Introduction - KTH · 2015. 2. 28. · Quot functor is represented by a scheme given as a disjoint union of projective schemes [Gro95]. When X! Sis locally of nite type and separated,

WEIL RESTRICTION AND THE QUOT SCHEME

ROY MIKAEL SKJELNES

Abstract. We introduce a concept that we call module restric-tion, which generalizes the classical Weil restriction. After havingestablished some fundamental properties, as existence and etaleness,we apply our results to show that the Quot functor Quotn

FX/Sof

Grothendieck is representable by an algebraic space, for any quasi-coherent sheaf FX on any separated algebraic space X −→ S.

Introduction

The main novelty in this article is the introduction of the modulerestriction, which is a generalization of the classical Weil restriction.Our main motivation for introducing the module restriction is given byour application to the Quot functor of Grothendieck.

If FX is a quasi-coherent sheaf on a scheme X −→ S, then theQuot functor Quot

FX/Sparametrizes quasi-coherent quotients of FX

that are flat and with proper support over the base. When the schemeX −→ S is projective, the base Noetherian and with FX coherent, theQuot functor is represented by a scheme given as a disjoint union ofprojective schemes [Gro95]. When X −→ S is locally of finite type andseparated, Artin showed that the Quot functor is representable by analgebraic space ([Art69] and erratum in [Art74]).

When the fixed sheaf FX = OX is the structure sheaf of X, the Quotfunctor is referred to as the Hilbert functor HilbX/S.

Grothendieck who both introduced the Quot functor and showedrepresentability in the projective situation, also pointed out the con-nection between the Hilbert scheme and the Weil restriction [Gro95, 4.Variantes]. If f : Y −→ X is a morphism with X separated, there isan open subset ΩY→X of HilbY/S from where the push-forward map f∗is defined. The fibers of f∗ : ΩY→X −→ HilbX/S are identified with theWeil restrictions.

However, even though the Weil restriction appears naturally in con-nection with Hilbert schemes, there does not seem to exist any descrip-tion of the more general situation with the Quot scheme replacing theHilbert scheme. The purpose of this article is to give such a descrip-tion with the Quot functor in the zero-dimensional case. We say that

Date: 120215.2010 Mathematics Subject Classification. 14A20, 14C05, 14D22.Key words and phrases. Weil restriction, Quot scheme, Fitting ideals.

1

2 ROY MIKAEL SKJELNES

a quasi-coherent and finite type sheaf E on g : X −→ S is finite, flatof relative rank n, if the support Supp(E ) is affine over S, and whereg∗E is locally free of rank n. We denote by Quotn

FX/Sthe functor

parametrizing quotients of FX that are finite, flat and of rank n.The Weil restrictions are generalized in [Ols06] and [Ver72], but those

are not suited for the present discussion. The generalization we under-take here is in the direction from ideals to modules, and is as follows.Fix a homomorphism of A-algebras f : B −→ R, and a B-module M .The module restriction M odMB→R parametrizes, as a functor from A-algebras to sets, the R-module structures extending the fixed B-modulestructure on M .

More specifically, let µ : B −→ EndA(M) denote the A-algebra ho-momorphism corresponding to theB-module structure onM . Then themodule restriction M odMB→R parametrizes A-algebra homomorphismsξ : R −→ EndA(M) such that ξ f = µ where f : B −→ R is the fixedstructure map.

The appearance of the non-commutative ring EndA(M) is natural inthis context, and is also the cause of some of the technical complicationsthat have to be addressed.

When M is finitely generated and projective as an A-module, weshow that the module restriction M odMB→R is representable by an A-algebra. We show representability by constructing the representingobject in the free algebra situation, and using Fitting ideals in thegeneral situation.

These observations are summarized by the following result.

Theorem 1. Let X −→ S be a separated morphism of schemes (oralgebraic spaces) and let C ohnX/S denote the stack of quasi-coherentsheaves on X that are finite, flat of relative rank n over the base S.For any affine morphism f : Y −→ X the push-forward map

(?) f∗ : C ohnY/S −→ C ohnX/S

is schematically representable.

The fibers of the push-forward map (?) are the module restrictionsparametrizing sheaves F on Y that are finite, flat and of relative rankn over the base, such that the push-forward f∗F is isomorphic to afixed sheaf E on X.

When the morphism f : Y −→ X is etale, then the the push-forwardmap (?) is not in general etale. The map (?) is etale when restrictedto the open substack UY→X consisting of sheaves F ∈ C ohnY/S on Y ,such that the induced map of supports

Supp(F ) −→ Supp(f∗F )

is an isomorphism.The requirement concerning the support of the sheaves, highlights a

difference between the Quot functor and the Hilbert functor. Fibers of

WEIL RESTRICTION AND THE QUOT SCHEME 3

(?) over E = OZ , structure sheaves of closed subschemes Z ⊆ X, areetale when f : Y −→ X is etale.

Let f∗ : UY→X −→ C ohnX/S also denote the restriction of (?) tothe open substack where the induced map of supports is an isomor-phism. If we denote Z = Supp(E ), the support of a given element E inC ohnX/S, then we show that the fiber f−1∗ (E ) equals the Weil restrictionof Y ×X Z −→ Z. Thus, the Weil restriction appears naturally in themore general context with the Quot functors as well. Even though thesupport Z = Supp(E ) −→ S is not necessarily flat, it turns out thatin our situation the Weil restriction of Y ×X Z −→ Z still exists as ascheme.

Having established these technical results concerning the support,the representability of Quotn

FX/Sfollows easily. Let FY denote the pull-

back of the quasi-coherent sheaf FX along f : Y −→ X. There is anatural, forgetful, map Quotn

FX/S−→ C ohnX/S whose pull-back along

the map (?) restricted to UY→X , gives a representable, etale covering

ΩFY→X −→ Quotn

FX/S.

This etale cover specializes in the Hilbert functor situation, that is withFX = OX , to the classical cover mentioned earlier, with the fibers beingWeil restriction. We obtain the following result.

Theorem 2. Let X −→ S be a separated map of algebraic spaces, andFX a quasi-coherent sheaf on X. Then the Quot functor Quotn

FX/Sis

representable by an algebraic space.

In particular this generalizes the result about the representability ofthe Hilbert functor HilbnX/S described in [ES14]. See also the general-ization to Hilbert stacks in [Ryd11]. The result also extends the earliermentioned result of Artin in the sense that we do not assume the mapX −→ S to be of finite type, and there is no restriction on the base S.

Acknowledgments. Comments and corrections from Dan Laksov andDavid Rydh were important for the presentation of this manuscript.Discussion with Runar Ile about non-commutative ring theory werealso helpful and clarifying.

1. Fitting ideals

We will in this first section point out some facts about Fitting idealsthat we will use later on.

1.1. Conventions. A commutative ring A is always a unital commu-tative ring. The category of A-algebras, means the category of com-mutative A-algebras.


Lemma 1.2. Let E be a projective A-module of rank n. Let E −→ Qbe a quotient module, and let Fn−1(Q) ⊆ A denote the (n−1)’st Fittingideal of Q. Then the A-module map E −→ Q is an isomorphism if andonly if Fn−1(Q) = 0 is the zero ideal. In particular we have that a ringhomomorphism A −→ A′ will factorize via A/Fn−1(Q) if and only ifE⊗

AA′ −→ Q

⊗AA

′ is an isomorphism.

Proof. The statement can be checked locally on A, hence we may as-sume that E is free of finite rank n. The result then follows from thedefinition of the Fitting ideal.

