar multiplication” morphism

id : OZ → RH om(F, F )satisfying (tr)(id) = ·r. The desired isomorphism is given by g 7→ (E ⊗ tr)(g ⊗ F∨). Itsinverse is given by h 7→ (h⊗ F )(N ⊗ r−1id).

4.4. Remark. Similar hypotheses on invertibility of rank are needed in [HT].

6A complex is called perfect if it is locally quasi-isomorphic to a bounded complex of locally free sheavesof finite rank (vector bundles).

26 W. D. GILLAM

4.5. Lemma. Let A be an abelian category with enough injectives and let B be a fullabelian subcategory of A. Let f : E → F be a map in D(A) between complexes withcohomology in B and vanishing in positive degrees. Then the following are equivalent:

(1) H0(f) is an isomorphism.(2) Hom(F, I) → Hom(E, I) is an isomorphism for every I in B.

If these equivalent conditions are satisfied, then the following are equivalent:

(1) H−1(f) is surjective.(2) Ext1(F, I) → Ext1(E, I) is injective for every I in B.

Proof. This follows easily from the spectral sequence Extp(H−q(E), I) ⇒ Extp+q(E, I)using the Subtle Five Lemma.

4.6. Theorem. Suppose Y/S is flat, proper, and Gorenstein, E is a coherent sheaf on Yflat over S, f : π∗2E → F is a surjection of sheaves on X × Y with F flat over X andperfect. Assume that the rank of F is invertible on X × Y and that N := Ker f is alsoperfect. Then the functor

D(X) → Mod(Γ(X,OX))I 7→ HomD(X×Y )(N,π∗1I ⊗L F )

is represented by

E := RH om(Rπ1∗RH om(N,F ),OX) ∈ D(X).

In particular, the natural transformation β (4.1.3) may be viewed as a D(X) morphismβ : E → LX . If f defines a formally étale map from X to Quot E, then H0(β : E → LX)is an isomorphism and H−1(β : E → LX) is surjective.Proof. By the hypotheses on Y/S, there is an invertible sheaf ωY on Y and an isomorphism

RH omX(Rπ1∗A,B) = Rπ1∗RH omX×Y (A, π∗1B ⊗ π∗2ωY [d])(4.6.1)

(Grothendieck-Serre duality7) in D(X) natural in A ∈ D(X × Y ) and B ∈ D(X).8 Thereare natural isomorphisms

HomD(X×Y )(N, π∗1I ⊗L F ) = HomD(X×Y )(N ⊗L F∨, π∗1I)

= HomD(X×Y )(N ⊗L F∨ ⊗ π∗2ωY [d], π∗1I ⊗ π∗2ωY [d])= HomD(X)(Rπ1∗(N ⊗L F∨ ⊗ π∗2ωY [d]), I),

where the first isomorphism is obtained from Lemma 4.3, the second isomorphism simplyreflects the fact that ⊗π∗2ωY [d] is an automorphism of D(X×Y ) since π∗2ωY is an invertiblesheaf, and the third isomorphism is obtained from the Serre duality isomorphism (4.6.1)by applying RΓ and taking H0.

7To be safe, one should probably assume S is locally noetherian of finite Krull dimension, as thisassumption is often used in the usual treatments of duality theory [H].

8We may take this as the definition of “Gorenstein” for the proper morphism Y/S. The d here is alocally constant function on Y , or, by abuse of notation, its pullback to any base-change of Y .


Using Serre duality and the various perfection hypotheses, we obtain a sequence ofnatural isomorphisms

E = RH om(Rπ1∗RH om(N,F ),OX)= Rπ1∗(RH om(RH om(N,F ), π

∗2ωY [d]))

= Rπ1∗(RH om(RH om(N,F ),OX×Y )⊗ π∗2ωY [d])= Rπ1∗(RH om(N

∨ ⊗L F,OX×Y )⊗ π∗2ωY [d])= Rπ1∗(N ⊗L F∨ ⊗ π∗2ωY [d]).

Putting this together with the isomorphism from the previous paragraph proves the firstpart of the theorem.

The second statement follows from Theorem 4.2 and Lemma 4.5 (applied with A =Mod(OY ), B = quasi-coherent sheaves).

