ARTIN’S AXIOMS, COMPOSITION AND MODULI SPACES

JASON STARR

Abstract. We prove Artin’s axioms for algebraicity of a stack are compatible with composition of 1-morphisms. Consequently, some natural stacks are algebraic. One of these is a common generalization of Vistoli’s Hilbert stack and the stack of branchvarieties defined by Alexeev and Knutson.

1. Introduction

Many moduli functors in algebraic geometry, properly interpreted, are algebraic stacks, also called Artin stacks (please note, following Artin, we do not assume diagonal morphisms are quasi-compact). In [Art74], Artin gave axioms for algebraicity involving deformation-obstruction theory and compatibility with completion.

There exist natural stacks Y where the completion axiom fails but all other axioms hold. Sometimes there exists a stack X for which the completion axiom holds and a 1-morphism f : X → Y representable by algebraic stacks. Intuitively X should satisfy all Artin’s axioms, and thus be algebraic. In other words, Artin’s axioms should be compatible with composition of 1-morphisms of stacks. The difficulty is that, given a relative obstruction theory for f and an obstruction theory for Y, there may exist no “extension” obstruction theory for X.

Existence of an extension obstruction theory is circumvented using Propositions 2.9 and 2.10. The main result is the following version of [Art74, Theorem 5.3].

Proposition 1.1. Let S be an excellent scheme and let f : X → Y be a 1-morphism of limit preserving stacks in groupoids over (Aff/S) for the étale topology. Let O_{Y} be an obstruction theory for Y and let O_{f} be a relative obstruction theory for f. The stack X is algebraic if,

(1) Conditions [Art74, (S1,2)] hold for deformations and automorphisms of Y and X (or equivalently, Y and f).

(2) For any complete local O_{S}-algebra Â with residue field of finite type over S, the canonical map

X (Spec Â) → \underleftarrow{lim} X (Spec Â/𝔪^{n})

is faithful, and has a dense image, i.e., the projection to X (Spec Â/𝔪^{n}) is essentially surjective for every n.

(3) Automorphisms, deformations and obstructions of Y and f satisfy the conditions in Notations 2.4 and 2.21.

(4) If the object a_{0} of X (Spec A_{0}) is algebraic, and if ϕ is an automorphism of a_{0} inducing the identity in X (Spec k(y)) for a dense set of finite type points y of Spec A_{0}, then ϕ equals Id_{a_{0}} on a non-empty open subset of Spec A_{0}.

A consequence is algebraicity of some natural stacks.

Proposition 1.2. Let S be an excellent scheme and let Y be a limit preserving algebraic stack over (Aff/S) with finite diagonal. The stack H parametrizing triples (X, L, g) of a proper algebraic space X, a 1-morphism g : X → H, and an invertible, g-ample O_{X}-module L is a limit preserving algebraic stack over (Aff/S) with quasicompact, separated diagonal.

This stack H is a common generalization of Vistoli’s Hilbert stack, [Vis91], and the stack of branchvarieties defined by Alexeev and Knutson, [AK06]. The proof of Proposition 1.2 gives a new proof of algebraicity in each of these special cases.

Acknowledgments. I thank Martin Olsson for pointing out [Ols04, Appendix A]. I thank Valery Alexeev and Allen Knutson for pointing out that Proposition 1.2 is not a formal consequence of Vistoli’s Hilbert stack.

2. Artin’s axioms and obstruction theory

All hypotheses regarding obstructions in [Art74] trace back to the proofs of [Art74, Proposition 4.3 and Theorem 4.4]. This section describes lifting properties, why the lifting properties are compatible with composition, how Artin uses these lifting properties to prove openness of versality, and finally, how Artin uses obstruction theory to prove the lifting properties.

2.1. Relative deformation situations. Relative deformation situations and obstruction theories are studied in [Ols04, Appendix A]. Following are the basic definitions.

Let S be a locally Noetherian algebraic space (for later applications, it will be necessary that S is excellent). An infinitesimal extension is a surjective homomorphism of Noetherian O_{S}-algebras with nilpotent kernel, A′ ⟶^{q} A. An extension pair is a pair of infinitesimal extensions (A′ ⟶^{q} A ⟶^{q_{0}} A_{0}) such that A_{0} is reduced and the kernel M of q is annihilated by the kernel of q_{0} ∘ q.

A morphism of infinitesimal extensions (u′, u) is a Cartesian diagram of O_{S}- algebras,

A′ ⟶^{q_{A}} A
u′↓  ↓^{u}
B′ ⟶^{q_{B}} B

whose rows are infinitesimal extensions. Morphisms of extension pairs (u′, u, u_{0}) are defined analogously.

Let X be a stacks in groupoids over (Aff/S) for the étale topology. An infinitesimal extension over X is a datum (q, a) of a reduced infinitesimal extension A ⟶^{q_{0}} A_{0} and an object a of X (Spec A). A morphism of infinitesimal extensions over X,

(u, u_{0}, ϕ) : (A ⟶^{q_{A,0}} A_{0}, a) → (B ⟶^{q_{B,0}} B_{0}, b),

is a morphism (u, u_{0}) of infinitesimal extensions together with a morphism ϕ : b → a in X mapping to u^{∗} : Spec B → Spec A.

Definition 2.1. A deformation situation over X is a datum (A′ ⟶^{q} A ⟶^{q_{0}} A_{0}, a) of an extension pair (q, q_{0}) and an object a of X (Spec A).

A morphism of deformation situations over X is a datum (u′, u, u_{0}, ϕ) of a morphism (u′, u, u_{0}) of extension pairs together with a morphism ϕ : b → a in X mapping to u^{∗}.

By the axioms for a stack, given a deformation situation (q, q_{0}, a) and a morphism of extension pairs (u′, u, u_{0}), there exists a morphism ϕ so that (u′, u, u_{0}, ϕ) is a morphism of deformation situations over X. A clivage normalisé determines a choice of ϕ, cf. [Gro03, Définition VI.7.1]. From this point on, a clivage normalisé is assumed given. The image of (u′, u, u_{0}, ϕ) is called the base change of (q, q_{0}, a) by (u′, u, u_{0}).

Let f : X → Y be a 1-morphism of stacks in groupoids over (Aff/S). An infinitesimal extension over f is a datum (q̃, a) of a morphism q̃ of Y mapping to a reduced infinitesimal extension A ⟶^{q_{0}} A_{0} and an object a of X (Spec A) mapping to the target of q̃. Equivalently, it is a 1-morphism Spec A → Y and an infinitesimal extension over the 2-fibered product Spec A ×_{Y} X. Morphisms of infinitesimal extensions over f, deformation situations over f, morphisms of deformation situations over f, and base change are defined analogously.

If Y satisfies the Schlessinger-Rim criterion [Art74, (S1)], there are well-defined relative analogues of the Schlessinger-Rim criterion for f. Lemma 2.2. [Ols04, §A.14] Let Y be a stack in groupoids over (Aff/S) satisfying [Art74, (S1)]. Let f : X → Y be a 1-morphism of stacks in groupoids over (Aff/S).

(i) The stack in groupoids X satisfies [Art74, (S1)] if and only if f satisfies the relative analogue of [Art74, (S1)].

(ii) Let g : Z → Y be a 1-morphism of stacks in groupoids. If X and Z satisfy [Art74, (S1)], then also the 2-fibered product X ×_{Y} Z satisfies [Art74, (S1)].

Proof. This is largely verified in [Ols04, §A.14]. The details are left to the reader. □

Let f : X → Y be a 1-morphism of stacks in groupoids over (Aff/S), each of which satisfies [Art74, (S1)]. For each reduced O_{S}-algebra A_{0} and object a_{0} of X (Spec A_{0}), [Ols04, §A.15] gives a natural 7-term exact sequence of automorphism and deformation groups. This implies the following. Lemma 2.3. [Ols04, §A.15]

(i) Assuming Y satisfies [Art74, (S2)], X satisfies [Art74, (S2)] if and only if f satisfies the relative analogue of [Art74, (S2)].

(ii) If g : Z → Y is a 1-morphism of stacks in groupoids such that X , Y and Z each satisfy [Art74, (S1,2)], then also X ×_{Y} Z satisfies [Art74, (S1,2)].

Proof. This is largely verified in [Ols04, §A.15]. The details are left to the reader.