1.3. Rank of projective modules. The rank of a finitely gener-ated projective module E is constant on the connected componentsof Spec(A). We will employ the following notation. Let I ⊆ E be asubmodule of a finitely generated and projective module E. We let

Fitt(I) := FrkE−1(E/I) ⊆ A

denote the Fitting ideal we obtain by assigning on each connected com-ponent of Spec(A) the Fitting ideal Fn−1(E/I), where n is the rank ofE on that particular component.

1.4. Trace map. Let E be anA-module. The dual module HomA(E,A)we will denote by E?. The trace map is the induced A-module homo-morphism

(1.4.1) Tr : E⊗

AE? // A .

Lemma 1.5. Let I ⊆ E be an inclusion of A-modules, where E isfinitely generated and projective. Then we have the identity of idealsTr(I

⊗AE

?) = Fitt(I) in A, where Tr denotes the trace map 1.4.1.

Proof. Both the Fitting ideal and the trace map Tr commute with basechange, and we may therefore assume that E is free as an A-module.There the statement is clear.

1.6. Closed conditions. Let F be a co-variant functor from the cate-gory (or a subcategory) of A-algebras to sets. We say that F is a closedcondition on A if the functor F is representable by a quotient algebraof A.

Proposition 1.7. Let ξ : R −→ E be an A-module homomorphism.Assume that E is finitely generated and projective as an A-module.

(1) Let I ⊆ R be a submodule. Then ξ factorizing via the quotientmap R −→ R/I is a closed condition on A.

(2) Let ξ′ : R −→ E be another A-module homomorphism. Then ξbeing equal to ξ′ is a closed condition on A.


Proof. In the first situation consider the Fitting ideal Fitt(I1), of thequotient module of E given by I1 = ξ(I). In the second situationconsider the Fitting ideal Fitt(I2), of the quotient module of E givenby the A-submodule

I2 = ξ(x)− ξ′(x) | x ∈ R.

It then follows from Lemma 1.2 that assertions 1 and 2 are representedby the quotient algebras A/Fitt(I1) and A/Fitt(I2), respectively.

1.8. Algebras. By an A-algebra E, with E not necessarily commuta-tive, we mean a unital ring homomorphism c : A −→ E from a com-mutative ring A, to an associative, unital ring E, and where the imagec(A) is contained in the center of E [Bou98].

Corollary 1.9. Let ξ : R −→ E be an A-algebra homomorphism be-tween two not necessarily commutative, A-algebras. Assume that E isfinitely generated and projective as an A-module.

(1) Let I ⊆ R be a two-sided ideal. Then ξ factorizing via thequotient map R −→ R/I is a closed condition on A.

(2) Let ξ′ : R −→ E be another A-algebra homomorphism. Then ξbeing equal to ξ′ is a closed condition on A.

Proof. In both cases the question is whether a submodule of E is zeroor not, and then the statement follows from the proposition.

Proposition 1.10. Let ξi : Ri −→ E be two A-algebra homomorphismsbetween not necessarily commutative A-algebras (i = 1, 2). Assume thatE is finitely generated and projective as an A-module. Then the condi-tion that ξ1 commutes with ξ2 is a closed condition on A. In particular,if R1 and R2 are commutative, then the two A-algebra homomorphismsξi : Ri −→ E factorizing via R1

⊗AR2 is a closed condition on A.

Proof. We consider the Fitting ideal Fitt(I), where I ⊆ E is the A-submodule

I = ξ1(x)ξ2(y)− ξ2(y)ξ1(x) | x ∈ R1, y ∈ R2.

The condition that ξ1 commutes with ξ2 is that the module I is the zeromodule. Hence, by Lemma 1.2 we get that A/Fitt(I) represents thiscondition. That two A-algebra homomorphisms ξi : Ri −→ E (i = 1, 2)commute is, by the universal property of the tensor product, equivalentto have an A-algebra homomorphism R1

⊗AR2 −→ E.

2. Parametrizing algebra homomorphisms

2.1. Preliminaries. If V is an A-module, we let SA(V ) denote thesymmetric quotient algebra of the tensor algebra TA(V ) =

⊕n≥0 V

⊗n.


Let A −→ R and A −→ E be two, not necessarily commutative,A-algebras. We consider the functor HomA(R,E), that to each com-mutative A-algebra A′, assigns the set

(2.1.1) HomA-alg(R,E⊗

AA′) .

Remark 2.2. Since E is an A-algebra, the tensor product E⊗

AA′

exists and is an A′-algebra.

Proposition 2.3. Let A be a commutative ring, and let A −→ E bean A-algebra, where E is not necessarily commutative. Assume that Eis finitely generated and projective as an A-module. For any A-moduleV we have that the A-algebra SA(V

⊗AE

?) represents HomA(R,E),with R = TA(V ).

Proof. An A-algebra homomorphism uA′ : TA(V ) −→ E⊗

AA′ is de-

termined by an A′-linear map u1 : V⊗

AA′ −→ E

⊗AA

′. Since E isfinitely generated and projective, the A′-linear map u1 is equivalentwith an A′-linear map ϕ1 : V

⊗AA

′⊗A′ (E

⊗AA

′)? −→ A′. More-over, the canonical map

HomA(E,A)⊗

AA′ // HomA′(E

⊗AA

′, A′)

is an isomorphism ([Bou98, 4.3. Proposition 7]). Therefore ϕ1 corre-sponds to an A-algebra homomorphism ϕ : SA(V

⊗AE

?) −→ A′. Seee.g. [Die62], or [Bou98].

Corollary 2.4. Let A −→ R and A −→ E be two, not necessarilycommutative, A-algebras. Assume that E is finitely generated and pro-jective as an A-module. Then HomA(R,E) is representable.

Proof. Write R = TA(V )/I, for some two-sided ideal I ⊆ TA(V ), forsome A-module V . By Proposition 2.3 the functor HomA(TA(V ), E)is representable by H = SA(V

⊗AE

?). Let µ : TA(V )⊗

AH −→E⊗

AH denote the universal map. We are interested describing thoseA-algebra maps H −→ A′ that factorize via R. The result now followsby applying Corollary 1.9 (1), to the H-algebra µ (and where the ob-jects in the category are H-algebras, and the morphisms are H-algebrahomomorphisms that also are A-linear).

2.5. Non-commutative Weil restrictions. Let g : A −→ B be a ho-momorphism of commutative rings. Let f : B −→ R and µ : B −→ Ebe two A-algebra homomorphisms, where R and E are not necessarilycommutative. Thus we fix the following data

(2.5.1) Ag //

c

@@@@@@@ Bf //

µ

R

E

where c = µ g.


Definition 2.6. Fixing the data as above, we consider the functorHomB/A(R,E), from the category of commutative A-algebras to sets,that for any A-algebra A′ assigns the set of A-algebra homomorphismsξ : R −→ E

⊗AA

′ that fits into the commutative diagram

Bf //

µ

R

ξ

Ecan // E

⊗AA

′.

Remark 2.7. If E and R were commutative, then the A′-valued points ξof HomB/A(R,E) would simply have been B-algebra homomorphisms.It is important for applications that we have in mind that we do notassume that µ : B −→ E is a B-algebra.

Remark 2.8. Let A′ be an A-algebra. An A-algebra homomorphismξ : R −→ E

⊗AA

′ is the same as having an A′-algebra homomorphismξ′ : R

⊗AA

′ −→ E⊗

AA′. Thus anA′-valued point ξ of HomB/A(R,E)

is the same as having an A′-algebra homomorphism ξ′ that fits into thecommutative diagram

B⊗

AA′ f⊗1 //

µ⊗1

R⊗

AA′

ξ′xxq qq

qq

E⊗

AA′.