4.7. Remark. Theorem 4.2 should be viewed as a manifestation of the following generalphilosophy of deformation theory. Let X → S be a morphism of schemes. An obstructiontheory E on X/S consists of the following data:

(1) an object E ∈ D(X) with quasi-coherent cohomology supported in non-positivedegrees,

(2) for each quasi-coherent sheaf I on X and each square-zero thickening i : X ↪→ X ′of S schemes with ideal I, an element ω = ω(E, I, i) ∈ Ext1X(E, I) whose vanishingis necessary and sufficient for the existence of a retract of i, and

(3) a simply transitive (torsorial) action of HomD(X)(E, I) = HomX(H0(E), I) on theset of retracts of i whenever ω = 0.

The element ω in (2) and the action in (3) are required to be natural in I, i in an obvioussense. The Fundamental Theorem (III.1.2.3) implies that every morphism X → S admitsa canonical obstruction theory with E = LX/S . A morphism of obstruction theories E → Fis a D(X) morphism ϕ : E → F such that for every I, i, the map

ϕ∗ : Ext1X(F, I) → Ext1X(E, I)takes ω(F, I, i) to ω(E, I, i), and the map

ϕ∗ : HomD(X)(F, I) → HomD(X)(E, I)is a bijection, which identifies the actions in (3) whenever i admits a retract.

The general philosophy is that every obstruction theory “arising in nature” admits amorphism to the canonical one. As far as I know, there is no “real counterexample” tothis philosophy, even though the morphism E → LX/S is sometimes difficult to produce.A large part of this paper and [HT] are devoted to constructing such morphisms.

J. Lurie’s program of derived algebraic geometry suggests the following motivation forthis philosophy. Whenever an obstruction theory E on X/S “arises in nature,” it arisesbecause there is a “derived scheme” X ′ over S (representing some derived version ofthe moduli problem represented by X/S) with π0(X

′) = X and a natural isomorphismE = LX′/S |X . That is, E is the (derived) restriction of the canonical obstruction theoryon the derived moduli problem to its 0th truncation. The map E → LX/S is then nothingbut the map LX′/S |X → LX induced by the truncation morphism X → X ′. In the caseof the Quot scheme, studied here, this philosophy could surely be made precise (with a

28 W. D. GILLAM

fair amount of work) using the study of the derived Quot scheme in [CFK1]—note the“formula” for its cotangent complex [CFK1, 0.3.2].

4.8. Lemma. Suppose Y/S is a flat, proper curve and E is a coherent sheaf on Y . Thenfor any quasi-compact scheme X and any exact sequence (4.1.2) with F flat over X, Nlocally free, and π1 : X × Y → X projective, the complex

E = RH om(Rπ1∗RH om(N,F ),OX) ∈ D(X)

is of perfect amplitude ⊆ [−1, 0].9

Proof. Since N is locally free we have RH om(N,F ) = H om(N,F ) = N∨⊗F . The sheafG := N∨⊗F is a coherent sheaf on X×Y flat over X. It suffices to show that the complexRπ1∗G is of perfect amplitude ⊆ [0, 1], which is a more-or-less standard argument; ourproof is adapted from the proof of [Beh, Proposition 5]. Let O(1) be an X-ample invertiblesheaf on X × Y . Using flatness of X × Y over X, quasi-compactness of X, and standardbase-change and semicontinuity arguments, we can find an n large enough that

(1) π∗1π1∗(G(n)) → G(n) is surjective,(2) R>0π1∗G(n) = 0, and(3) π1∗O(−n) = 0.

Set G′ := (π∗1π1∗(G(n)))(−n). By the vanishing (2), flatness of G over X, and Grauert’sCriterion, π1∗G(n) is a vector bundle on X, hence G

′ is a vector bundle on X × Y . Fromthe surjectivity in (1), we obtain a surjection G′ → G, whose kernel G′′ is flat over Xbecause G′ and G are flat over X. From the vanishing (3), it follows that π1∗G

′ = 0, henceπ1∗G

′′ = 0 as well. Since R>1π1∗ = 0 because Y/S is a curve (relative dimension one), theshort exact sequence

0 → G′′ → G′ → G → 0and the established vanishings yield

Rπ1∗G = [R1π1∗G

′′ → R1π1∗G′],

so the proof is complete because R1π1∗G′′ and R1π1∗G

′ are bundles by Grauert.

4.9. Corollary. (Assume Q ⊆ OS) Suppose Y/S is a flat, proper, Gorenstein curve,E is a vector bundle on Y , and X is a quasi-compact open subset of QuotE on whichthe universal kernel N ⊆ π∗2E is locally free (this always holds if Y/S is smooth) andX × Y → X is projective. Then the map

RH om(Rπ1∗RH om(N,F ),OX) → LXobtained from the reduced Atiyah class defines a perfect obstruction theory on X.