□

Notation 2.4. [Art74, (4.1)], [Ols04, §A.11] There are relative analogues of the conditions on automorphisms and deformations. (The relative analogues of conditions on obstructions are stated in Notation 2.21.) (4.1.i) The functors, Aut_{f}, resp. D_{f}, are compatible with étale localization: For

every morphism (ũ, ũ_{0}, ϕ) of infinitesimal extensions over f, if A_{0} is a finitetype O_{S}-algebra and u is étale, then the following associated natural transformations of functors are isomorphisms,

Aut_{f,a_{0}}(A_{0} + M) ⊗_{A_{0}} B_{0} → Aut_{f,b_{0}}(B_{0} + M ⊗_{A_{0}} B_{0}),
D_{f,a_{0}}(M) ⊗_{A_{0}} B_{0} → D_{f,b_{0}}(M ⊗_{A_{0}} B_{0}).

(4.1.ii) The functors Aut_{f} and D_{f} are compatible with completions: For every finite-type O_{S}-algebra A_{0} and every maximal ideal 𝔪 of A_{0}, the following natural maps are isomorphisms,

Aut_{f,a_{0}}(A_{0} + M) ⊗_{A_{0}} Â_{0} → \underleftarrow{lim} Aut_{f,a_{0}}((A_{0} + M)/𝔪^{n}),
D_{f,a_{0}}(M) ⊗_{A_{0}} Â_{0} → \underleftarrow{lim} D_{f,a_{0}}(M/𝔪^{n}M).

(4.1.iii) For every infinitesimal extension (q̃_{0}, a) over f with A a finite type O_{S}- algebra, there is an open dense set of points of finite type p ∈ Spec A_{0} so that the following maps are isomorphisms,

Aut_{f,a_{0}}(A_{0} + M) ⊗_{A_{0}} k(p) → Aut_{f,a_{0}}(k(p) + M ⊗_{A_{0}} k(p)),
D_{f,a_{0}}(M) ⊗_{A_{0}} k(p) → D_{f,a_{0}}(M ⊗_{A_{0}} k(p)).

Lemma 2.5. With notation as in Lemma 2.2 and Lemma 2.3, assume X and Y satisfy [Art74, (S1,2)] and assume automorphisms, respectively deformations, of Y satisfy [Art74, (4.1)]. Automorphisms, resp. deformations, of X satisfy [Art74, (4.1)] if and only if automorphisms, resp. deformations, of f satisfy 2.21. Also, if f : X → Y and g : Z → Y are 1-morphisms of stacks in groupoids over (Aff/S) whose automorphisms, resp. deformations, satisfy [Art74, (S1,2),(4.1)], then also the automorphisms, resp. deformations, of X ×_{Y} Z satisfy [Art74, (S1,2),(4.1)].

Proof. This also largely follows from [Ols04, §A.15]. The details are left to the reader. □

2.2. Lifting properties. Let (B′ ⟶^{q} B ⟶^{q_{0}} B_{0}) be an extension pair and let (q̃, q̃_{0}, b_{X}) be a deformation situation over f mapping to (q, q_{0}). The deformation situation is algebraic, resp. reduced, integral, if B_{0} is finite type, resp. reduced, integral.

Localization. The lifting properties involve localization. Let q : B′ → B be an infinitesimal extension. Images and inverse images of multiplicative systems under q are again multiplicative. Let S′ be a multiplicative subset of B′ and let S′_{q} be the multiplicative subset q^{−1}(q(S′)). The localization of B′ with respect to S′ equals the localization of B′ with respect to S′_{q}. Moreover, S^{−1}B′ → q(S)^{−1}B is an infinitesimal extension, and the associated graded pieces of the kernel are the localizations q(S)^{−1}(N^{i}/N^{i+1}). Therefore, the localizations of B′ are in 1-to-1 bijection with the localizations of B. Moreover, a localization of B is finitely generated if and only if the associated localization of B′ is finitely generated.

Let (B′ \underrightarrow{q̃_{B}} B \underrightarrow{q̃_{B,0}} B_{0}, b_{X}) be an integral deformation situation. Let A_{0} be the fraction field of B_{0}, and let A and A′ be the associated localizations of B and B′ respectively. Let A′ ⟶^{q_{A}} A ⟶^{q_{A,0}} A_{0} denote the associated extension pair. Denote by (a′_{Y} \underrightarrow{q̃_{A}} a_{Y} ⟶^{q_{A,0}} a_{Y,0}, a_{X}) the base change over A′, etc., of the deformation situation (q̃_{B}, q̃_{B,0}, b).

Definition 2.6. A generic lift of the deformation situation (q̃_{B}, q̃_{B,0}, b_{X}) is a morphism a′_{X} ⟶^{r_{A}} a_{X} in X over q̃_{A}. An integral lift of the generic lift is a morphism b′_{X ,new} ⟶^{r_{B}} b_{X} in X satisfying the following conditions.

(i) The image of r_{B} in (Aff/S) is an infinitesimal extension B′_{new} ⟶^{q_{B,new}} B

sitting between B′ and q_{A}^{−1}(B).
(ii) The image q̃_{B,new} of r_{B} in Y is the base-change of q̃_{B}.

(iii) And the base-change of r_{B} by (q_{B,new}, q_{B,0}) → (q_{A}, q_{A,0}) is r_{A}.

Pushouts of M. Given a deformation situation (q̃, q̃_{0}, b_{X}) over f mapping to an extension pair (B′ ⟶^{q} B ⟶^{q_{0}} B_{0}), for every surjection of B_{0}-modules, M → N, there is a surjection of O_{S}-algebras B′ ⟶^{u} B′_{N} whose kernel is the kernel of M → N. This gives a morphism of deformation situations,

(u′, Id, Id, Id) : (q̃, q̃_{0}, b_{X}) → (q̃_{N}, q̃_{0}, b_{X}),

such that the image of q̃_{N} is the extension B′_{N} → B and the image of u′ is u.

Let 𝔞 be a radical ideal in B_{0}. Let R denote the semilocalization of B_{0} at the generic points of 𝔞. Assume M is a finite type B_{0}/𝔞-module. A localization A_{0} of B_{0} is 𝔞-generic if Spec A_{0} ∩ Spec (B_{0}/𝔞) is dense in Spec (B_{0}/𝔞), i.e., A_{0} is isomorphic to a B_{0}-subalgebra of R. An 𝔞-generic quotient of M is a pair (A_{0}, N) of an 𝔞-generic localization A_{0} and a surjection M ⊗_{B_{0}} A_{0} → N. It is finite type if A_{0} is a finite type B_{0}-algebra. It is projective if N is a projective A_{0}/𝔞A_{0}- module. It is extending if for the associated deformation situation (q̃_{A,N}, q̃_{A,0}, a_{X}) over (A′_{N}, A, A_{0}), there is a lifting of q̃_{A,N} to X.

Definition 2.7. Let (q̃, q̃_{0}, b_{X}) be a deformation situation over f whose kernel M is a finite type B_{0}/𝔞-module for a radical ideal 𝔞. A generic extender is a finite type, projective, extending, 𝔞-generic quotient (A_{0}, N) such that for every finite type, projective, 𝔞-generic quotient (C_{0}, P), the quotient P ⊗_{C_{0}} R factors through the quotient N ⊗_{A_{0}} R.

A generic extender is compatible with étale extension if for every étale homomorphism v : B′ → B′_{ét}, the pair (A_{0} ⊗_{B_{0}} B_{ét,0}, N ⊗_{B_{0}} B_{ét,0}) is a v(𝔞)-generic extender for the base change of the deformation situation to B′_{ét}.

A generic extender is compatible with closed points if for every closed point A_{0} → k(y) of Spec A_{0} and every surjection M ⊗_{B_{0}} k(y) → N_{y}, there exists a lift q̃_{X ,N_{y}} of the base change of q̃ to A′_{N_{y}} if and only if N_{y} is a quotient of N ⊗_{A_{0}} k(y).

Lemma 2.8. Assume X and Y each satisfy [Art74, (S1)]. Then for every deformation situation and radical ideal 𝔞 there exists a generic extender.

Proof. For every finite type, projective, extending, 𝔞-generic quotient (C_{0}, P), consider the quotient M ⊗_{B_{0}} R → P ⊗_{C_{0}} R. This system of quotients has an inverse limit. Because M ⊗_{B_{0}} R has finite length, the inverse limit is equal to the inverse limit of a finite subsystem. Thus, it suffices to prove the following. For every pair of finite type, projective, extending, 𝔞-generic quotients (A_{0,1}, N_{1}) and (A_{0,2}, N_{2}), there exists a finite type, projective, extending, 𝔞-generic quotient (A_{0}, N) such that both quotients N_{1} ⊗_{A_{0,1}} R and N_{2} ⊗_{A_{0,2}} R factor through N ⊗_{A_{0}} R.

