So in the near future I want to write a bit more about Hilbert schemes in the stacks project. Now it feels a bit wrong to say “Hilbert space” for the… uh… Hilbert space of an algebraic space. My promethean colleague Davesh Maulik suggests using “Hilbert skeem” so that typographically at least we make the reader aware that the… uh… Hilbert skeem may not be a scheme. What do you think?

This is about example 4 from this post. It turns out that you can repair what I said there to make it work. The mistake was pointed out by David Rydh in the comments of that post. Thanks for Bhargav Bhatt for explaning how to repair it. Any mistakes are mine (please tell me if there are any).

Let X be an algebraic space over a base S (not necessarily flat). Consider diagrams


Y ---> X
|      |
v      v
T ---> S


where f : Y —> T is proper and flat. In this situation let C’ ∈ D(Y) be the cone of the map

L_{X/S} ⊗ O_Y —> L_{Y/T}.

Then I claim there is a canonical map C’ —> f^*L_{T/S}[1] which controls the deformation theory of the diagram (i.e., we look at first order thickenings T’ of T over S and flat deformations of Y to T’ mapping into S).

This is much better than the original suggestion in the post but it works for the same reason and the obstruction group doesn’t depend on the thickening T’ only on the ideal I defining T in T’.

The “real” reason this works is the following observation: We can think of the cone C’ as the cotangent complex of Y over the derived base change of X to T. Hence it is clear that EXt(C’, f^*I) computes obstructions, infinitesimal deformations, infinitesimal automorphisms of the morphism of Y into the derived base change. But since Y is a usual scheme and flat over T any (flat) deformation of Y to T’ is still a usual scheme, and morphisms from a usual scheme to a derived scheme map through pi_0(the derived scheme).

Jason Starr has privately emailed me something similar for the Quot scheme, which I haven’t fully understood yet.

Since the last update we have added the following material:

1. results on limits of algebraic spaces (including results on quasi-coherent modules),
2. results on coherent modules on locally Noetherian algebraic spaces, see
   Section Tag 07U9 and Section Tag 07UI,
3. devissage of coherent modules on Noetherian algebraic spaces, see Section Tag 07UN
4. a decent singleton algebraic space is a scheme (Lemma Tag 047Z),
5. a qc + qs algebraic space such that H^1 is zero on any quasi-coherent module is an affine scheme (Proposition Tag 07V6),
6. if X —> Y is a surjective integral morphism, X is an affine scheme, and Y an algebraic space, then Y is an affine scheme (as far as I know this result is due to David Rydh), see Proposition Tag 07VT,
7. the previous result in particular implies that if an algebraic space has a reduction which is a scheme then it is a scheme (you can find this in a paper by Conrad, Lieblich, and Olsson). This allowed us to significantly improve the exposition on thickenings of algebraic spaces which leads into the next item,
8. pushouts of algebraic spaces, see Section Tag 07SW,
9. this is applied to get a very general version of the Rim-Schlessinger condition for algebraic stacks, see Section Tag 07WM,
10. a section about what happens with deformation theory when you have a finite extension of residue fields (possibly inseparable), see Section Tag 07WW,
11. a (partial) solution to question 04PZ thanks to Philipp Hartwig, see Lemma Tag 07VM,
12. a bunch more stuff in the chapter on Artin’s Axioms including an approach to checking openness of versality which works exactly as explained here and here.

A short term goal is now to apply the results of the chapter on Artin’s Axioms to show that some natural moduli problems (in restricted generality) are representable by algebraic stacks or algebraic spaces. For example: Picard stacks, moduli of curves, moduli of canonically polarized smooth projective varieties, Hilbert schemes/spaces, Quot schemes/spaces, etc.

A longer term goal would be to get the most general results of this type, for example the stack (of flat families) of finite covers of P^n (this is a made up example). For the longer term goal I see no way around working with the full cotangent complex (and not the naive one). Do you?