Proof. The parenthetical remark follows from the fact that if C is a smooth curve (over afield), then OC is a sheaf of PIDs, so any (quasi-coherent) subsheaf of a vector bundle is avector bundle. If N is locally free, then from the exact sequence (4.1.2) it is clear that Fhas finite perfect amplitude (⊆ [−1, 0] in fact), so the Corollary follows immediately fromTheorem 4.6 and Lemma 4.8.

9Here we mean “perfect amplitude ⊆ [−1, 0]” in the strong sense of being globally isomorphic to atwo-term complex of vector bundles supported in degrees −1, 0. In Behrend’s terminology [Beh], E admitsglobal resolutions. If one only wants this locally, then the hypothesis “X quasi-compact” is superfluous.


4.10. Remark. If Y is smooth, then π1 : X × Y → X is also smooth, so flatness of F(hence N) over X implies perfection of F and N [SGA6.III.3.6]. It may be possible toweaken some of the perfection hypotheses.

4.11. Example. When Y is a smooth projective curve over S = SpecC and E is a vectorbundle on Y , the perfect obstruction theory on (components X of) QuotE provided byCorollary 4.9 is used in [MO] and [G2] to calculate and study deformation invariants ofQuotE and their relationship to various problems of enumerative geometry and curvecounting.

4.12. Example. Let S := Mg,m ⊗Z C be the Artin stack of m-marked nodal curves ofgenus g and let Y → S be the (flat, proper, Gorenstein) universal curve. Set E := OnY . LetX = Qg,mG((r, n), d) be the moduli space of stable quotients in the sense of [MOP] andlet Q be the Quot functor (4.1.1). The map X → Q determined by the universal quotientsequence on X is formally étale because the stability condition for stable quotients [MOP,2.2] is defined in terms of X-relative ampleness of a certain line bundle on the universalcurve C = X×Y , which is an open condition, and a certain local-freeness condition on theuniversal quotient F which is also clearly open. The universal kernel N on C is a vectorbundle [MOP, 2.2], hence Corollary 4.9 provides a perfect obstruction theory E → LX/S[MOP, Theorem 2].

References

[Beh] K. Behrend, Gromov-Witten invariants in algebraic geometry. Invent. Math. 127 (1997) 601-617.[BF] K. Behrend and B. Fantechi, The intrinsic normal cone. Invent. Math. 128 (1997) 45-88.[CFK1] I. Ciocan-Fontanine and M. Kapranov, Derived Quot schemes. Ann. Sci. ENS 34 (2001) 403-440.[CFK2] I. Ciocan-Fontanine and M. Kapranov, Virtual fundamental classes via dg-manifolds.Geom. Topol.

13(3) (2009) 1779-1804.[G1] W. Gillam, Localization of ringed spaces. Adv. Pure Math. 1(5) (2011) 250-263.[G2] W. Gillam, Maximal subbundles, Quot schemes, and curve counting. arXiv:1103.2169.[Gro] A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique IV:

Les schémas de Hilbert. Séminaire Bourbaki 221 (1960-61).[H] R. Hartshorne, Residues and Duality. Lec. Notes Math. 20, Springer-Verlag, 1966.[HT] D. Huybrechts and R. Thomas, Deformation-obstruction theory for complexes via Atiyah and

Kodaira-Spencer classes. Math. Ann. 346 (2010) 545-569.[Ill] L. Illusie, Complexe cotangent et déformations, I. Lec. Notes Math. 239, Springer-Verlag, 1971.[MO] A. Marian and D. Oprea, Virtual intersection numbers on the Quot scheme and Vafa-Intriligator

formulas. Duke Math. J. 136 (2007) 81-113.[MOP] A. Marian, D. Oprea, and R. Pandharipande, The moduli space of stable quotients. Geom. Topol.

15 (2011) 1651-1706.[SGA6] A. Grothendieck, P. Berthelot, and L. Illusie, Séminaire de Géométrie Algébrique du Bois-Marie.

Lec. Notes Math. 255, Springer-Verlag, 1966-67.

Department of Mathematics, Brown University, Providence, RI 02912

E-mail address: [email protected]

· 2013. 9. 19. · created date: 1/13/2012 3:47:51 am

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