Replace B_{0} by the finite type, 𝔞-generic localization A_{0,1} ⊗_{B_{0}} A_{0,2} and replace B and B′ by the associated localizations. Consider the induced map M → N_{1} ⊕ N_{2}. After a further finite type, 𝔞-generic localization, the cokernel P is a projective B_{0}/𝔞-module. Denote by N the image of M in N_{1} ⊕ N_{2}. Of course B′_{N} equals the fiber product B′_{N_{1}} ×_{B′_{P}} B′_{N_{2}}.

Let D denote the functor D_{f,q̃_{0},b_{X ,0}}. For i = 1, 2, let q̃_{X ,i} : b_{X ,i} → b_{X} be a lift to X of the base change of q̃ over B′_{N_{i}}. The base change of b_{X ,i} over B′_{P} is a lift to X of the base change of q̃ over B′_{P}. Therefore the 2 base changes differ by an element d in D(P).

Because N_{1} ⊕ N_{2} → P is a surjective map of B_{0}/𝔞-modules whose image is a projective, it is split. Therefore D(N_{1} ⊕ N_{2}) → D(P) is surjective. So there exist elements d_{i} of D(N_{i}) for i = 1, 2 such that d is the image of (d_{1}, d_{2}). After translating b_{X ,1}, resp. b_{X ,2}, by d_{1}, resp. d_{2}, the base changes to B′_{P} agree. Therefore by [Art74, (S1)], there exists an element b_{X} over B′_{N} whose base change to B′_{N_{i}} equals b_{X ,i} for i = 1, 2. □

2.3. Lifting properties and compositions. Let e : W → X and f : X → Y be 1-morphisms of stacks in groupoids over (Aff/S).

Proposition 2.9. If for both e and f generic lifts of integral, algebraic deformation situations have integral lifts, then the same holds for f ∘ e.

Proof. Given an integral, algebraic deformation situation (q̃_{Y,B}, q̃_{Y,B,0}, b_{W}) over f∘e and a generic lift a′_{W} → a_{W}, then e(a′_{W}) → e(a_{W}) is a generic lift of the integral, algebraic deformation situation (q̃_{Y,B}, q̃_{Y,B,0}, e(b_{W})) over f. By hypothesis, this has an integral lift r_{f}. After replacing B′, this integral lift gives a deformation situation (r_{f}, q̃_{X ,B,0}, b_{W}) over e. And a′_{W} → a_{W} is a generic lift. By hypothesis, there exists an integral lift r_{e}. This is also an integral lift for the original generic lift over f ∘ e. □

Proposition 2.10. Assume each of W, X and Y satisfy [Art74, (S1,2)] and their deformations satisfy [Art74, (4.1.i),(4.1.iii)]. If for both e and f there exist generic extenders of algebraic deformations compatible with étale extensions and with closed points, the same holds for f ∘ e.

Proof. Let (q̃_{Y}, q̃_{Y,0}, b_{W}) be a deformation situation over f ∘ e whose associated extension pair is B′ ⟶^{q} B ⟶^{q_{0}} B_{0}. Assume the kernel M of q is a module over B_{0}/𝔞 for a radical ideal 𝔞. By Lemma 2.8, there exists a generic extender (A_{0}, N_{f∘e}) for the deformation situation over f ∘ e. The problem is to prove the generic extender is compatible with étale extensions and closed points. The main issue is to find an appropriate finite type, 𝔞-generic localization of A_{0}.

First of all, by [Art74, (4.1.iii)], there is a finite type, 𝔞-generic localization of B_{0} so that D_{f}(M) and D_{e∘f}(M) are compatible with closed points. Replace B_{0} by this localization and replace B and B′ by the associated localizations. Denote by q̃_{W,N_{f∘e}} a lift to W of the base change of q̃_{Y} to B′_{N_{f∘e}}. Replace B_{0} by A_{0} and replace B and B′ by the associated localizations. By hypothesis there exists a generic extender (A_{0}, N_{f}) for f compatible with étale extensions and with closed

points. Replace B_{0} by A_{0}, etc. Let b′_{X} \underrightarrow{q̃_{X}} b_{X} denote a lift to X of the base change of q̃_{Y} to B′_{N_{f}}. Because the base change of q̃_{Y} to B′_{N_{f∘e}} lifts to e(q̃_{W,N_{f∘e}}), N_{f∘e} is a quotient of N_{f}.

“Calibrate” the lifts q̃_{W,N_{f∘e}} and q̃_{X} as follows. Denote by q̃_{X ,N_{f∘e}} the base change of q̃_{X} to B′_{N_{f∘e}}. The difference of e(q̃_{W,N_{f∘e}}) and q̃_{X ,N_{f∘e}} is an element d of D_{f,q̃_{W,0},b_{X ,0}}(N_{f∘e}). After a further finite type, 𝔞-generic localization, the surjection N_{f} → N_{f∘e} is split. Therefore there exists an element d′ of D_{f,q̃_{W,0},b_{X ,0}}(N_{f}) mapping to d. Replace q̃_{X} by its translation by d. Then e(q̃_{W,N_{f∘e}}) equals q̃_{X ,N_{f∘e}}.

Claim 2.11. It suffices to prove the proposition after replacing M by the quotient N_{f}, replacing B′ by B′_{N_{f}}, etc.

For every étale extension B′ → B′_{ét} and every projective, extending quotient M ⊗_{B_{0}} B_{ét,0} → N_{ét} for f ∘ e over B′_{ét}, also N_{ét} is extending for f over B′_{ét}. By hypothesis, N_{f} is compatible with étale extension. Thus M_{ét} is a quotient of N_{f}.

Similarly, for every closed point y of Spec B_{0}, and every surjection M ⊗_{B_{0}} k(y) → N_{y}, if the base change of q̃_{Y} to B′_{N_{y}} lifts to W, then it also lifts to X. By hypothesis, N_{f} is compatible with closed points. Thus N_{y} is a quotient of N_{f}. This proves Claim 2.11. Therefore, without loss of generality, replace M by the quotient N_{f}, replace B′ by B′_{N_{f}}, etc.

Here is the problem: the lift q̃_{X} is not unique. For every lift, by hypothesis, there is a finite type, 𝔞-generic localization and a generic extender over this localization which is compatible with etale extensions and finite points. However, the finite type localization would appear to depend on the choice of lift. As there may be infinitely many lifts, it is a priori possible the coproduct of these localizations is not finite type.

Here is the remedy. Let D denote the module D_{f,q̃_{Y,0},b_{X ,0}}(B_{0}/𝔞). Localize so that D is a finite projective B_{0}/𝔞-module. Denote the module Hom_{B_{0}}(D, B_{0}/𝔞) by D^{∨}. Then D_{f,q̃_{Y,0},b_{X ,0}}(D^{∨}) is canonically isomorphic to Hom_{B_{0}}(D, D). Let β denote the object of X over B_{0} + D^{∨} corresponding to Id_{D}. Of course (B_{0} + D^{∨}) ×_{B_{0}} B′

equals B′ + D^{∨}. By [Art74, (S1)], there exists an extension b′_{X ,D} \underrightarrow{q̃_{X ,D}} b_{X} of q̃_{X} over B′ + D^{∨} such that the image over B′ is q̃_{X} and the image over B_{0} + D^{∨} is β.

Here is the point: for every lift b′_{X ,1} \underrightarrow{q̃_{X ,1}} b_{X} of q̃_{Y}, the difference of b′_{X ,1} and b_{X ,1} is an element of D_{f,q̃_{Y,0},b_{X ,0}}(M), i.e., an element d of Hom_{B_{0}}(D^{∨}, M). There is a unique surjection of B′-algebras, B′ + D^{∨} → B′ whose restriction to D^{∨} is d : D^{∨} → M. The image of b′_{X ,D} equals b′_{X ,1}.

By hypothesis, for the ideal 𝔞 the deformation situation (q̃_{X ,D}, q̃_{X} , b_{W}) over e has a generic extender (A_{0}, N_{D}) compatible with étale extensions and closed points. Let D^{∨} → E be the fiber coproduct of M ⊕ D^{∨} → N_{D} and the projection pr_{D^{∨}} : M ⊕ D^{∨} → D^{∨}. After a further finite type, 𝔞-generic localization, E is projective.

Claim 2.12. The quotient M ⊕ D^{∨} → N_{D} is the direct sum of the quotient M → N_{f∘e} and the quotient D^{∨} → E.