[Edit July 5, 2012: Jason Starr points out that in his preprint on “Artin’s Axioms” in Remark 4.5 he proves the stack mentioned above is an algebraic stack without using the full cotangent complex.]

Thanks to Frank Gounelas for alerting me to the existence of a competing stacks project:

This post continues the discussion started here.

Traditionally, an obstruction theory for a moduli problem is a way of computing infinitesimal automorphism groups, infinitesimal deformation spaces, and an obstruction space for a given moduli problem using cohomology. Moreover, in all cases where this can be done (as far as I know) these groups are computed as consecutive cohomology groups of a particular sheaf, or complex of sheaves, or sometimes consecutive ext groups. Let me give some examples.

Let A’ \to A be a surjection of rings over some base ring Λ whose kernel is an ideal I having square zero.

1. If Y is a smooth proper algebraic space over A, then
   1. an obstruction to lifting Y to a smooth proper space over A’ lies in H^2(Y, T_{Y/A} ⊗ I),
   2. if Y has a lift Y’ then the set of isomorphism classes of lifts is principal homogeneous under H^1(Y, T_{Y/A} ⊗ I),
   3. the infinitesimal automorphism group of Y’ over Y is H^0(Y, T_{Y/A} ⊗ I)

   You can work this example out by yourself using just Cech cohomology methods.

2. If Y’ is a flat proper algebraic space over A’ and F is a finite locally free O_Y-module where Y = Y’ ⊗ A, then
   1. an obstruction to lifting F to a locally free module over Y’ lies in H^2(Y, End(F) ⊗ I)
   2. if F has a lift F’ then the set of isomorphism classes of lifts is principal homogeneous under H^1(Y, End(F) ⊗ I)
   3. the infinitesimal automorphism group of F’ over F is H^0(Y, End(F) ⊗ I)

   Again a Cech cohomology computation will show you why this is true.

3. If X’ is an algebraic space flat over A’ and f : Y —> X is a morphism of algebraic spaces with Y flat and proper over A, then
   1. an obstruction to lifting f to f’ : Y’ —> X’ with Y’ flat over A’ lies in Ext^2(L_{Y/X}, O_Y ⊗ I)
   2. if f has a lift f’ then the set of isomorphism classes of lifts is principal homogeneous under Ext^1(L_{Y/X}, O_Y ⊗ I)
   3. the infinitesimal automorphism group of f’ over f is Ext^0(L_{Y/X}, O_Y ⊗ I)

   For this one I recommend looking in Illusie.

4. If X’ is an algebraic space over A’ (not necessarily flat) and f : Y —> X is a morphism of algebraic spaces with Y flat and proper over A and X = X’ ⊗ A. Denote g : Y —> X’ the composition of f and the closed immersion X —> X’. Let C ∈ D(Y) be the cone of the map g^*L_{X’/A’} —> L_{Y/A}. Then
   1. an obstruction to lifting f to f’ : Y’ —> X’ with Y’ flat over A’ lies in Ext^2(C, O_Y ⊗ I)
   2. if f has a lift f’ then the set of isomorphism classes of lifts is principal homogeneous under Ext^1(C, O_Y ⊗ I)
   3. the infinitesimal automorphism group of f’ over f is Ext^0(C, O_Y ⊗ I)

   For this one, I haven’t written out all the details. Note that the obstruction space maps to Ext^2(L_{Y/A}, O_Y ⊗ I) and the obstruction in A maps to the obstruction to lifting Y to a flat space over A’. Once we have chosen a Y’ the obstruction of A is lifted to an element of

   Ext^1_{O_Y}(g^*L_{X’/A’}, O_Y ⊗ I) =
   Ext^1_{g^{-1}O_{X’}}(g^{-1}L_{X’/A’}, O_Y ⊗ I) =
   Ext^1_{g^{-1}O_{X’}}(L_{g^{-1}O_{X’}/A’}, O_Y ⊗ I) =
   Exal_{A’}(g^{-1}O_{X’}, O_Y ⊗ I)

   which measures the obstruction to lifting f^# to a map g^{-1}O_{X’} —> O_{Y’}, i.e., measures the obstruction to lifting f to a morphism Y’ —> X’. Changing the choice of Y’ alters this obstruction by the corresponding element of Ext^1(L_{Y/A}, O_Y ⊗ I). A similar story goes for the other groups.