Remark 2.9. When A = B, then the functor HomA/A(R,E) is simplythe functor HomA(R,E) discussed in 2.1.1.

Remark 2.10. With R and commutative and with E = B, then thefunctor HomB/A(R,B) is by definition the Weil restriction RB/A(R),see e.g. [BLR90].

Proposition 2.11. Let A −→ B be a homomorphism of commutativerings. Let A −→ D and let µ : B −→ E be A-algebra homomorphisms,where E is finitely generated and projective as an A-module. ThenHomB/A(B

⊗AD,E) is representable.

Proof. By Lemma 2.4 the functor HomA(D,E) is representable. LetH be the representing object, and let u : D

⊗AH −→ E

⊗AH be the

universal map. Let µ ⊗ 1: B⊗

AH −→ E⊗

AH denote the inducedmap we get from the fixed A-algebra homomorphism µ : B −→ E. ByProposition 1.10 there is a quotient algebra H −→ H/I representingthe closed condition where µ ⊗ 1 and u commute. We have that therestrictions of the two maps to H/I factorize as

µ⊗ u : B⊗

AD⊗

AH/I −→ E⊗

AH/I .

It follows that H/I represents HomB/A(B⊗

AD,E).


Corollary 2.12. Let A −→ B −→ R be homomorphism of commuta-tive rings, and let µ : B −→ E be an A-algebra homomorphism, wherethe A-algebra E is not necessarily commutative, but is finitely generatedand projective as an A-module. Then HomB/A(R,E) is representable.

Proof. Write R as a quotient TA(V )⊗

AB/I for some A-module V ,and some two-sided ideal I ⊆ TA(V )

⊗AB. By the proposition we

have that HomB/A(TA(V )⊗

AB,E) is representable. Let H denotethe representing object, and let u : TA(V )

⊗AB

⊗AH −→ E

⊗AH

denote the universal element. The result then follows by Corollary 1.9(1).

2.13. Weil restrictions and Fitting ideals. Let A −→ B be a ho-momorphism of commutative rings, and assume that B is finitely gener-ated and projective as an A-module. Let V be an A-module. By Propo-sition (2.3) we have that SA(V

⊗AB

?) represents HomA(TA(V ), B).Let

u : TA(V ) −→ B⊗

A SA(V⊗

AB?)

denote the universal map. We get by extension of scalars an inducedB-algebra homomorphism

(2.13.1) uB : B⊗

A TA(V ) −→ B⊗

A SA(V⊗

AB?) .

For any ideal I ⊆ B⊗

A TA(V ), we let uB(I) denote the SA(V⊗

AB?)-

module generated by the image of I.

Corollary 2.14. Let g : A −→ B be a homomorphism of commutativerings, where B is finitely generated and projective as an A-module. Letf : B −→ R be homomorphisms of rings. Write R = B

⊗A TA(V )/I

as a quotient of the full tensor algebra, where V is some A-module.Then the Weil restriction RB/A(R) = HomB/A(R,B) is represented bythe A-algebra

SA(V⊗

AB?)/Fitt(uB(I)) ,

where uB is the universal map 2.13.1.

Proof. By Proposition 2.3 we have that the A-algebra SA(V⊗

AB?)

represents HomA(TA(V ), B). Since B is commutative, it follows thatSA(V

⊗AB

?) also represents HomB/A(B⊗

A TA(V ), B). Then, finally,the result follows from Corollary 1.9 (1).

Remark 2.15. The defining properties of the full tensor algebra TA(V )as well as the symmetric quotient SA(V ) are well-known, and can befound in e.g. [Die62] and [Bou98]. The situation with the Weil re-striction as in Corollary 2.14, can be found in e.g. [BLR90, Theorem7.4].


Example 2.16. We will in this example explicitly describe how to com-pute the Fitting ideals needed to describe the algebra HomA(R,E),given a presentation of R = TA(V )/I. Assume that the A-algebra E isfree as an A-module with basis e1, . . . , en, and let V be a free A-modulewith basis tii∈I . For any monomial f = ti1 ⊗ · · · ⊗ tip in TA(V ), weconsider the element fE in E

⊗A SA(V

⊗AE

?) given as

(2.16.1) fE =( n∑k=1

ek ⊗ (ti1 ⊗ e?k))· · ·( n∑k=1

ek ⊗ (tip ⊗ e?k)),

where e?1, . . . , e?n is the dual basis of E?. The element fE = u(f), where

u is the universal map

u : TA(V ) −→ E⊗

A SA(V⊗

AE?).

Describing the correspondence given by u for monomials will suffice todescribe the correspondence for arbitrary elements. We have a uniquedecomposition

fE =n∑k=1

ek ⊗ fEk ,

with fEk ∈ SA(V⊗

AE?), for k = 1, . . . , n. If we expand the defining

expression 2.16.1 of fE we get that

fE =∑

1≤ki≤ni=1,...,p

ek1 · · · ekp ⊗ (ti1 ⊗ e?k1) · · · (tip ⊗ e?kp).

Each monomial expression ek1 · · · ekp in the free A-module E, can bewritten

∑nj=1m

j(k)ej for some mj(k) ∈ A, with j = 1, . . . , n, and eachordered tipple k = k1, . . . , kp. Therefore we get that

fE =n∑j=1

ej ⊗( ∑1≤ki≤ni=1,...,p

(ti1 ⊗ e?k1) · · · (tip ⊗ e?kp) ·mj(k)

).

In particular we have that

fEj =∑

1≤ki≤ni=1,...,p

(ti1 ⊗ e?k1) · · · (tip ⊗ e?kp) ·mj(k),

for each j = 1, . . . , n. Thus, if we have a two-sided ideal in TA(V )generated by an element f , then SA(V

⊗AE

?)/(fE1 , . . . , fEn ) represents

HomA(TA(V )/(f), E).

3. Module restrictions

In this section we will introduce the module restriction, which is themain novelty of the article. From now on, the algebras A −→ B −→ Rare all assumed to be commutative.


3.1. Module structure. Recall that an A-module structure on anAbelian group M is to have a ring homomorphism ρ : A −→ EndZ(M).The image of ρ will factorize via the subring of A-linear endomorphismsEndA(M), making EndA(M) an A-algebra: The ring EndA(M) is unitaland associative, and the image of the map can: A −→ EndA(M) liesin the center.

3.2. Extension of module structures. Let g : A −→ B be a homo-morphism of rings. If M is an A-module, then a B-module structureon the set M , extending the fixed A-module structure, is a B-modulestructure on M that is compatible with the A-module structure. Thatis, a B-module structure on M extending the A-module structure isa ring homomorphism µ : B −→ EndA(M) making the commutativediagram

Bµ // EndA(M)

A

g

OOcan

::uuuuuuuuuu

.

Remark 3.3. If we have anA-algebra homomorphism µ : B −→ EndA(M),then the map will factorize via EndB(M). In particular we have thatEndB(M) is a B-algebra, but in general EndA(M) is not a B-algebra.

Definition 3.4. Let g : A −→ B and f : B −→ R be commutativealgebras and homomorphisms, and let M be a B-module. We define thefunctor M odMB→R, from the category of A-algebras to sets, by assigningto each A-algebra A′ the set

M odMB→R(A′) =

R⊗

AA′-module structures on M

⊗AA

′, extendingthe fixed B

⊗AA

′-module structure on M⊗

AA′.

We call this functor the module restriction.

Theorem 3.5. Let g : A −→ B and f : B −→ R be homomorphismof commutative rings, and let M be a B-module. Assume that M ,considered as an A-module, is projective and finitely generated. Thenthe functor M odMB→R is naturally identified with HomB/A(R,E), where

E = EndA(M), and in particular the functor M odMB→R is representable.