The claim holds if and only if it holds after a finite type, 𝔞-generic localization. Let K_{D} denote the kernel of the surjection M ⊕ D^{∨} → N_{D}. Let K_{f∘e} denote the

kernel of the surjection M → N_{f∘e}. The composition of pr_{M} : M ⊕ D^{∨} → M and M → N_{f∘e} is extending. Therefore, after a further 𝔞-generic localization, it factors through N_{D}, i.e., the image pr_{M}(K_{D}) is contained in K_{f∘e}. On the other hand, the quotient M/pr_{M}(K_{D}) is extending. After a finite type, 𝔞-generic localization, it is projective.

Because M → M/pr_{M}(K_{D}) is extending and locally free, after a further localization it is a quotient of N_{f∘e}, i.e., K_{f∘e} is contained in pr_{M}(K_{D}). Therefore pr_{M}(K_{D}) equals K_{f∘e}.

Denote by K′_{D} the kernel of the surjection K_{D} → K_{f∘e}. There is a projection pr_{K,D^{∨}} : K′_{D} → D^{∨}. After a finite type, 𝔞-generic localization, the cokernel E is projective. The composition K_{D} ⟶^{pr_{D^{∨}}} D^{∨} → E factors through K_{D} → K_{f∘e}. Denote this morphism by δ : K_{f∘e} → E. The claim is equivalent to the vanishing of δ.

By way of contradiction, assume δ is not zero. Then there exists a surjection E → Q such that K_{f∘e} → Q is surjective and Q is a projective B_{0}/𝔞-module of length 1. Denote by K_{f∘e,Q} the kernel of K_{f∘e} → Q. Take the quotient of M by K_{f∘e,Q}. There is an induced isomorphism K_{f∘e}/K_{f∘e,Q} → Q. Inverting this isomorphism and composing with the inclusion gives a map of B_{0}/𝔞-modules, i : Q → M/K_{f∘e,Q}. Define i′ : Q → M to be any B_{0}/𝔞-morphism lifting i. Define ϕ : D^{∨} → M to be the composition of D^{∨} → Q and the negative of i′. Define (Id_{M}, ϕ) : M ⊕ D^{∨} → M to be the induced surjection. The induced morphism (Id_{M}, ϕ) : K_{D} → M annihilates K′_{D}. So there is an induced map from K_{f∘e} to M. The restriction to K_{f∘e,Q} is the inclusion. So there is an induced map from K_{f∘e}/K_{f∘e,Q} to M/K_{f∘e,Q}. But, by construction, this is the zero map. Therefore (Id_{M}, ϕ)(K_{D}) is strictly contained in K_{f∘e}. The quotient M → M/(Id_{M}, ϕ)(K_{D}) is extending, but does not factor through M → N_{f∘e}. This contradiction proves that δ is zero, i.e., it proves Claim 2.12.

Claim 2.13. After this sequence of localizations, the generic extender M → N_{f∘e} for f ∘ e is compatible with étale extensions and with closed points.

Étale extensions. By hypothesis, the generic extender M ⊕ D^{∨} → N_{D} is compatible with étale extensions. By Claim 2.13, this surjection is split. It follows that M → M_{f∘e} is also compatible with étale extensions.

Closed points. Let B_{0} → k(y) be a closed point, let M ⊗_{B_{0}} k(y) → N_{y} be a surjection, and let q̃_{W,y} be a lift to W of the base change of q̃_{Y} to B′_{N_{y}}. Because D_{f}(M) is compatible with closed points, there is a morphism d : D^{∨} → M so that e(q̃_{W,y}) is the base change of q̃_{X ,D} associated to the composition of (Id_{M}, d) : M ⊕ D^{∨} → M and M → N_{y}. Because M ⊕ D^{∨} → N_{D} is compatible with closed points, N_{y} is a quotient of N_{D}, i.e., the kernel of M → N_{y} contains (Id_{M}, d)(K_{D}). Because K_{D} is the direct sum K_{f∘e} ⊕ K′_{D}, in particular (Id_{M}, d)(K_{D}) contains (Id_{M}, d)(K_{f∘e} ⊕ 0), i.e., K_{f∘e}. Thus N_{y} factors through M → N_{f∘e}. Therefore M → N_{f∘e} is compatible with closed points. □

2.4. Openness of versality. Let X be a stack in groupoids over (Aff/S) for the étale topology.

Definition 2.14. The stack X satisfies openness of versality if for every finite type O_{S}-algebra R and every object v of X (Spec R), if v is formally versal at p, then there is an open neighborhood of p in Spec R on which v is formally smooth. Openness of versality is also compatible with étale extensions if for every étale morphism e : Spec R^{∗} → Spec R, for every point p^{∗} of Spec R^{∗}, if v is formally versal at e(p^{∗}), then e^{∗}v is formally smooth on an open neighborhood of p^{∗} in Spec R^{∗}. Theorem 2.15. [Art74, Proposition 4.3 and Theorem 4.4] Assume X is limit preserving, deformations of X satisfy [Art74, (S1,2), (4.1)], every generic lift of an integral, algebraic deformation situation over X has an integral lift, and every algebraic deformation situation over X has a generic extender compatible with étale extensions and with closed points, cf. Definitions 2.6 and 2.7. Then X satisfies openness of versality compatibly with étale extensions.

Remark 2.16. The hypotheses of Theorem 2.15 do not include existence or properties of an obstruction theory for X. In practice, an obstruction theory is used to verify existence of integral lifts and generic extenders, etc.

Proof. The entire proof will not be repeated. We only indicate the modifications necessary to replace existence and properties of obstruction theory by the hypotheses above.

First of all, the proof of [Art74, Proposition 4.3] uses obstructions only at the end of the proof. In fact, obstructions are used precisely to prove existence of generic extenders compatibly with étale extensions.

Next, in the proof of [Art74, Theorem 4.4], obstructions are used to prove formal versality is stable under generization. The arguments on [Art74, p. 176] use obstructions precisely to deduce existence of an integral lift b′ of the generic lift a′ of the deformation situation (B′ ⟶^{q} B ⟶^{q_{0}} B_{0}, b). By [Art74, Theorem 3.3], formal versality of v at p implies versality of v at p. Thus, there is a morphism from R to the Henselization of the localization of B′ at p inducing a lift from v to the “Henselization” of b′ at p, compatibly with the map from v to the “Henselization” of b at p. Because X is limit preserving, this morphism factors through an étale extension B′ → B′_{new}. Replace the original deformation situation by the base change by this étale extension.

Because the base change of v by R → B′ agrees with b′ after localizing at p, and by the same sort of arguments as above, after replacing B′ by a subring B′[t^{−1}M] of A′, the base change of v equals b′. So the composition of R → B′ with the localization B′ → A′ gives a morphism v → a′ making the diagram commute. Therefore v is formally versal at the point x.

The final use of obstructions in the proof of [Art74, Theorem 4.4] is at the bottom of p. 177 and the top of p. 178. Here obstructions are used to construct a generic extender (A_{0}, N) compatibly with closed points. Replace B_{0} by A_{0}, and replace B, B′ by the associated localizations. By the hypothesis on S, the quotient N has nonzero fiber N ⊗_{B_{0}} k(y) for a dense set of points. Therefore the generic fiber of N is nonzero. Let M ⟶^{ϕ} B_{0} be any surjection factoring through M → N. Denote by B′ → B^{∗} the associated surjection. This has the desired property. The rest of the proof goes through precisely as in [Art74]; there is no further use of obstructions. □

Let there be given a sequence of 1-morphisms of stacks in groupoids over (Aff/S),

X_{n} ⟶^{f_{n−1}^{n}} X_{n−1} ⟶^{f_{n−2}^{n−1}} … ⟶^{f_{1}^{2}} X_{1} ⟶^{f_{0}^{1}} (Aff/S).

Assume each X_{i} satisfies [Art74, (S1,2)].

Corollary 2.17. For every i = 1, …, n, assume X_{i} is limit preserving. Also, for every i = 1, …, n, assume f_{i−1}^{i} satisfies the relative version of [Art74, (S1,2)] and the conditions of Notation 2.4. Finally, for every i = 1, …, n, assume every generic lift of an integral, algebraic deformation situation over f_{i−1}^{i} has an integral lift, and every algebraic deformation situation over f_{i−1}^{i} has a generic extender compatible with étale extensions and with closed points. Then every X_{i} satisfies openness of versality compatibly with étale extensions.

Proof. This follows immediately from Lemmas 2.2, 2.3, 2.5 and Propositions 2.9, 2.10. □

2.5. Artin’s representability theorems. Artin’s approximation theorem has been generalized by Conrad and de Jong, [CdJ02]. This gives the following version of Artin’s representability theorem.

