Let us say that a Noetherian local ring is a strong complete intersection if it is of the form S/(f_1, …, f_r) where S is a regular local ring and f_1, …, f_r is a regular sequence. It turns out that if R = S/I = S’/I’ where S, S’ are regular local rings, then I is generated by a regular sequence if and only if I’ is (this is not a triviality!). But, as there exist Noetherian local rings which are not the quotient of any regular local rings, this definition of a strong complete intersection does not make a whole lot of sense. Note that it is clear that if R is a strong complete intersection, then so is R_p for any prime ideal p of R.

The Cohen structure theorem tells us we can write the completion of any Noetherian local ring as the quotient of a regular local ring. Thus we say a Noetherian local ring is a complete intersection if its completion is a strong complete intersection. By the Cohen structure theorem we can write the completion as a quotient of a regular local ring, so this definition makes sense.

The problem: Why is the localization of a complete intersection at a prime a complete intersection?

The solution to this conundrum comes from a theorem of Avramov: If R —> R’ is a flat local homomorphism of Noetherian local rings, and if R’ is a complete intersection, then R is a complete intersection. How do we use this? Suppose that R is a complete intersection and p a prime ideal of R. Let R’ be the completion of R. As R —> R’ is faithfully flat, we can find a prime p’ of R’ lying over p. By assumption R’ is a strong complete intersection, hence R’_{p’} is a strong complete intersection. Since the local ring map R_p —> R’_{p’} is flat, Avramov’s theorem kicks in and we see that R_p is a complete intersection!

Cool, no?

This post is a brief review of deformation theory. I personally learned how to compute deformation spaces of singularities from Jozef Steenbrink, Theo de Jong and Duco van Straten (it is very enjoyable to compute deformation spaces of singularities late at night provided one has a large supply of either coffee or beer).

Consider a field k. Set R = k[x_1, …, x_n]. Let I be an ideal of R. Choose a set of generators f_1, …, f_r of I. Denote Rel the module of relations, i.e., such that we have a short exact sequence

0 —> Rel —> R^{oplus r} —> I —> 0

Suppose that A is an Artinian local ring with residue field k. Set R_A = A[x_1, …, x_n]. An embedded deformation of R/I over A is an ideal I_A ⊂ R_A such that I_A \otimes_A k = I and such that R_A/I_A is flat over A. It turns out that this is equivalent to the following:

1. I_A can be generated by elements f’_1, …, f’_r whose reductions modulo m_A are equal to f_1, …, f_r, and
2. for any (g_1, …, g_r) ∈ Rel there exist g’_1, …, g’_r in R_A such that ∑ g’_if’_i = 0.

If you’ve ever tried to compute the deformation space of a singularity then you’ve seen this. In particular, if A = k[ε] is the ring of dual numbers, then f_i’ = f_i + ε h_i and condition 2 implies that f_i → h_i defines a map from I to R/I. Thus the tangent space to this deformation functor is

T^1 = Hom_R(I, R/I) = Hom_R(I/I^2, R/I)

Note that this is typically an infinite dimensional space (except if R/I is Artinian), which make sense because we are doing embedded deformations. To get the first order deformation space of R/I as a singularity we can divide out by the module of derivations of R over k, but for this blog post we prefer not to do so.

Next, I want to consider the obstruction space to our given deformation problem. Suppose that we have a small extension A —> B of Artinian local rings with kernel K. Suppose that f’_1, …, f’_r is a bunch of elements of R_A whose images in R_B do define an embedded deformation. To see if f’_1, …, f’_r is also a deformation we need to check if relations G = (g_1, …, g_r) ∈ Rel lift to relations among the f’_1, …, f’_r. By the assumption that we have a deformation over B we know that we can pick (g’_1, …, g’_r) such that ∑ g’_i f’_i is an element Ob(G) of KR_A = R \otimes_k K. Picking different choices of g’_i changes Ob(G) by an element of KI. Hence the obstruction is a well defined map

Rel —> R/I \otimes_k K

Note that the trivial relations f_if_j = f_jf_i of course do lift to relations among the f’_i. Hence we see that the obstruction map is an element of

Ob ∈ Hom_R(Rel/TrivRel, R/I) \otimes_k K

where TrivRel ⊂ Rel is the module of trivial relations. However, the obstruction map above depends on the initial choice of lifts f’_i to elements of R_A (with more or less given images in R_B as we’re given the deformation over B). Altering the choice of these f’_i modifies Ob by an element of Hom_R(R^{oplus r}, R/I) \otimes_k K. Combining all of the above we see that

T^2 = Coker(Hom_R(R^{oplus r}, R/I) —> Hom_R(Rel/TrivRel, R/I))

Again this is usually an infinite dimensional vector space over k.