Proof. The B-module structure on M is given by an A-algebra homo-morphism µ : B −→ EndA(M). Let A′ be an A-algebra, and let ξ bean A′-valued point of the module restriction M odMB→R. Then we havethat the A′-valued point ξ is a A′-algebra homomorphism making thefollowing commutative diagram

(3.5.1) R⊗

AA′ ξ // EndA′(M

⊗AA

′)

B⊗

AA′

f⊗id

OO

µ⊗id // EndA(M)⊗

AA′,

ν

OO


where ν is the canonical map. Since M is finitely generated and pro-jective, the map ν is an isomorphism ([Bou98, 4.3. Proposition 7]). Inother words, we have a natural identification of functors

M odMB→R = HomB/A(R,E),

with E = EndA(M). As M is projective and finitely generated, sois E, and the statement about representability follows from Corollary2.14.

4. Module restriction in a geometric context

In this section we will set our result about module restrictions in ageometric context.

4.1. Relative rank and support. Let E be quasi-coherent sheaf ofmodules on an algebraic space X. If E is of finite type, then the supportis the closed subspace Supp(E ) ofX determined by the annihilator idealann(E ).

Definition 4.2. Let g : X −→ S be a morphism of algebraic spaces,and let E be a sheaf on X that is quasi-coherent and of finite type. Wesay that E is finite, flat of relative rank n over S, if E is flat over S,and Supp(E ) is affine over S, and the locally free OS-module g∗E hasconstant rank n.

Proposition 4.3. Let f : Y −→ X be an affine morphism of S-spaces,with X separated. Let E be a sheaf on Y that is finite, flat of relativerank n over S. Then also f∗E on X, is finite, flat of relative rank n.

Proof. We may assume that S = Spec(A), and then by restricting tothe support of E we may assume that Y = Spec(B). We have by as-sumption an inclusion B −→ EndA(M), where M is the B-module Mcorresponding to E . Since M is finitely generated as an A-module, wehave that Spec(B) −→ Spec(A) is integral. As X is separated, we get[Gro61, Proposition 6.1.5] that f : Spec(B) −→ X is integral. In par-ticular f is quasi-compact, and we let X ′ denote the scheme-theoreticimage of f . Then f : Spec(B) −→ X ′ is still integral, and surjective,hence X ′ = Spec(B′) is affine by Chevalleys Theorem [Ryd15, The-orem 8.1]. We have that f∗E is quasi-coherent and that it vanisheson the open complement X \ X ′. Since M is finitely generated as anA-module, it is also finitely generated as an B′-module. Hence we havethat f∗E is of finite type, and the support, given by the annihilatorideal, is a closed subscheme of X ′ = Spec(B). Thus the support of f∗Eis affine, and the result follows.


4.4. The stack of quasi-coherent sheaves. We denote by C ohnX/Sthe stack ([LMB00] and [Lie06]) of quasi-coherent and finite type sheaveson X, that are finite, flat and of relative rank n over S. A morphism be-tween two objects E and F in C ohnX/S(T ), where T −→ S is a schemeover S, is an OX×ST -module isomorphism ϕ : E −→ F . If T −→ S isa morphism then we have an isomorphism of stacks

(4.4.1) C ohnX×ST/T' C ohnX/S ×S T.

Corollary 4.5. Let f : Y −→ X be an affine morphism of S-spaceswith X is separated. Then we have the push-forward gives a mapf∗ : C ohnY/S −→ C ohnX/S.

Proof. Let T −→ S be a morphism with T a scheme, and let E be anelement of C ohnY/S(T ). To check that f∗E is an element of C ohnX/S(T ),we may assume that T an affine scheme, and the result follows.

Theorem 4.6. Let f : Y −→ X and g : X −→ S be morphisms ofaffine schemes X, Y and S. Then we have that the push-forward mapf∗ : C ohnY/S −→ C ohnX/S is schematically representable.

Proof. We want to see that for arbitrary scheme T , the fiber product

(4.6.1) T ×C ohnX/S

C ohnY/S

is representable by a module restriction. We may assume that T isaffine, and by 4.4.1, that T = S. Let E in C ohnX/S(S) be the element

corresponding to a given map S −→ C ohnX/S. Let S = Spec(A), X =

Spec(B), and let µ : B −→ EndA(M) be the A-algebra homomorphismcorresponding to the B-module structure on M , where the modulecorresponds to the sheaf E . Then M is projective and finitely generatedas an A-module, and we have by Theorem 3.5 the A-algebra M odMB→Rwhere Y = Spec(R). We have a natural map

α : Spec(M odMB→R) // S ×C ohnX/S

C ohnY/S

given as follows. Let u′ : Spec(A′) −→ Spec(M odMB→R) be a morphismof affine schemes over S, and let M ′ = M

⊗AA

′. By the definingproperties of the module restriction M odMB→R, the morphism u′ cor-responds to an A-algebra homomorphism ξ : R −→ EndA′(M

′) thatcomposed with B −→ R equals the induced map B −→ EndA′(M

′).Then α(u′) = (s′,FY ′ , id), where s′ : S ′ = Spec(A′) −→ S is the struc-ture map, and where FY ′ is the sheaf on Y ′ = Y ×S S ′ correspondingto the module given by ξ. The map α is a monomorphism, and weneed to see that it also is essentially surjective.

Let (s′,F , ψ) be a S ′ = Spec(A′)-valued point of the fiber product4.6.1. Then s′ : S ′ −→ S is a morphism that composed with the fixedsection of C ohnX/S is given by the induced A′-algebra homomorphism

µ ⊗ 1: B′ −→ EndA(M)⊗

AA′, where B′ = B

⊗AA

′. The sheaf


F is given by an A′-algebra homomorphism ξ′ : R′ −→ EndA′(N),for some projective A′-module N , and where R′ = R

⊗AA

′. Finallyψ : M ′ −→ N is an isomorphism of B′-modules - an in particular anisomorphism of A′-modules. We then get an induced A′-algebra iso-morphism ψ : EndA′(M

′) −→ EndA′(N) that respect the B′-modulestructures. That is we have a commutative diagram of A′-algebras

B′µ⊗1 //

f

EndA′(M′)

ψ

R′ξ′

// EndA′(N).

We then have an A′-algebra homomorphism ξ = ψ−1 ξ′ that extendsthe map µ ⊗ 1. By the defining properties of the module restrictionM odMB→R there exists a unique u′ : S ′ −→ Spec(M odMB→R) correspond-ing to ξ. Thus α(u′) is isomorphic to (s′,F , ψ), and we have shownthat α is essentially surjective.

Corollary 4.7. Let f : Y −→ X be an affine morphism of S-spaces,with X separated. Then the push-forward map

f∗ : C ohnY/S // C ohnX/S

is schematically representable.

Proof. Let E be an element of C ohnX/S(T ), where T −→ S is a scheme.We may assume that T = S, and that the base S is affine. We mayfurthermore replace X with the support of Supp(E ) = Spec(B), andthen replace Y by Spec(B′) = Y ×S X. The result then follows fromthe theorem.

Lemma 4.8. Let X −→ S be a separated map of algebraic spaces, andlet f : Y −→ X be an affine morphism. Then the natural morphism

C ohnY/S ×C ohnX/S

C ohnY/S −→ C ohnY/S ×S C ohnY/S

is a closed immersion.