Corollary 2.18. [Art74, Corollary 5.2], [LMB00, Corollaire 10.8], [CdJ02, Theorem 1.5] Assume S is an excellent scheme. Let X be a limit preserving stack in groupoids over (Aff/S) with the étale topology. Then X is an algebraic stack if

(1) X is relatively representable, i.e., the diagonal 1-morphism Δ : X → X ×X is representable by locally finitely presented algebraic spaces.

(2) The Schlessinger-Rim criteria [Art74, (S1,2)] hold.

(3) If Â is a complete local O_{S}-algebra with residue field of finite type over S, then X (Spec Â) → \underleftarrow{lim} X (Spec Â/𝔪^{n}) has a dense image.

(4) The stack X satisfies openness of versality compatibly with étale extensions.

Applying Theorem 2.15, this gives the following version. Theorem 2.19. [Art74, Theorem 5.3] Assume S is an excellent scheme. Let X be a limit preserving stack in groupoids over (Aff/S) with the étale topology. Then X is an algebraic stack if

(1) The Schlessinger-Rim criteria [Art74, (S1,2)] hold. Also, for every algebraic element a_{0} of X (Spec A_{0}) and every finite A_{0}-module M, Aut_{a_{0}}(A_{0} + M) is a finite A_{0}-module.

(2) For any complete local O_{S}-algebra Â with residue field of finite type over S, the canonical map

X (Spec Â) → \underleftarrow{lim} S(Spec Â/𝔪^{n})

is faithful, and has a dense image, i.e., the projection to X (Spec Â/𝔪^{n}) is essentially surjective for every n.

(3) D and Aut_{a_{0}}(A_{0} + M) satisfy [Art74, (4.1)], every generic lift of an integral, algebraic deformation situation over X has an integral lift, and every algebraic deformation situation over X has a generic extender compatible with étale extensions and with closed points.

(4) If the object a_{0} of X (Spec A_{0}) is algebraic, and if ϕ is an automorphism of a_{0} inducing the identity in X (Spec k(y)) for a dense set of finite type points y of Spec A_{0}, then ϕ equals Id_{a_{0}} on a non-empty open subset of Spec A_{0}.

2.6. Relative obstruction theory. Relative obstruction theories are studied in [Ols04, Appendix A]. Let f : X → Y be a 1-morphism between stacks in groupoids over (Aff/S) both satisfying Schlessinger’s conditions [Art74, (S1,2)]. Definition 2.20. [Ols04, Def. A.10] A relative obstruction theory for f consists of the following:

(i) an assignment to each reduced infinitesimal extension (q̃_{0}, a) over f of an A_{0}-linear functor,

O_{q̃_{0},a} : (finite A_{0} − modules) → (finite A_{0} − modules),

(ii) an assignment to each morphism (ũ, ũ_{0}, ϕ) of infinitesimal extensions over f of a natural transformation of A_{0}-linear functors

θ_{ũ,ũ_{0},ϕ} : O_{q̃_{A,0},a}(M) ⊗_{A_{0}} B_{0} ⇒ O_{q̃_{B,0},b}(M ⊗_{A_{0}} B_{0}),

(iii) and an assignment to each deformation situation (q̃, q̃_{0}, a) over f of an

element o_{q̃,q̃_{0},a} of O_{q̃,a}(M) satisfying the following axioms.

(i) For every deformation situation (q̃, q̃_{0}, a) over f, o_{q̃,q̃_{0},a} equals 0 if and only if there exists a morphism a′ → a in X mapping to q̃.

(ii) For every morphism (u′, ũ, ũ_{0}, ϕ) of extension pairs over f, θ_{ũ,ũ_{0},ϕ}(o_{q̃_{A},q̃_{A,0},a})

equals o_{q̃_{B},q̃_{B,0},b}.

Notation 2.21. [Art74, (4.1)], [Ols04, §A.11] As in Notation 2.4, there are relative analogues of the conditions on obstruction theory. (4.1.i) The functor O_{f} is compatible with étale localization: For every morphism

(ũ, ũ_{0}, ϕ) of infinitesimal extensions over f, if A_{0} is a finite-type O_{S}-algebra and u is étale, then the following associated natural transformation of functors is an isomorphism,

O_{f,q̃_{A,0},a_{0}}(M) ⊗_{A_{0}} B_{0} → O_{f,q̃_{B,0},b_{0}}(M ⊗_{A_{0}} B_{0}).

(4.1.iii) For every infinitesimal extension (q̃_{0}, a) over f with A a finite type O_{S}- algebra, there is an open dense set of points of finite type p ∈ Spec A_{0} so that the following map is injective

O_{q̃_{0},a}(M) ⊗_{A_{0}} k(p) → O_{q̃_{0},a}(M ⊗_{A_{0}} k(p)).

2.7. Obstructions and lifting properties. Properties of obstructions imply the hypotheses of Theorem 2.15. Lemma 2.22. [Art74, Lemma 4.6] Let f : X → Y be a 1-morphism of stacks in groupoids over (Aff/S). Assume both satisfy [Art74, (S1,2)]. Let O_{f} be a relative obstruction theory for f. If X is limit preserving, if D_{f} satisfies (4.1.i) of Notation 2.4 and if O_{f} satisfies (4.1.i) of Notation 2.21, then every generic lift of an integral, algebraic deformation situation over f has an integral lift.

Proof. Denote by M the kernel of q̃_{B}. The obstruction to finding a morphism r_{B} over q̃_{B} is an element o of the obstruction group O = O_{f,q̃_{B,0},b_{X}} (M). By hypothesis, there is a lift after localizing to the total rings of fractions. These localizations are limits of finite type localizations. Because X is limit preserving, there exists a lift after a finite type localization. Thus, the image of o in the obstruction group of this finite type localization is zero. Because a finite type localization is an étale morphism, (4.1.i) of Notation 2.21 implies the obstruction group of the localization is the localization of the obstruction group. Therefore, the image of o in O ⊗_{B_{0}} A_{0} is zero. So there exists a nonzero element t of B_{0} such that t · o is zero in O.

Let B′_{new} be the subring of A′ generated by B′ and by t^{−1}M inside M ⊗_{B_{0}} A_{0}, i.e., the kernel of q_{A}. Denote by q_{B,new} the restriction of q_{A} to B′_{new} considered as a morphism with target B. The infinitesimal extension B′_{new} ⟶^{q_{B,new}} B gives an deformation situation (q_{B,new}, q_{B,0}) which is the image of an obvious morphism from (q_{B}, q_{B,0}). The base change of (q̃_{B}, q̃_{B,0}, b_{X}) by this morphism is an deformation situation (q̃_{B,new}, q̃_{B,0}, b_{X}) over f. By Axiom (ii) of Definition 2.20, the obstruction o_{new} of this deformation situation is the image of o in O_{f,q̃_{B,0},b_{X}} (t^{−1}M). There is an isomorphism t^{−1}M → M sending t^{−1}m to m. This defines an isomorphism O_{f,q̃_{B,0},b_{X}} (t^{−1}M) → O_{f,q̃_{B,0},b_{mcX}} (M), and the image of o_{new} is exactly t · 0, which is zero. Therefore o_{new} is zero. By Axiom (i) of Definition 2.20, there is a morphism r_{B}^{∗} of X over q̃_{B,new}.

Replace B′ by B′_{new}, etc. By construction, r_{B}^{∗} satisfies Axioms (i) and (ii) of an integral lift. The base change of r_{B}^{∗} over A′ → A is a morphism r_{A}^{∗} over q̃_{A}. By [Art74, (S1,2)], there exists an element δ of D_{f,a_{X ,0}}(M ⊗_{B_{0}} A_{0}) such that δ · r_{A}^{∗} equals r_{A}. By the same type of argument as above, using (4.1.i) of Notation 2.4 for D_{f} and that X is limit preserving, there exists a nonzero element t of A_{0} and an element d of D_{f,b_{X ,0}}(M) such that tδ = d. Thus, after replacing B′ by an extension B′ → B′_{new} as above and replacing r_{B}^{∗} by its base change over B′_{new}, there exists an element d of D_{f,b_{X ,0}}(M) such that r_{A} equals d · r_{A}^{∗}. Define r_{B} to be d · r_{B}^{∗}. This is an integral lift of r_{A}. □ Lemma 2.23. [Art74, Proof of Theorem 4.4] Let f : X → Y be a 1-morphism of stacks in groupoids over (Aff/S). Assume both satisfy [Art74, (S1,2)]. Let O_{f} be a relative obstruction theory for f. If O_{f} satisfies (4.1.i) and (4.1.iii) of Notation 2.21, then every algebraic deformation situation over f has a generic extender compatible with étale extension and with closed points.