I’m going to try and say something intelligent about this obstruction space in a future blog post. But for the moment I just make the comment that Rel/TrivRel is equal to the first Koszul homology group for the ring R and the sequence of elements f_1, …, f_r.

Let R be a Noetherian ring and let I, J be ideals of R. Then Tor^R_*(R/I, R/J) is a differential graded algebra (with zero differential). How does one get this algebra structure?

In a paper published in 1957, John Tate came up with the following strategy: Try to find a strictly commutative differential graded R-algebra A endowed with divided powers (as in this post) together with a given augmentation ε : A —> R/I such that

1. H_i(A) = 0 for i > 0 and H_0(A) = R/I, and
2. A is obtained from R by successively adjoining divided power variables.

The first condition means that A is quasi-isomorphic to R/I as a dga (with divided powers) and the second implies that A is a free resolution of R/I as an R-module. Hence we see that Tor_*(R/I, R/J) is the homology of A \otimes_R R/J = A/JA which is a dga with divided powers.

Tate shows that you can construct such dga resolutions of R/I by successively adjoining variables to kill cycles; starting with the Koszul complex for a set of generators of I. In the book by Gulliksen and Levin it is checked that the dga which Tate gets is endowed with divided powers.

I’d just like to make here the observation that this also determines divided powers on Tor^R_*(R/I, R/J). This despite the problem that in general the homology of a dga with divided powers isn’t endowed with divided powers as I mentioned here.

Namely, let B be a dga with divided powes. It turns out that the only obstruction to defining γ_n on H_*(B) is that it may happen that y ∈ B of even degree is a coboundary but γ_n(y) isn’t.  For example if B is the divided power algebra over F_2 on x in degree 1 and y in degree 2 and d(x) = y, then γ_2(y) isn’t a coboundary! But, if there exists a surjection φ : A —> B of dgas with divided powers where A is such that H_i(A) = 0 for i > 0, then this disaster doesn’t happen. The reason is that writing y = d(x) and x = φ(x’) for some x’  ∈ A, then y’ = d(x’) is a coboundary in A, hence γ_n(y’) is a cocycle in A by the compatibility of divided powers with d, hence γ_n(y’) = d(x”) as H_i(A) = 0, hence γ_n(y) = d(φ(x”)).

And of course, in the situation of Tate’s construction above, we have that A/JA is the quotient of a dga acyclic in positive degrees!

Consider a differential graded algebra (A, d) sitting in homological degrees 0, 1, 2, … and with d : A_n —> A_{n – 1}. Then the cohomology H(A) is also a differential graded algebra (with zero differential of course).

We say that (A, d) is strictly commutative if xy = (-1)^e yx with e = deg(x)deg(y) and x^2 = 0 when x has odd degree. In this case H(A) is a strictly commutative differential graded algebra.

We say that (A, d) is a strictly commutative differential graded algebra endowed with divided powers if for every homogeneous element x of A in even degree d we have divided powers γ_n(x) of degree nd satisfying the usual rules for divided powers, and satisfying the compatibility

d(γ_n(x)) = d(x) γ_{n – 1}(x), for all n > 1

with the differential. Then H(A) is a strictly commutative differential graded algebra endowed with divided powers, right?

Wrong! Can you spot the mistake?

Are you attending the conference Moduli Spaces and Moduli Stacks in May? Or will you come by Columbia University in the next few months? If so, then you can order a Stacks Project T-shirt and pick it up here at the math department.

The picture shows me wearing the T-shirt; on the back it has “The Stacks Project” in white letters. Since I am trying to avoid spam orders, I will not put a link to the order form here, but I suggest you follow the link to the moduli conference above to find the link over there.