Proof. We may assume that S = Spec(A) is affine. By 4.4.1 it sufficesto check closedness for S-valued points. Let E1 and E2 be two S-valuedpoints of C ohnY/S such that h∗E1 = h∗E2, where h : Y −→ S is the

structure map. Let M be the A-module of global sections Γ(S, h∗E1).Let Spec(R) ⊆ Y denote the support of E1. Then the restriction of thesheaf Ei (i = 1, 2) to Spec(R) is given by an A-algebra homomorphismξi : R −→ EndA(M). The support of f∗E1 on X is affine (Proposition4.3), say given as Spec(B) ⊆ X. Denote by ϕ : B −→ R the inducedA-algebra homomorphism. We then have that the restriction of f∗Ei to


Spec(B) is given by the composite A-algebra homomorphism (i = 1, 2)

Rξi

$$JJJJJJJJJJ

B

ϕ

OO

// EndA(M).

By Corollary 1.9 the equality of the two maps ξ1 ϕ and ξ2 ϕ is aclosed condition on A. Applying the same argument to the supportSpec(R′) ⊆ Y of E2 gives us a closed condition on S, on where thetwo sheaves f∗E1 and f∗E2 are equal. We have then shown that a S-valued point of the fiber product to the right in the statement, factoringthrough the fiber product to the left, is a closed condition on S.

5. Weil restriction revisited

We will in this section define an open subfunctor of the modulerestriction that inherits properties as etaleness. This open subfunctoris important for our application given in the last section.

Proposition 5.1. Let f : B −→ R be homomorphism of commutativeA-algebras, and let RB/A(R) = HomB/A(R,B) denote the Weil restric-tion functor. Then we have that

(1) If f : B −→ R is of finite presentation, then RB/A(R) is offinite presentation.

(2) If f : B −→ R satisfies the infinitesimal lifting property foretaleness (respectively smoothness), then the functor RB/A(R)satisfies the corresponding infinitesimal lifting property.

Proof. We will use the functorial characterization [Gro66, 8.14.2.2] toprove the first assertion. If we have a directed system of A-algebras(and A-algebra homomorphisms) Aαα∈A , then we obtain an inducedinjective map

(5.1.1) lim→

RB/A(R)(Aα) −→ RB/A(R)(lim→

(Aα)).

We need to see that this map 5.1.1 is a bijection, that is we needto check surjectivity. Let lim→Aα = A′, and let ξ′ ∈ RB/A(R)(A′).By definition we have that f ξ′ equals the canonical map B −→B⊗

AA′. Hence we have that ξ′ is a B-algebra homomorphism. As

lim→(B⊗

AAα) = B⊗

AA′, and as f : B −→ R is of finite presenta-

tion, we have a bijection

lim→

(HomB-alg(R,B⊗

AAα)) −→ HomB-alg(R,B⊗

AA′) .

Consequently ξ′ = ξα is a sequence of B-algebra homomorphismsξα ∈ RB/A(R)(Aα). Thus the map 5.1.1 is a bijection, and RB/A(R) isof finite presentation. To check the two remaining assertions, let A′ be


an A-algebra, and N ⊆ A′ a nilpotent ideal. Let ξN be an A′/N -valuedpoint of RB/A(R). We then obtain the following commutative diagram

RξN// B

⊗AA

′/N

B

f

OO

// B⊗

AA′.

OO

Now, as N ⊆ A′ is nilpotent, the kernel of the canonical map

B⊗

AA′ −→ B

⊗AA

′/N = B/NB

is nilpotent. Then if f : B −→ R has an infinitesimal lifting property,we obtain a lifting ξ′ : R −→ B

⊗AA

′ of ξN . Then ξ′ is an A′-valuedpoint of RB/A(R), and we have proved the last assertions.

Remark 5.2. It is well-known that the Weil restriction RB/A(R) inheritsproperties as etaleness, smoothness, if B −→ R is etale, respectivelysmooth. In [BLR90], e.g., these and other properties are shown for theWeil restriction, however with some assumptions that does not applyin our context.

Example 5.3. We give here an example showing that if B −→ R isetale, then HomB/A(R,E) will not necessarily satisfy the infinitesimallifting property. That fact was pointed out to us by Dan Laksov, whothereby corrected an error we had in a earlier version of this article.Consider first the matrices

x =

[0 ε0 0

]and y =

[a1 + a2ε b1 + b2εc1 + c2ε d1 + d2ε

],

where the entries of the matrices are in some ring A, where ε is anon-zero element such that ε2 = 0. Since

xy =

[a1ε b1ε0 0

]and yx =

[0 b1ε0 c1ε

],

these matrices do not in general commute. However, when we set ε = 0,the matrix x becomes the zero matrix, and the reduced matrices clearlycommute. Therefore we have the following. Let A = k[ε]/(ε2), over afield k. Let B = A[X] denote the polynomial ring in the variableX over A, and let M = A

⊕A. The matrix x gives a B-module

structure on M by sending the variable X to the matrix x. Thus wehave an A-algebra homomorphism µ : B −→ EndA(A

⊕A) = E. We

let R = A[X, Y ]/(Y 2−1), which is etale over B when the characteristicof k is different from two. We let furthermore A′ = A, and the nilpotent


ideal N = (ε) ⊆ A. We then have the following commutative diagram

A // B = A[X]f //

µ

R = A[X, Y ]/(Y 2 − 1)

ξ

EndA(A

⊕A) // Endk(k

⊕k) = E

⊗A k,

where ξ : R −→ Endk(k⊕

k) is determined by sending X to 0 and

sending Y to the endomorphism given by the matrix y =

[0 −1−1 0

].

Any lifting of y to an element in EndA(A⊕

A) is of the form

y =

[a2ε −1 + b2ε

−1 + c2ε d2ε

]with elements a2, b2, c2, d2 in k. From the considerations above we havethat no such lifting will commute with the matrix x. Therefore there ex-ist no B-algebra homomorphism ξ : R −→ EndA(A

⊕A) that extends

ξ. Thus, even if f : B −→ R is etale, the A-algebra HomB/A(R,E) isnot necessarily etale.

Example 5.4. Our next example shows that even if B −→ R is of finitepresentation, the A-algebra representing HomB/A(R,E) is not of finitepresentation. It follows, though, from the constructions given in theproofs (of the representability) that if B −→ R is of finite type, thenthe A-algebra HomB/A(R,E) is of finite type.

Let A = k[wi]i≥0 be the polynomial ring in a countable number ofvariables w1, w2, . . . over some ring k. Let M = A

⊕A be the free

A-module of rank 2. From the polynomial ring in one variable A[T ]over A, we obtain an A[T ]-module structure on M by sending T to thematrix

t =

[0 11 0

].

Let B = A[Xi]i≥1 be the polynomial ring in the variables X1, X2, . . . ,over A. For each i we consider the matrix

xi =

[wi 00 wi+1

].

Since the diagonal matrices commute, we get an B-module structure onM by sending the variable Xi to the matrix xi. One checks that the twoA-algebra homomorphisms µ : B −→ E = EndA(M) and u : A[T ] −→E commute if and only if wi = wi+1, for all i = 1, 2, . . .. Thus, theclosed condition on A over where the two maps µ and u commuteis given by A/(wi − wi+1)i≥1, which is not of finite presentation. Asfinite presentation is preserved under specialization, we get that the A-algebra HomB/A(B

⊗AA[T ], E) can not be of finite presentation either.


5.5. Isomorphic image functor. Let f : B −→ R be homomor-phisms of commutative A-algebras, and let µ : B −→ E be a homo-morphism of A-algebras, where E is not necessarily commutative. Wewill define a subfunctor

(5.5.1) ImEB=R ⊆ HomB/A(R,E).

Recall that an A′-valued point of HomB/A(R,E) is an A′-algebra ho-momorphism ξ′ that fits into the following commutative diagram

(5.5.2) R⊗

AA′

ξ′

&&MMMMMMMMMM

B⊗

AA′

OO

µ⊗1//

f⊗1

OO

E⊗

AA′.