Proof. Because of (4.1.i), we may replace B_{0} by any finite type localization (and replace B and B′ by the associated localizations). In particular, assume M is a finite projective B_{0}-module. Similarly, assume P = O_{f,q̃_{0},b_{X}} (B_{0}) is a finite projective B_{0}- module.

By B_{0}-linearity of O_{f}, O_{f,q̃_{0},b_{X}} (M) is canonically isomorphic to M ⊗_{B_{0}} P. Moreover, for every surjection M → M′, the map M ⊗_{B_{0}} P → O_{b,q̃_{0},b_{X}} (M′) factors through the surjection M ⊗_{B_{0}} P → M′ ⊗_{B_{0}} P. By (4.1.iii), after replacing B_{0} by a further finite type localization, the induced map M′ ⊗_{B_{0}} P → O_{f,q̃_{0},b_{X}} (M′) is injective whenever M′ is a direct sum of copies of k(y) for y a closed point.

The obstruction class o associated to (q̃, q̃_{0}, b_{X}) gives a B_{0}-linear map, B_{0} → M⊗_{B_{0}}P. By adjunction, this is equivalent to a B_{0}-linear map, Hom_{B_{0}}(P, B_{0}) → M. Define M → N to be the cokernel of this map. For every surjection M → N′, the image of o in N′ ⊗_{B_{0}} P is zero if and only if N′ is a quotient of N. Because the image of o in N ⊗_{B_{0}} P is zero, the image is also zero in O_{f,q̃_{0},b_{X}} (N). Therefore there exists a morphism b_{X ,N} → b_{X} in X mapping to q̃_{N}. So N is a generic extender.

For a quotient M′ of M ⊗_{B_{0}} k(y). the induced map M′ ⊗_{B_{0}} P → O_{f,q̃_{0},b_{X}} (M′) is injective. Therefore the image of o in O_{f,q̃_{0},b_{X}} (M′) is zero if and only if M′ is a quotient of N. Thus N is compatible with closed points.

Finally, compatibility with étale extensions follows from (4.1.i). □

Corollary 2.24. Assume Y is limit preserving and satisfies [Art74, (S1,2)] and f is representable by limit preserving algebraic stacks. Then X is limit preserving and satisfies [Art74, (S1,2)]. Also every generic lift of an integral, algebraic deformation situation over f has an integral lift, and every algebraic deformation situation over f has a generic extender compatible with étale extension and with closed points. Finally, if Y is relatively representable, then X is relatively representable.

Proof. Let A equal lim A_{i} and let a_{X} be an object of X (Spec A). Denote by a_{Y} the image f(a_{X}). Because Y is limit preserving, there exists an i and an object a_{Y,i} of Y(Spec A_{i}) whose base change is isomorphic to a_{Y}. Form the 2-fibered product,

X_{a_{Y} ,i} = X ×_{f,Y,a_{Y,i}} Spec A_{i}.

By hypothesis, this is a limit preserving. And a_{X} gives an object of X_{a_{Y} ,i}(Spec A). Thus there exists a j and an object a_{X ,j} of X_{a_{Y} ,i}(Spec A_{j}) whose base change is isomorphic to a_{X}. Therefore X is limit preserving. By a similar argument, X satisfies [Art74, (S1,2)].

Given an algebraic deformation situation over f, (q̃, q̃_{0}, b_{X}), the 2-fibered product

X_{b′_{Y}} = X ×_{f,Y,b′_{Y}} Spec B′

is a limit preserving algebraic stack. Every base change of this deformation situation over f is the projection of a deformation situation over X_{b′_{Y}}. Thus it suffices to prove algebraic deformation situations over X_{b′_{Y}} have integral lifts and generic extenders, etc.

By [Art74, p. 182], [Ols05a, Remark 1.7], X_{b′_{Y}} has an obstruction theory such that [Art74, (S1,2), (4.1)] are satisfied for automorphisms, deformations and obstructions. By Lemmas 2.22 and 2.23, algebraic deformation situations over X_{b′_{Y}} have integral lifts, generic extenders, etc.

Finally, if Y is relatively representable, then X ×_{Y} X → X × X is representable. Since f is relatively representable, X → X ×_{Y} X is representable. Therefore the composition X → X × X is representable, i.e., X is relatively representable. □

3. The stack of algebraic spaces

Denote by X the category whose objects are all pairs (U, X) of an affine scheme U and a U-algebraic space X, and whose morphisms are all pairs,

(f_{U}, f_{X}) : (U′, X′) → (U′′, X′′),

of a morphism f_{U} : U′ → U′′ of affine schemes and an isomorphism of U′-algebraic spaces, f_{X} : X′ → U′ ×_{U′′} X′′. The identity morphisms and the composition law are the obvious ones. There is a functor X → (Aff/Spec ℤ). This functor is a stack for the étale topology, and even the fpqc topology on (Aff), cf. [LMB00, (3.4.6), (9.4)].

Claim 3.1. The diagonal 1-morphism Δ : X → X × X is not representable.

Proof. Let k be any field. Consider the 1-morphism ζ : Spec (k) → X associated to the object (Spec (k), 𝔸_{k}^{1}). There is an associated 1-morphism (ζ, ζ) : Spec (k) → X × X. The 2-fibered product of (ζ, ζ) and Δ is the stack associated to the contravariant functor I on (Aff/Spec (k)) associating to every k-algebra A the set

I(A) := Isom_{A}(A[t], A[t]).

The claim is that I is not an algebraic space.

Consider the subset,

T = {ϕ ∈ I(k[ϵ]/ϵ^{2})|ϕ ≡ Id(modϵ)}.

By direct computation, there is an isomorphism from the k-vector space k[t] to T via f ↦ (t ↦ t+ϵf). The dimension of k[t] as a k-vector space is countably infinite. Therefore the dimension of T as a k-vector space is countably infinite.

On the other hand, for every algebraic space J over k, and every k-point p of J, the set,

T_{J,p} = {ϕ ∈ J(k[ϵ]/ϵ^{2})|ϕ ≡ p(modϵ)},

is isomorphic to the k-vector space,

Hom_{k}(Ω, k),

where Ω is the k-vector space Ω_{J/k}/𝔪_{p}Ω_{J/k}. If T′ is infinite-dimensional, then Ω is also infinite-dimensional. For an infinite-dimensional k-vector space Ω, the dimension of T′ = Hom_{k}(Ω, k) is uncountable. Since the dimension of T is countably infinite, I is not an algebraic space over k. □

Denote by X_{pr,f,lpf} the category whose objects are all pairs (U, X) of an affine scheme U and a proper, flat, locally finitely presented U-algebraic space. This is a full subcategory of X. The restriction of F is a functor F : X_{pr,f,lpf} → (Aff). As with X, X_{pr,f,lpf} is a stack for the étale and fpqc topology.

Proposition 3.2. The stack X_{pr,f,lpf} is limit preserving.

Proof. This follows from results in [Gro67, §8]. □

Proposition 3.3. The diagonal morphism Δ : X_{pr,f,lpf} → X_{pr,f,lpf} × X_{pr,f,lpf} is representable, separated and locally finitely presented.

Proof. This follows from [Art69, Theorem 6.1]. □

Denote by π : V_{pr,f,lpf} → X_{pr,f,lpf} the universal 1-morphism representable by proper, flat, locally finitely presented algebraic spaces.

Proposition 3.4. Deformations and automorphisms of X_{pr,f,lpf} satisfy [Art74, (S1,2), (4.1)]. There is an obstruction theory O for X_{pr,f,lpf} satisfying [Art74, (4.1)].

Proof. The existence of an obstruction theory satisfying [Art74, (S1)] follows from [Ill71, Proposition III.2.1.2.3] using the relative cotangent complex of π. The condition [Art74, (S2)] for deformations and the analogous condition for automorphisms follow from coherence of the cohomology sheaves of the cotangent complex [Ill71, Corollaire II.2.3.7] together with the finiteness theorem, [BGI71, Théorème III.2.2].

Condition [Art74, (4.1.i)] follows from [Ill71, II.1.2.3.5] and standard results about cohomology and flat base change. Condition [Art74, (4.1.ii)] follows from the theorem on formal functions, [Knu71, Theorem V.3.1]. Condition [Art74, (4.1.iii)] follows from generic flatness, [Gro67, Théorème 6.9.1], and the semicontinuity theorem [Gro63, Théorème 7.7.5]. □

However X_{pr,f,lpf} does not satisfy Axiom 3 of [Art74, Corollary 5.2].

Claim 3.5. There does not exist a pair (Q, f) of an algebraic space Q and a representable, faithful, smooth 1-morphism f : Q → X_{pr,f,lpf}. In fact, X_{pr,f,lpf} does not satisfy Axiom 3 of [Art74, Corollary 5.2].