In each of the cases above I think we can get a naive obstruction theory (as defined in the previous post). Essentially, each time the groups look like Ext^i(C, I), i = 0, 1, 2 for some object C of the derived category of some Y endowed with a proper flat morphism p : Y —> Spec(A). and you can take E = Rp_*(C ⊗ ω^*_{Y/A}) where ω^*_{Y/A} is the relative dualizing complex. [Edit June 28, 2012: This doesn’t work for case 4 because as Daivd Rydh points out below, the cone C may depend on A’. Thus you would have to allow for E to depend on the thickening… Ugh!]

Working dually. Folklore says that as soon as you can write down such a sequence of cohomology groups, then a naive obstruction theory should exist. The idea for the rest of this post is that you can try to axiomatize this. As stated here it only applies to cases 1 and 2 above; with some modifications it works in case 3 if you assume Y projective over A.

Let X be a category fibred in groupoids on (Sch/Λ). Let us define a dual naive obstruction theory as being given by the following data

1. for every object x of X over a Λ-algebra A we get K_x* ∈ D(A),
2. for any surjection A’ —> A with square zero kernel I and x over A an element ξ ∈ H^2(K_x^* ⊗ I),
3. for any surjection A’ —> A with square zero kernel I and liftable x over A, a free transitive action of H^1(K_x^* ⊗ I) on the set of isomorphism classes of lifts,
4. for any surjection A’ —> A with square zero kernel I and x’ over A’, an identification of H^0(K_x^* ⊗ I) with the infinitesimal automorphisms of x’ over x.

We impose some axioms on these data; we refrain from listing them all here. An important axiom is functoriality: if we have A —> B and x over A with base change y to B, then K_x^* ⊗_A B = K_y^*. We will describe two other key axioms. Suppose that we have a pair (A, x) and three surjections A_i —> A, i = 1, 2, 3 with square zero kernels I_i. Moreover, suppose we have maps

A_1 —> A_2 —> A_3

which induce a short exact sequence 0 —> I_1 —> I_2 —> I_3 —> 0. Denote

∂ : H^n(K_x^* ⊗ I_3) —> H^{n + 1}(K_x^* ⊗ I_1)

the boundary operator on cohomology. Then, we require (using the functoriality axiom to identify some of the groups):

1. given lifts x_3 and x_3′ over A_3 differing by θ ∈ H^1(K_x^* ⊗ I_3) the obstructions to lifting x_3 and x_3′ to A_1 differ by ∂(θ) in H^2(K_x^* ⊗ I_1),
2. given a lift x_2 over A_2 and an infinitesimal automorphism θ ∈ H^0(K_x^* ⊗ I_3) of x_2|_{Spec(A_3)}, the obstruction to lifting θ to an infinitesimal automorphism of x_2 is ∂(θ) in H^2(K_x^* ⊗ I_1).

Now, I believe (I worked it out on the blackboard here yesterday but it got erased) that given such a theory one can construct a (somewhat canonical) element

ξ(A, x) ∈ H^1(K_x^* ⊗ NL_{A/Λ})

which describes all the categories of lifts Lift(x, A’) for all surjections A’ —> A as above. Moreover, if K_x^* is a perfect complex, then we can set E = RHom_A(K_x^*, A) and use evaluation to get E —> NL_{A/Λ} and obtain a naive obstruction theory as in the previous post.

Let S be a scheme. Let X be a category fibred in groupoids over (Sch/S). In Artin’s work on algebraic stacks there is a notion of an obstruction theory for X. Artin splits the discussion into infinitesimal deformations and obstructions. Ideally we’d like to handle both at the same time. Sometimes the naive cotangent complex can be used to handle this.