For any A-algebra A′, we let

ImEB=R(A′) = ξ′ ∈ HomB/A(R,E)(A′) | µ⊗ 1 = ξ′ f ⊗ 1.

The functor ImEB=R parametrizes A-algebra homomorphisms R −→ E

such that the image of R equals the image of B −→ E.

Proposition 5.6. Let A −→ B −→ R be homomorphism of com-mutative rings, and let µ : B −→ E be an A-algebra homomorphismwith E not necessarily commutative. Assume that E is finitely gen-erated and projective as an A-module, and that B −→ R is of finitetype. Then the functor ImE

B=R is representable by an open subschemeof HomB/A(R,E).

Proof. By Corollary 2.12 we have that the functor HomB/A(R,E) isrepresentable, and we let H be the A-algebra representing the func-tor. Let ξ : R

⊗AH −→ E

⊗AH denote the universal element of

HomB/A(R,E), and let Im(ξ) ⊆ E⊗

AH denote the image of ξ. Letfurthermore, BH denote the image of µ ⊗ 1: B

⊗AH −→ E

⊗AH.

We have the inclusion of BH-modules BH ⊆ Im(ξ) ⊆ E⊗

AH. Anyelement x ∈ E

⊗AH gives by multiplication, an H-linear endomor-

phism on E⊗

AH. The Cayley-Hamilton theorem guarantees that theelement x will satisfy its characteristic polynomial, and consequentlythat x is integral over H. From this we deduce the following two con-sequences. Firstly, since R is finite type over B we have that Im(ξ) isa finitely generated BH-module. In particular the quotient BH-moduleIm(ξ)/BH has closed support Z ⊆ Spec(BH) given by the annihilatorideal annBH

(Im(ξ)/BH). Secondly, as BH is integral over H, the corre-sponding morphism g : Spec(BH) −→ Spec(H) is closed. Thus g(Z) isthe closed subscheme given by the ideal annBH

(Im(ξ)/BH)∩H. And asg−1(g(Z)) = Z it is clear that the open subscheme U = Spec(H)\g(Z)represents ImE

B=R.

Remark 5.7. Similar result can be found in [Ryd08].


5.8. Properties of the Isomorphic image functor. We keep thenotation introduced above (5.5), and assume that the A-algebra E isfinitely generated and projective as an A-module. Let H be the A-algebra representing HomB/A(R,E), and let ξ : R

⊗AH −→ E

⊗AH

denote the universal element. We then have the following commutativediagrams

(5.8.1) R⊗

AH// RH

""EEEEEEEEE

B⊗

AH

OO

// BH

OO

i // Im(ξ) // E⊗

AH,

where BH is the image of µ⊗1: B⊗

AH −→ E⊗

AH, where the mapi is the natural inclusion, and where RH = R

⊗B BH .

Proposition 5.9. Let A −→ B −→ R be homomorphism of commu-tative rings, and let µ : B −→ E be an A-algebra homomorphism, withE not necessarily commutative. Assume that E is finitely generatedand projective as an A-module, and that B −→ R is of finite type.Let H be the A-algebra representing HomB/A(R,E). Let BH denotethe image of the composite map µ⊗ 1: B

⊗AH −→ E

⊗AH, and let

RH = R⊗

B BH . Then we have that the functor ImEB=R equals the Weil

restriction RBH/A(RH). In particular we have that the Weil restrictionRBH/A(RH) is representable by a scheme.

Proof. By the finite type assumption of B −→ R we have by Proposi-tion 5.6 that the functor ImE

B=R is represented by an open subschemeU ⊆ Spec(H). When we restrict the diagram 5.8.1 to the open sub-scheme U ⊆ Spec(H), we get by definition that the map of OU -modules

(5.9.1) i|U : BH |U −→ Im(ξ)|U

is surjective. As U ⊆ Spec(H) is an open immersion, and in particular aflat map that preserves injectivity, we get that the restriction morphism5.9.1 is an isomorphism. Composition of the restriction of the universalmap

ξ|U : R⊗

AH|U// E⊗

AH|U

with the inverse of 5.9.1, induces a map RH|U −→ BH|U . In other wordsan U -valued point of the Weil restriction RBH/A(RH). And conversely,any A′-valued point of the the Weil restriction, composed with theinduced map i ⊗ 1: BH

⊗AA

′ −→ E⊗

AA′, is an A′-valued point of

ImEB=R.

Remark 5.10. Note that the A-algebra BH is not assumed to be finitelygenerated or projective as an A-module. Those conditions are requiredfor the representability of the Weil restriction in e.g. [BLR90].


Corollary 5.11. Let A −→ B −→ R be homomorphism of commuta-tive rings, and let µ : B −→ E be an A-algebra homomorphism, withE not necessarily commutative. Assume that E is finitely generatedand projective as an A-module, and that B −→ R is etale (smooth).Then the scheme representing the functor ImE

B=R is etale (respectivelysmooth).

Proof. If f : B −→ R is etale, or smooth, then in particular it is of finitetype. Hence, by Proposition 5.6, the functor ImE

B=R is representable bya scheme. Moreover, by Proposition 5.9 we have that ImE

B=R equals theWeil restriction RBH/A(RH), where we use the notation of Proposition5.9. As f : B −→ R is etale, or smooth, then we have by base changethat BH −→ RH is etale, or respectively smooth. The result thenfollows by Proposition 5.1.

Corollary 5.12. Let f : B −→ R be an A-algebra homomorphism.Let M be an B-module, and let E = EndA(M). Assume that M isfinitely generated and projective as an A-module. Then we have thatfor any A-algebra A′, the set of A′-valued points of ImE

B=R correspondsto R′ = R

⊗AA

′-module structures on M⊗

AA′, extending the fixed

B′ = B⊗

AA′-module structure, and such that the induced map of

supports

B′/ annB′(M⊗

AA′) −→ R′/ annR′(M

⊗AA

′)

is an isomorphism.

Proof. As M projective and finitely generated A-module, we have thatEndA(M)

⊗AA

′ = EndA′(M⊗

AA′), for any A-algebra A′. Then, for

a given A-algebra A′, we have that the kernel of µ ⊗ 1: B⊗

AA′ −→

E⊗

AA′ is the annihilator ideal. Thus, B′/ annB′(M

⊗AA

′) is theimage of µ ⊗ 1, with B′ = B

⊗AA

′. If ξ′ is an A′-valued point ofImE

B=R, then we have by definition that

B′/ annB′(M⊗

AA′) = Im(µ⊗ 1) = Im(ξ′) = R′/ annR′(M

⊗AA

′),

where R′ = R⊗

AA′.

5.13. Situation with commutative rings. We will end this sectionby showing that when the module M actually is an algebra, then ourfunctors specializes to the ordinary Weil restriction.

Proposition 5.14. Let A −→ B −→ R be homomorphisms of com-mutative rings. Let E = EndA(B), and let µ : B −→ E denote thecanonical map. We have the equality of functors

RB/A(R) = ImEB=R = HomB/A(R,E).