Proof. If X_{pr,f,lpf} did satisfy Axiom 3, then every proper algebraic space over a field would have an effective, formal, versal deformation. By [Art69, Theorem 1.6], every proper algebraic space over a field does have a formal, versal deformation. However, it is not always effective. The following example of a projective, smooth variety in characteristic 0 with no effective, formal, versal deformation is well-known.

Let k be an uncountable, characteristic 0, algebraically closed field. Let X be a smooth anticanonical divisor in ℙ^{1} × ℙ^{2}. This is a K3 surface together with an elliptic fibration pr_{ℙ^{1}} : X → ℙ^{1}.

Because the Schlessinger-Rim criteria [Art74, (S1,2)] hold for X, there is a complete, local k-algebra R and a formal, versal deformation (X_{n})_{n≥0} of X over R. Because K3 surfaces are unobstructed, R is formally smooth, i.e., R is a power series ring. Also there is a canonical isomorphism,

Hom_{k}(𝔪/𝔪^{2}, k) ≅ H^{1}(X, T_{X}).

For every invertible sheaf L on X, there is an associated first Chern class C_{1}(L) in H^{1}(X, Ω_{X}). This defines an injective group homomorphism,

C_{1} : N^{1}(X) → H^{1}(X, Ω_{X}),

where N^{1}(X) is the group of numerical equivalence classes of invertible sheaves. There is a cup-product pairing,

H^{1}(X, Ω_{X}) × H^{1}(X, T_{X}) → H^{2}(X, O_{X}).

Because X is a K3 surface, or equivalently by adjunction for the inclusion of X in ℙ^{1} ×ℙ^{2}, there exists an isomorphism of the dualizing sheaf ω_{X} with O_{X}. Using this isomorphism, the cup-product pairing above is equivalent to the pairing for Serre duality, which is a perfect pairing.

Let L be an invertible sheaf on X and let θ : 𝔪/𝔪^{2} → k be an element in H^{1}(X, T_{X}), considered as a k-algebra homomorphism R/𝔪^{2} → k[ϵ]/ϵ^{2}. Denote by X_{θ} the Abelian scheme over k[ϵ]/ϵ^{2} given by θ(X_{2}). The invertible sheaf L is the restriction of an invertible sheaf on X_{θ} only if C_{1}(L) ∪ θ is 0 in H^{2}(X, T_{X}). If L is not numerically trivial, then C_{1}(L) is nonzero. Then, because the cup product pairing is nondegenerate, set of θ for which L extends over X_{θ} is a proper k-subspace V_{L} of H^{1}(X, T_{X}).

By the theorem of the base, the group of N^{1}(X) of numerical equivalence classes of invertible sheaves is a finitely generated Abelian group. Therefore, the set of subspaces V_{L} arising from invertible sheaves as above is countable. Because k is uncountable, there exists an element θ of H^{1}(X, T_{X}) contained in none of the countably many proper subspaces V_{L}. Therefore every invertible sheaf on X_{θ} is numerically trivial.

Because R is a power series ring, there is a local k-algebra homomorphism t : R → k \begin{matrix}[ \\ [ & ϵ & ] \\ ] \end{matrix} whose associated map 𝔪/𝔪^{2} → k is θ. If (R/𝔪^{n}, X_{n}) comes from a proper algebraic space X_{R} over Spec (R), then t(X_{R}) is a proper, flat algebraic space over Spec k \begin{matrix}[ \\ [ & ϵ & ] \\ ] \end{matrix}. By [Knu71, Corollary 5.20], there is a dense open subspace U of t(X_{R}) which is an affine scheme. Denote by D′ the complement of U in t(X_{R}). Denote by D the closure in t(X_{R}) of the generic fiber of D′ → Spec k \begin{matrix}[ \\ [ & ϵ & ] \\ ] \end{matrix}. Since the closed fiber of t(X_{R}) is smooth, t(X_{R}) is smooth over Spec k \begin{matrix}[ \\ [ & ϵ & ] \\ ] \end{matrix}. Also, t(X_{R}) is separated. Therefore, essentially by [Gro63, Théorème 5.10.5], D is a Cartier divisor in X_{R}. Moreover, it is flat over Spec k \begin{matrix}[ \\ [ & ϵ & ] \\ ] \end{matrix}. Denote by L the associated invertible sheaf. By the paragraph above, the restriction of L to X is numerically trivial.

Because the generic fiber of t(X_{R}) is a proper, positive-dimensional algebraic space, U is not all of the generic fiber, i.e., the generic fiber of D′ is not empty. Because D is proper over Spec k \begin{matrix}[ \\ [ & ϵ & ] \\ ] \end{matrix} and its image contains the generic point, the closed fiber of D is not empty. Because D is flat over Spec k \begin{matrix}[ \\ [ & ϵ & ] \\ ] \end{matrix}, the closed fiber of D is not all of X. Thus it is a nonempty curve in X. But no nonempty curve in X is numerically trivial: it has positive intersection number with either the pullback of O_{ℙ^{1}}(1) or the pullback of O_{ℙ^{2}}(1). This contradicts that the restriction of L to X is numerically trivial. The contradiction proves there exists no proper algebraic space X_{R} giving the formal, versal deformation of X. □

Remark 3.6. The counterexample also proves there is no effective, formal, versal deformation of the pair (X, pr_{ℙ^{1}}^{∗}O_{ℙ^{1}}(1)). If there were, then the argument above would prove there is a proper, flat algebraic space t(X_{R}) over Spec k \begin{matrix}[ \\ [ & ϵ & ] \\ ] \end{matrix} and an invertible sheaf M on t(X_{R}) restricting to pr_{ℙ^{1}}^{∗}O_{ℙ^{1}}(1) such that the restriction to X of every invertible sheaf on t(X_{R}) is numerically equivalent to pr_{ℙ^{1}}^{∗}O_{ℙ^{1}}(d) for some d. Because h^{1}(X, pr_{ℙ^{1}}^{∗}O_{ℙ^{1}}(1)) is zero, every global section of pr_{ℙ^{1}}^{∗}O_{ℙ^{1}}(1) is the restriction of a global section of M. Because pr_{ℙ^{1}}^{∗}O_{ℙ^{1}}(1) is generated by global sections, there exists a global section of M that does not vanish identically on D. The generic fiber of the zero locus of this section is a nonempty curve not contained in D. Because it is proper, it is also not contained in U. Thus it has positive intersection number with D. On the other hand, C_{1}(pr_{ℙ^{1}}^{∗}O_{ℙ^{1}}(1))^{2} is zero.

This proves the restriction of L is not numerically equivalent to pr_{ℙ^{1}}^{∗}O_{ℙ^{1}}(1). The contradiction proves there is no effective, versal, deformation of (X, pr_{ℙ^{1}}^{∗}O_{ℙ^{1}}(1)).

The following is also noteworthy.

Claim 3.7. The diagonal morphism is not quasi-compact.

Proof. Let k be an algebraically closed field and let (E, 0) be an elliptic curve over Spec (k) whose automorphism group is ℤ/2ℤ. Let X = E × E. For the object (Spec (k), X), the fiber in the diagonal is the scheme I = Isom_{k}(X, X). The scheme I surjects smoothly to X by associating to each map f the image f(0). The kernel is isomorphic to the discrete, non-quasi-compact k-group scheme GL_{2}(ℤ) by associating to f the unique matrix

( \begin{matrix}a & b \\ c & d \end{matrix} ) ,

in GL_{2}(ℤ) such that f(x, y) = (ax + by, cx + dy). □

4. Variants of the stack of algebraic spaces

As above, denote by π : V_{pr,f,lpf} → X_{pr,f,lpf} the universal 1-morphism representable by proper, flat, locally finitely presented algebraic spaces. Denote by Coh_{π} the category whose objects consist of triples (U, X, E) of an object (U, X) of X_{pr,f,lpf} and a locally finitely presented O_{X}-module E which is O_{U}-flat, cf. [LMB00, 2.4.4]. Morphisms in Coh_{π} are data (f_{U}, f_{X}, f_{E}) of a morphism (f_{U}, f_{X}) in X_{pr,f,lpf} and an isomorphism f_{E} from E′ to the pullback of E′′. There is a forgetful functor G : Coh_{π} → X_{pr,f,lpf}. Proposition 4.1. [LMB00, Théorème 4.6.2.1] The category Coh_{π} is a stack for the étale and fpqc topologies on (Aff). Moreover, the functor G is representable by limit preserving algebraic stacks with quasi-compact, separated diagonal.