Recall that the naive cotangent complex NL_{A/R} is the truncation τ_{≥ -1}L_{A/R} which is very weasy to work with, see Definition Tag 07BN. We can extend the definition of NL to schemes, algebraic spaces, and algebraic stacks (either by truncating the cotangent complex or by a direct construction we’ll come back to in the future).

Let’s define a naive obstruction theory for X over S as a rule which associated to every pair (T, x) where T is an affine scheme over S and x an object of X over T a map ξ : E —> NL_{T/S} in D(T) with the following properties:

1. the construction (E, ξ) is functorial in (T, x),
2. given a first order thickening T’ of T we have x lifts to x’ over T’ ⇔ the image of ξ in Hom(E, NL_{T’/T}) is zero,
3. the set of lifts x’ is principal homogeneous under Hom(E, NL_{T/T’}[-1]),
4. given two sections a,b : T’ —> T the lifts a^*x and b^*x differ by the element δ o ξ where δ = a – b : NL_{T/S} —-> NL_{T’/T}[-1] (see below), and
5. given a lift x’ then Inf(x’/x) = Hom(E, NL_{T/T’}[-2])

where Inf(x’/x) is the group of infinitesimal automorphisms of x’ over x. Note that NL_{T/T’} = I[1] where I is the ideal sheaf of T in T’ so the groups above are just Ext^{-1}(E, I), Hom(E, I), Ext^1(E, I). The map δ = a – b in 3 is just the composition NL_{T/S} —> Ω_{T/S} —> I associated to the difference between the ring maps a, b : O_T —> O_{T’}.

The motivation for this definition is the nonsensical formula “E = x^*NL_{X/S}”. It is nonsensical since we didn’t assume anything on X beyond being a category fibred in groupoids (b/c we’d like to use a naive obstruction theory to prove X is an algebraic stack). Thus a naive obstruction theory is an additional part of data. Of course, even a given algebraic stack X can have many different (naive) obstruction theories.

Example: If X is the stack whose category of sections over a scheme T is the category of families of smooth proper algebraic spaces of relative dimension d over T and x = (f : P —> T) then we can take E = Rf_*(ω_{P/T} ⊗ Ω^1_{P/T})[d – 1] and E —> NL_{T/S} the Kodaira spencer map.

Observations: (1) You do really have to take Rf_* because if P = P^1_T then in order for 3 to work you need E to be a rank 3 sheaf sitting in degree 1. (2) In order to define the Kodaira-Spencer map you use the triangle f^*NL_{T/S} —> NL_{P/S} —> NL_{P/T} and relative duality for f. (3) Using a bit of cohomology and base change, you can set E = dual perfect complex to Rf_*(T_{P/T}) and construct ξ whilst avoiding relative duality.

Versality. Now suppose that S is locally Noetherian and T of finite type over S. Let t be a closed point of T. Then we can ask if x is versal at t as defined in the chapter on Artin’s Axioms. If X has a naive obstruction theory, then (I haven’t checked all the details) x is versal at t if and only if

1. H^0(E ⊗ κ) —> H^0(NL_{T/S} ⊗ κ) is injective, and
2. H^{-1}(E ⊗ κ) —> H^{-1}(NL_{T/S} ⊗ κ) is surjective

where κ = κ(t).

Openness of versality. We’d like to show that if conditions i and ii hold, then the same is true in an open neighbourhood of t. Let C be the cone on the map ξ : E —> NL_{T/S}. Then conditions i and ii are equivalent to H^{-1}(C ⊗ κ) = 0. Provided that C has finite type cohomology modules, this condition then holds on an open neighbourhood of t, see Lemma Tag 068U as desired.

This is as it should be!

Let k be a field. Let G be a group scheme over k which is locally of finite type. Then X = [Spec(k)/G] is an algebraic stack over k (see for example Lemma Tag 06PL).