Proof. We first note that the map µ : B −→ EndA(B) is universallyinjective: Let A′ be an A-algebra. We have a natural maps

B⊗

AA′ µ⊗1 // EndA(B)

⊗AA

′ // EndA′(B⊗

AA′) ,

where the composite is the canonical map µ′ : B′ −→ EndA′(B′), where

B′ = B⊗

AA′. The canonical map µ′ is injective, and it follows that

B⊗

AA′ −→ EndA(B)

⊗AA

′ is injective for all A-algebras A′.We then prove the two equalities in the proposition. Let ξ′ be an

A′-valued point of HomB/A(R,E). It is readily verified that ξ′ will

factorize through Im(µ⊗ 1). The equality of ImEB=R = HomB/A(R,E)

then follows from the definitions.Since µ ⊗ 1 is furthermore injective we may identify B

⊗AA

′ withits image in E

⊗AA

′. We then have that the map ξ′ equals a mapR⊗

AA′ −→ B

⊗AA

′. That is an A′-valued point of the Weil restric-tion RB/A(R). The equality of the two functors RB/A(R) and ImE

B=R

then follows.

6. Representability of the Quot functor

In this last section we will apply our result to show that the Quotfunctor of finitely supported quotients is representable.

6.1. The stack of isomorphic supports. Let f : Y −→ X be mor-phism of S-spaces, with X separated. For each scheme T , and for anyelement E in C ohnY/S(T ) we have an induced map on supports

(6.1.1) fT | : Supp(E ) −→ Supp(f∗E ).

Let

UY→X ⊂ C ohnY/S

denote the substack, whose objects are OYT -modules E ∈ C ohnY/S(T )such that the induced map on supports is an closed immersion.

Proposition 6.2. Let f : Y −→ X be a S-morphism that is affineand of finite type, and where X is separated. Then the induced map ofstacks

UY→X −→ C ohnY/S

is a representable open immersion. If furthermore f : Y −→ X is etale(smooth), then the induced composite map

UY→X −→ C ohnY/S −→ C ohnX/S

is etale (respectively smooth).

Proof. The results follows by Proposition 5.6 and Corollary 5.11.


6.3. The Quot stack. Fix a quasi-coherent sheaf FX on an algebraicspace X −→ S. For any S-scheme T we let FXT

denote the pull-backof FX along the first projection pX : X ×S T −→ X. The T -valuedpoints of the quot stack QuotnFX/S

are all OX×ST -module morphismsq : FXT

−→ E , from FXTto a quasi-coherent sheaf E on X ×S T ,

where E is finite, flat of relative rank n over T . A morphism betweentwo objects q : FXT

−→ E and q′ : FXT−→ E ′ is an OX×ST -module

isomorphism ϕ : E −→ E ′ such that q′ = ϕ q.

Remark 6.4. Note that the maps q : FXT−→ E are not assumed to

be surjective, and in particular the T -valued points of the quot stackQuotnFX/S

are not quotients of FX . The definition of the quot stackis motivated by the definition of the Hilbert stack in [Ryd11], and in[Ols06].

6.5. Identification of pull-backs. We will in the sequel of this articlereturn to a particular situation that we describe below. Let T −→ Sbe a morphism. Then we have the following Cartesian diagram

(6.5.1) Y ×S TfT //

pY

X ×S TpX

Y

f // X.

For any sheaf FX on X there is a canonical identification between thetwo sheaves f ∗TFXT

= f ∗Tp∗XFX and p∗Y f

∗FX . We will denote both thesetwo sheaves with FYT . We have, furthermore, a natural map

c : QuotnFX/S// C ohnX/S

that takes a T -valued point of the quotient stack q : FXT−→ E to the

sheaf E .

Lemma 6.6. Let f : Y −→ X be a morphism of S-spaces, with X sep-arated. For any quasi-coherent sheaf FX on X, we have the Cartesiandiagram

QuotnFX/Sc // C ohnX/S

QuotnFY /S

f∗

OO

c // C ohnY/S.

f∗

OO

Proof. We first establish the map f∗ from QuotnFY /Sto QuotnFX/S

. Letq : FYT −→ E be a T -valued point of QuotnFY /S

. The canonical mapFXT

−→ fT∗f∗TFXT

, where we use the notation of 6.5.1, combined withthe identification f ∗TFXT

= FYT , gives the composition

(6.6.1) FXT// fT∗f

∗TFXT

= fT∗FYT // fT∗E .


By Corollary 4.5 the above sequence is a T -valued point of QuotnFX/S.

Now as we have have the map, the proof is a formal consequence ofadjunction.

6.7. The Quot functor. Let X −→ S be a morphism of algebraicspaces, and let FX be a quasi-coherent sheaf on X. We define theQuot functor Quotn

FX/Sas the functor that to each S-scheme T −→ S

assigns the set of surjective OXT-module maps q : FXT

−→ E , where Eis finite, flat and of relative rank n. Two surjective maps q : FXT

−→ Eand q′ : FXT

−→ E ′ are considered as equal if their kernels coincide assubsheaves of FXT

. This definition of QuotnFX/S

extends the one given

by Grothendieck ([Gro95], [Art69]) for projective schemes X −→ S.Let q : FXT

−→ E be a T -valued point of QuotnFX/S

. We define

ι(E ) = OX×ST/kerq.

This determines a map ι : QuotnFX/S

−→ QuotnFX ,S.

6.8. Subfunctors of the Quot functor. We will define two sub-functors of the Quot functor, and then in the next lemma relate thesesubfunctors with the stacks introduced earlier. Let f : Y −→ X be amorphism of S-spaces with X separated, and let FX be a quasi-coherentsheaf on X. We want to consider the two following subfunctors

ΩFY→X ⊆ ωFY→X ⊆ Quotn

FY /S.

We define ωFY→X as the subfunctor of QuotnFY /S

whose T -valued points

are surjective OYT -module maps q : FYT −→ E , where E is finite, flatof rank n over T , such that the induced map of OXT

-modules

f∗(q) : FXT// fT∗E

is surjective, where f∗ is the push forward map 6.6.1. And we defineΩFY→X with the further requirement that the induced map of supports

(6.1.1) is a closed immersion.

Lemma 6.9. Let f : Y −→ X be a map of S-spaces, with X separated.For any quasi-coherent sheaf FX on X we have that ωFY→X is an opensubfunctor of Quotn

FY /S. Moreover, we have the following Cartesian

diagrams

UY→X // C ohnY/Sf∗ // C ohnX/S

QuotnFY /Sf∗ //

c

OO

QuotnFX/S

c

OO

ΩFY→X

//

OO

ωFY→X//

OO

QuotnFX/S

i

OO


In particular we have that

(1) If f : Y −→ X is an affine morphism, then the induced mor-phism QuotnFY /S

−→ QuotnFX/Sis schematically representable.

(2) If f : Y −→ X is affine and of finite type, then ΩFY→X is an open

subfunctor of QuotnFY /S

, and ΩFY→X −→ Quotn

FX/Sis schemati-

cally representable.(3) Finally, if f : Y −→ X is affine and etale, then

ΩFY→X −→ Quotn

FX/S

is a schematically representable, etale morphism.

Proof. If E is a T -valued point of QuotnFY /S

, then fT∗E has affine sup-

port over T (Proposition 4.3). The surjectivity of the induced mapf∗(q) : FXT

−→ fT∗E is then an open condition on the base, since con-sidered as an OT -module E is finitely generated. This shows that ωFY→Xis open in Quotn

FY /S.

The three statements (1), (2) and (3) all will follow from Corollary4.7 and Proposition 6.2 if we prove that the diagrams are Cartesian.The upper right diagram is Cartesian by Lemma 6.6. We consider thelower right diagram.

We start by noting that there is a natural map α : ωFY→X −→ P ,where P is the fiber product in question. If q : FYT −→ E is a T -valuedpoint of ωFY→X , then by assumption f∗(q) : FXT

−→ fT∗E is surjective.So α(q) = (f∗(q), i(q), id). The map α is full and faithful.