Proof. By the proof of the first half of [LMB00, Théorème 4.6.2.1], the diagonal of Coh_{X/U} is representable by separated, finitely presented algebraic spaces. Note the first half of the proof uses only properness of X/U and does not use cohomological flatness in dimension 0. To be absolutely precise, one has to generalize [Gro63, Corollaire 7.7.8] to the case when f : X → Y is a proper morphism of algebraic spaces. Using [Art74], it is clear how to do this (perhaps [Gro63, Remarques 7.7.9.iii] anticipates this generalization).

The proof that Coh_{X/U} is a locally finitely presented Artin stack uses [Art74, Theorem 5.3]. Existence of an obstruction theory for Coh_{X/U} such that automorphisms, deformations and obstructions satisfy [Art74, (S1,2), (4.1)] is just as in Proposition 3.4, replacing [Ill71, Proposition III.2.1.2.3] by [Ill71, Proposition IV.3.1.5]. Finally, compatibility with completions follows from the Grothendieck existence theorem, [Knu71, Theorem V.6.3]. □

Denote by X_{pol} the full subcategory of Coh_{π} of triples (U, X, L) where L is an invertible sheaf on X ample relative to U.

Proposition 4.2. The category X_{pol} is a limit preserving algebraic stack with quasicompact, separated diagonal.

Proof. By [Gro63, Théorème 4.7.1], the inclusion functor X_{pol} → Coh_{π} is representable by open immersions. Therefore X_{pol} → X_{pr,f,lpf} is representable by limit preserving algebraic stacks. By Proposition 2.9, Proposition 2.10, Corollary 2.24, Proposition 3.2 and Proposition 3.4, X_{pol} satisfies Axioms 1,2 and 4 of Corollary 2.18.

The proof of Axiom (3) uses the Grothendieck existence theorem. Let Â be a complete local algebra and let (X_{n}, L_{n}) be a compatible collection of objects of X_{pol} over Spec Â/𝔪^{n}. For d sufficiently large, L_{0}^{⊗d} is very ample and h^{1}(X_{0}, L_{0}^{⊗d}) is zero. The compatible system of Â/𝔪^{n}-modules H^{0}(X_{n}, L_{n}^{⊗d}) defines a finite free Â-module. Choosing a basis for this module, there are induced closed immersions X_{n} → ℙ_{Â/𝔪^{n}}^{N} such that the pullback of O_{ℙ^{N}} (1) is L_{n}^{⊗d}. By the Grothendieck existence theorem, [Gro63, Corollaire 5.1.8], there exists a closed subscheme X of ℙN ; Â whose reductions give the compatible family X_{n}. By the theorem on formal functions, [Gro63, Théorème 4.1.5], H^{0}(X, L^{⊗e}) equals \underleftarrow{lim} H^{0}(X_{n}, L_{n}^{⊗e}) for every e. Because the higher cohomologies vanish, and because the X_{n} are flat over Spec Â/𝔪^{n}, also H^{0}(X, L^{⊗e}) is flat over Â. Therefore X is flat over Spec Â. Finally, by [Gro63, Corollaire 5.1.6], there exists an invertible sheaf L on X whose reductions give L_{n}. By [Gro63, Théorème 4.7.1], this is ample. □

Remark 4.3. There are, of course, other ways to verify the proposition. One can use the existence of Hilbert schemes to give a smooth morphism from a scheme to X_{pol}.

Let S be an excellent scheme. Let Y be an limit preserving, separated algebraic stack over (Aff/S) whose diagonal is representable by finite morphisms. Define H to be the category whose objects are 4-tuples (U, X, L, g) where (U, X) is an object of X_{pr,f,lpf}, L is an invertible sheaf on X, and g : X → Y is a 1-morphism. A datum is required to satisfy the condition that L is g-relatively ample, i.e., for every affine scheme Spec A and object a of Y(SPA), the pullback of L to the 2-fibered product Spec A ×_{a,Y,g} X is ample.

Morphisms in H are data (f_{U}, f_{X}, f_{L}, f_{g}) of a morphism (f_{U}, f_{X}, f_{L}) in Coh_{π} together with a 2-isomorphism from the 1-morphism g′ : X′ → Y to the composite 1-morphism

X′ ⟶^{f_{X}} X′′ ×_{U′′} U′ ⟶^{pr_{X′′}} X′′ ⟶^{g′′} Y.

Proposition 4.4. The category H is a limit preserving algebraic stack over (Aff/S) with quasi-compact, separated diagonal.

Proof. There is a 1-morphism T : H → C≀⟨_{π}. In fact this factors through the open substack of data (U, X, L) such that L is an invertible sheaf. By [Ols05b, Theorem 1.1], and using [Gro63, Théorème 4.7.1] (locally over the Hom stack, for a quasi-compact smooth cover of Y whose image contains the local image of g ), the 1-morphism T is representable by limit preserving algebraic stacks. Moreover, the diagonal morphism associated to T is quasi-compact and separated. By Proposition 4.1, the composite G ∘ F : H → X_{pr,f,lpf} is representable by limit preserving algebraic stacks with quasi-compact, separated diagonal.

By Proposition 2.9, Proposition 2.10, Corollary 2.24, Proposition 3.2 and Proposition 3.4, H satisfies Axioms 1,2 and 4 of Corollary 2.18. It only remains to verify Axiom (3).

To prove Axiom (3), apply the straightforward analogue of the argument from Proposition 4.2. Namely, given a compatible family (X_{n}, L_{n}, g_{n}) over Spec Â/𝔪^{n}, first form the sequence of sheaves E_{d,n} = (g_{n})_{∗}L_{n}^{⊗d} on Spec Â/𝔪^{n} ×_{S} Y. For d sufficiently large, the natural map g_{n}^{∗}E_{n,d} → L_{n}^{⊗d} is surjective for every d. For d perhaps larger, the induced morphism X_{n} → ProjSym(E_{n,d}) is a closed immersion. These statements are local on Y, and thus can be checked after base change by a smooth morphism Spec B → Y whose image contains g_{n}(X_{n}). Then the statements follow from the usual versions for schemes.

The sheaves E_{n,d} are coherent with proper support. Therefore, by the analogue of the Grothendieck existence theorem for stacks, [OS03, Proposition 2.1], there exists a coherent sheaf with proper support E_{d} on Spec Â ×_{S} Y whose reductions are the sheaves E_{n,d}. Denote by P the stack ProjSym(E_{d}) (which can be constructed by flat descent, for example). The projection,

pr : P → Spec Â ×_{S} Y,

is representable by proper, finitely presented algebraic spaces. Therefore P is a limit preserving algebraic stack with quasi-compact, separated diagonals. There are natural closed immersions X_{n} → Spec Â/𝔪^{n} ×_{Spec Â} P. Again by the Grothendieck existence theorem [OS03, Proposition 2.1], there exists a closed substack X of P whose reductions are X_{n}. Since the reductions are proper, flat, finitely presented algebraic spaces over Spec Â/𝔪^{n}, by the same argument as in the proof of Proposition 4.2, X is a proper, flat, finitely presented algebraic space over Spec Â. Define g to be the restriction of pr to X. By the Grothendieck existence theorem for algebraic spaces, [Knu71, Theorem V.6.3], there exists an invertible sheaf L on X whose reductions are the sheaves L_{n}. By [Gro63, Théorème 4.7.1] (applied after base change to a quasi-compact smooth cover of Y), L is g-ample. Thus (X, L, g) is an object of H(Spec Â) whose reductions are the objects (X_{n}, L_{n}, g_{n}). □

Remark 4.5. There is a stack H_{finite} closely related to H whose objects are data (U, X, g) as above such that O_{X} is g-ample. There is a 1-morphism H_{finite} sending (U, X, g) to (U, X, O_{X}, g). This morphism is representable by affine morphisms by the same proof as for the relative representability of Coh_{π} in [LMB00, Théorème 4.6.2.1]. Obviously, if O_{X} is g-ample if and only if g is representable by finite morphisms.

Denote by g : X → H_{finite} ×_{S} Y the universal morphism. There is an open substack of X where g is unramified: namely the complement of the support of the sheaf of relative differentials of g. The complement of this open stack is proper over H_{finite}. Its image in H_{finite} is proper. The complement of the image equals Vistoli’s Hilbert stack, cf. [Vis91].

Similarly, using [Gro67, Théorème 12.1.1], there is an open substack of H_{finite} over which the fibers of X are geometrically reduced and equidimensional. In the special case that Y is a projective scheme over S, this locus is the stack of branchvarieties of Alexeev and Knutson, cf. [AK06].

References

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