Let x_0 be the obvious 1-morphism Spec(k) —> X. Let’s look at the associated deformation problem, which in the stacks project is a category cofibred in groupoids

F = F_{X, k, x_0} —> C_k = (Artinian local k-algebras with residue field k)

see Section Tag 07T2. OK, so I was thinking about tangent spaces earlier today and it occurred to me that it is already somewhat fun to consider the example above. Namely, what is the tangent space TF in the situation above?

Your initial reaction might be “it is zero”. If you are a characteristic zero person, then you would be right, but before you read on: can you prove it?

Yeah, so the answer is that it is zero if G is a smooth group scheme over k (which is always the case in characteristic zero, see Lemma Tag 047N). Triviality of TF means that for every pair (T, t_0) where T is a G-torsor over Spec(k[ε]) and t_0 ∈ T(k), there exists a t ∈ T(k[ε]) which reduces to t_0 modulo ε. A torsor for a smooth group scheme is smooth. Hence the infinitesimal lifting criterion of smoothness implies that TF = 0.

But what if G isn’t smooth? In that case TF is always nontrivial. Namely, if TF = 0, then Spec(k) —> X is smooth (argument omitted) which isn’t true because Spec(k) x_X Spec(k) = G. I think that in general

dim(TF) + dim(G) = embedding dimension of G

but I haven’t tried to prove it or look it up. As usual, I welcome suggestions, comments, references, etc.

Since the last update we have added the following material:

1. A chapter on crystalline cohomology. This is based on a course I gave here at Columbia University. The idea, following a preprint by Bhargav and myself, is to develop crystalline cohomology avoiding stratifications and linearizations. This was discussed here, here, and here. I’m rather happy with the discussion, at the end of the chapter, of the Frobenius action on cohomology. On the other hand, some more work needs to be done to earlier parts of the chapter.
2. An example showing the category of p-adically complete abelian groups has kernels and cokernels but isn’t abelian, see Section Tag 07JQ.
3. Strong lifting property smooth ring maps, see Lemma Tag 07K4.
4. Compact = perfect in D(R), see Proposition Tag 07LT.
5. Lifting perfect complexes through thickenings, see Lemma Tag 07LU.
6. A section on lifting algebra constructions from A/I to A, culminating in
   • Elkik’s result (as improved by others) that a smooth algebra over A/I can be lifted to a smooth algebra over A, see Proposition Tag 07M8.
   • Given B smooth over A and a section σ : B/IB —> A/I then there exists an etale ring map A —>A’ with A/I = A’/IA’ and a lift of σ to a section B ⊗ A’ —> A’, see Lemma Tag 07M7.
7. We added some more advanced material on Noetherian rings; in particular we added the following sections of the chapter More on Algebra:
   • Some results on power series rings is a short technical section on properties of power series rings over fields and Cohen rings,
   • Permanence of properties under completion contains a discussion of properties of local rings which are preserved under completion,
   • Permanence of properties under henselization contains a discussion of properties of local rings which are preserved under henselization,
   • Filed extensions, revisited talks about p-bases,
   • The singular locus talks about the singular locus of Spec of a Noetherian ring,
   • Regularity and derivations shows that the existence of derivations
     sometimes helps to prove rings are regular,
   • Formal smoothness and regularity shows that A —> B is formally smooth (in m-adic topology) if and only if A —> B is regular (due to Andr\’e IIRC),
   • G-rings contains generalities about G-rings,
   • Excellent rings introduces them and doesn’t do much else besides.
8. You’re going to laugh, but we now finally have a proof of Nakayama’s lemma.
9. We started a chapter on Artin’s Axioms but it is currently almost empty.
10. We made some changes to the results produced by a tag lookup. This change is a big improvement, but I’m hoping for further improvements later this summer. Stay tuned!
11. We added some material on pushouts; for the moment we only look at pushouts where one of the morphisms is affine and the other is a thickening, see Section Tag 07RS for the case of schemes and see Section Tag 07SW for the case of algebraic spaces.
12. Some quotients of schemes by etale equivalence relations are schemes, see Obtaining a scheme.
13. We added a chapter on limits of algebraic spaces. It contains absolute Noetherian approximation of quasi-compact and quasi-separated algebraic spaces due to David Rydh and independently Conrad-Lieblich-Olsson, see Proposition Tag 07SU.