Let (qX , s, ψ) be a T -valued point of the fiber product P , whereqX : FXT

−→ E is surjective, s : FYT −→ F is a map of OY×ST -modules,E and F are finite, flat of relative rank n over the base, and where ψ isan isomorphism of OX×ST -modules, making the commutative diagram

fT∗f∗TFXT

= fT∗FYT // fT∗F

FXT

qX //

OO

E .

ψ

OO

The pull-back f ∗T (qX) composed with the adjoint of ψ gives a map

qY : FYT = f ∗TFXT// f ∗TE // F .

We claim that qY is surjective. To see this we may assume restrictourselves to the support Y ′ = Supp(F ) of F . By definition Y ′ isaffine, and in fact integral over the base. Then f : Y ′ −→ X ×S Tis integral [Gro61, Proposition 6.1.5], and in particular affine. Thensurjectivity of qY follows from surjectivity of qX . Thus, qY is a T -valued point of Quotn

FY /S, and in fact a T -valued point of ωFY→X . We

then have that f∗(qY ) is isomorphic to (qX , s, ψ), and thus that α isessentially surjective. The leftmost diagram is proven to be Cartesianin a similar way.


Theorem 6.10. Let X −→ S be a separated morphism of algebraicspaces, and let FX be a quasi-coherent sheaf on X. For each integer n,the functor Quotn

FX/Sis representable by a separated algebraic space.

Proof. As the Quot functor commutes with base change, we may reduceto the case with the base S being an affine scheme S = Spec(A).Moreover, any T -valued point q : FXT

−→ E of QuotnFX/S

is such that

the support Supp(E ) is affine over the base. Hence the support of thequotient E is contained in an open quasi-compact U ⊆ X. Thereforewe have that

limU⊆X

open, q-compact

QuotnFU/S

= QuotnFX/S

.

Hence it suffices to show the theorem for X −→ S being quasi-compact.With X −→ S quasi-compact we can find an affine scheme Y −→ Swith an etale, affine and surjective map f : Y −→ X. With affineschemes Y −→ S we have that Quotn

FY /Sis represented by a scheme

([GLS07]). By Lemma 6.9 (2) we get that ΩFY→X is open in Quotn

FY /S,

hence a scheme. Lemma 6.9 (3) gives that the induced map

(6.10.1) ΩFY→X

// QuotnFX/S

is representable and etale.We then want to see that the map (6.10.1) is surjective. Let k be

a field and FXk−→ E be a Spec(k)-valued point of Quotn

FX/Swhere

we use the notation Xk = X ×S Spec(k). We want to show that thereexists a separable field extension k −→ L such that the correspondingSpec(L)-valued point lifts to ΩF

Y→X .The reduced support Z = | Supp(E ) | is a disjoint union of a finite

set of points, given by finite field extensions k −→ ki with i = 1, . . . ,m.Then f−1(Spec(ki)) is also a finite union of points tmi

ji=1 Spec(Lji), withki −→ Lji a finite separable field extension for ji = 1, . . . ,mi. Thereexists a finite separable field extension k −→ L such that the inducedmap ki

⊗k L −→ Lji

⊗k L splits, for all i = 1, . . . ,m and all ji =

1, . . . ,mi. Then

f−1(Z)×Spec(k) Spec(L) // Z ×Spec(k) Spec(L)

has a section, and we have that the corresponding Spec(L)-valued pointof Quotn

FX/Slifts to ΩF

Y→X . We then have proven surjectivity, and

consequently, accordingly to definition in [RG71], that QuotnFX/S

is an

algebraic space. That the algebraic space representing QuotnFX/S

is

separated follows from Lemma 4.8 and the Cartesian diagrams 6.9.

Remark 6.11. With FX = OX the structure sheaf on X, the Quotfunctor Quotn

FX/Sis the Hilbert functor HilbnX/S. The situation with

the Hilbert scheme was considered in [ES14], and a similar approach


for Hilbert stacks was done in [Ryd11]. Note that when FX = OX

then we get by Proposition 5.14 that ωFY→X = ΩFY→X , and that the

situation covered in the present article generalizes the Hilbert schemeconstruction, see Proposition 7.2 in [ES14].

Remark 6.12. The separated assumption of X −→ S is a necessarycondition for representability [LS08]. On the other hand there existexamples of separated schemes X −→ S for which the Quot functor isnot represented by a scheme [Knu71], but only an algebraic space. Thuswhen considering representability, the setting with separated algebraicspaces X −→ S is the natural one.

Remark 6.13. The above result in its generality is not covered by theresult of Artin [Art69]; we have no restriction on the base space S, andwe do not assume that X −→ S is of locally finite type.

References

[Art69] M. Artin, Algebraization of formal moduli. I, Global Analysis (Papers inHonor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 21–71. pages1, 22, 25

[Art74] , Versal deformations and algebraic stacks, Invent. Math. 27(1974), 165–189. pages 1

[BLR90] Siegfried Bosch, Werner Lutkebohmert, and Michel Raynaud, Neron mod-els, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results inMathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin,1990. pages 7, 8, 15, 18

[Bou98] Nicolas Bourbaki, Algebra I. Chapters 1–3, Elements of Mathematics(Berlin), Springer-Verlag, Berlin, 1998, Translated from the French,Reprint of the 1989 English translation. pages 5, 6, 8, 11

[Die62] Jean Dieudonne, Algebraic geometry, Department of Mathematics Lec-ture Notes, No. 1, University of Maryland, College Park, Md., 1962. pages6, 8

[ES14] Torsten Ekedahl and Roy Skjelnes, Recovering the good component of theHilbert scheme, Ann. of Math. (2) 179 (2014), no. 3, 805–841. pages 3,24, 25

[GLS07] Trond Stølen Gustavsen, Dan Laksov, and Roy Mikael Skjelnes, An ele-mentary, explicit, proof of the existence of Quot schemes of points, PacificJ. Math. 231 (2007), no. 2, 401–415. pages 24

[Gro61] A. Grothendieck, Elements de geometrie algebrique. II. Etude globale

elementaire de quelques classes de morphismes, Inst. Hautes Etudes Sci.Publ. Math. (1961), no. 8, 222. pages 11, 23

[Gro66] Alexander Grothendieck, Elements de geometrie algebrique. IV. Etude

locale des schemas et des morphismes de schemas. III, Inst. Hautes EtudesSci. Publ. Math. (1966), no. 28, 255. pages 14

[Gro95] , Techniques de construction et theoremes d’existence en geometriealgebrique. IV. Les schemas de Hilbert, Seminaire Bourbaki, Vol. 6, Soc.Math. France, Paris, 1995, pp. Exp. No. 221, 249–276. pages 1, 22

[Knu71] Donald Knutson, Algebraic spaces, Lecture Notes in Mathematics, Vol.203, Springer-Verlag, Berlin, 1971. pages 25

[Lie06] Max Lieblich, Remarks on the stack of coherent algebras, Int. Math. Res.Not. (2006), Art. ID 75273, 12. pages 12


[LMB00] Gerard Laumon and Laurent Moret-Bailly, Champs algebriques, Ergeb-nisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Mod-ern Surveys in Mathematics [Results in Mathematics and Related Areas.3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000. pages 12

[LS08] Christian Lundkvist and Roy Skjelnes, Non-effective deformations ofGrothendieck’s Hilbert functor, Math. Z. 258 (2008), no. 3, 513–519. pages25

[Ols06] Martin C. Olsson, Hom-stacks and restriction of scalars, Duke Math. J.134 (2006), no. 1, 139–164. pages 2, 21

[RG71] Michel Raynaud and Laurent Gruson, Criteres de platitude et de projec-tivite. Techniques de “platification” d’un module, Invent. Math. 13 (1971),1–89. pages 24

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