Enjoy!

The last result mentioned will allow us to replicate many results for quasi-compact and quasi-separated algebraic spaces that we’ve already proven for schemes. Most of the results I am thinking of are contained in David Rydh’s papers, where they are proven actually for algebraic stacks. I think there is some merit in the choice we’ve made to work through the material in algebraic spaces first, namely, it becomes very clear as we work through this material how very close (qc + qs) algebraic spaces really are to (qc + qs) schemes.

Hupeffoolly thees duesn’t feeulete-a uny cupyreeghts! Bork Bork Bork!

Geefee a cless ooff elgebreeec fereeeties, it is reesuneble-a tu esk iff zeere-a ere-a oonly fineetely muny members de-afined oofer a geefee fineete-a fiild. Vheele-a thees is cleerly zee cese-a vhee zee epprupreeete-a mudoolee foonctur is buoonded, metters ere-a oofftee nut su seemple-a. Fur ixemple-a, cunseeder zee cese-a ooff ebeleeun fereeeties ooff a geefee deemensiun g. Zeere-a is nu seengle-a mudoolee spece-a peremetereezing zeem; rezeer, fur iech integer d ≥ 1 zeere-a is a mudoolee spece-a peremetereezing ebeleeun fereeeties ooff deemensiun g veet a pulereezeshun ooff degree-a d. It is neferzeeless pusseeble-a tu shoo (see-a [Z, Zeeurem 4.1], [Mee, Curullery 13.13]) thet zeere-a ere-a oonly fineetely muny ebeleeun fereeeties oofer a geefee fineete-a fiild, up tu isumurpheesm. Unuzeer netoorel cless ooff fereeeties vhere-a thees deeffficoolty ereeses is zee cese-a ooff K3 soorffeces. Es veet ebeleeun fereeeties, zeere-a is nut a seengle-a mudoolee spece-a boot rezeer a mudoolee spece-a fur iech ifee integer d ≥ 2, peremetreezing K3 soorffeces veet a pulereezeshun ooff degree-a d.

Wirklich, Ich sollte Deutsch verwenden!

Furthering the quest of making this the most technical blog with the highest abstraction level, I offer the following for your perusal.

Let (X, O_X) be a ringed space. For any open U and any f ∈ O_X(U) there is a largest open U_f where f is invertible. Then X is a locally ringed space if and only if

U = U_f ∪ U_{1-f}

for all U and f. Denoting j : U_f —> U the inclusion map, there is a natural map

O_U[1/f] —-> j_*O_{U_f}

where the left hand side is the sheafification of the naive thing. If X is a scheme, then this map is an isomorphism of sheaves of rings on U. Furthermore, we can ask if every point of X has a neighbourhood U such that

U is quasi-compact and a basis for the topology on U is given by the U_f

If X is a scheme this is true because we can take an affine neighbourhood. If U is quasi-affine (quasi-compact open in affine), then U also has this property, however, so this condition does not characterize affine opens.

We ask the question: Do these three properties characterize schemes among ringed spaces? The answer is no, for example because we can take a Jacobson scheme (e.g., affine n-space over a field) and throw out the nonclosed points. We can get around this issue by asking the question: Is the ringed topos of such an X equivalent to the ringed topos of a scheme? I think the answer is yes, but I haven’t worked out all the details.

You can formulate each of the three properties in the setting of a ringed topos. (There are several variants of the third condition; we choose the strongest one.) An example would be the big Zariski topos of a scheme.