On any ringed topos there is a notion of a quasi-coherent sheaf, see Definition Tag 03DL. The pullback of a quasi-coherent module via any morphism of ringed topoi is quasi-coherent, see Lemma Tag 03DO.

Let (X, O_X) be a scheme. Let tau = fppf, syntomic, etale, smooth, or Zariski. The site (Sch/X)_{tau} is a ringed site with sheaf of rings O. The category of quasi-coherent O_X-modules on X is equivalent to the category of quasi-coherent O-modules on(Sch/X)_{tau}, see Proposition Tag 03DX. This equivalence is compatible with pullback, but in general not with pushforward, see Proposition Tag 03LC.

Let me explain this last point a bit. Suppose  f : X —> Y is a quasi-compact and quasi-separated morphism of schemes. Denote f_{big} the morphism of big tau sites. Let F be a quasi-coherent O_X-module on X. The corresponding quasi-coherent O-module F^a on (Sch/X)_{tau} is given by the rule F^a(U) = Γ(U, φ^*F) if φ : U —> X is an object of (Sch/X)_{tau}. In general, for a sheaf G on (Sch/X)_{tau} we have f_{big, *}G(V) = G(V \times_Y X). Hence we see that the restriction of f_{big, *}F^a to V_{Zar} is given by the (usual) pushforward via the projection V \times_Y X —> V of the (usual) pullback of F to V \times_Y X via the other projection. It follows from the description of quasi-coherent sheaves on (Sch/Y)_{tau} as associated to usual quasi-coherent sheaves on Y that f_{big, *}F^a is quasi-coherent on (Sch/Y)_{tau} if and only if formation of f_*F commutes with arbitrary base change. This is simply not the case, even for morphisms of varieties, etc.

On the other hand, we know that f_*F commutes with any flat base change (still assuming f quasi-compact and quasi-separated). Hence f_{big, *}F^a is a sheaf H on (Sch/Y)_{tau} such that H|_{V_{Zar}} and H|_{V_{etale}} are quasi-coherent. Moreover, the same argument shows that if G is any sheaf of O-modules on (Sch/X)_{tau} such that G|_{U_{Zar}} or G|_{U_{etale}} is quasi-coherent for every U/X then H = f_{big, *}G is a sheaf such that H|_{V_{Zar}} or H|_{V_{etale}} are quasi-coherent for any object V of (Sch/Y)_{tau}. Moreover, this property is also preserved by f_{big}^* as this is just given by restriction.

Thus a convenient class of O-modules on (Sch/X)_{tau} appears to be the category of sheaves of O-modules F such that F|_{U_{etale}} is quasi-coherent for all U/X. These “quasi quasi-coherent sheaves” are preserved under any pullback and pushforward along quasi-compact and quasi-separated morphisms. Via the approach I sketched here they give a notion of quasi quasi-coherent sheaves on the tau site of any algebraic stack with arbitrary pullbacks and pushforward along quasi-compact and quasi-separated morphisms. An interesting example of a quasi quasi-coherent sheaf is the sheaf of differentials Ω on the etale site that I mentioned in here.

Can anybody suggest a better name for these sheaves?

It is that exciting time of the year where some lucky few in the mathematical community get to choose a graduate school to go to. This post is for you guys. Here are my assumptions: You applied to a bunch of graduate schools and you got into a slightly smaller bunch of graduate schools. Now you think you have a problem: you have to choose one.

The first thing to realize is that this isn’t a problem at all. Very likely any choice you make is as good as any other: you are you no matter where you go. It is (in my opinion) a great privilege to be able to spend time doing math and your time in grad school is going to be perhaps the period in your life where you have the most time to do math ever. It is going to be wonderful!

On the other hand, the choice you make will likely have an enormous impact on what the rest of your life looks like. It will determine who your friends are, where you live, what you eat, etc, etc. Being a graduate student will put a new kind of psychological pressure on you and your time as a graduate student will some sometimes be horrible.

In other words, the choice you make will have an important impact on your life outside of math and I think actually that those consequences are possibly more important than the purely mathematical ones.

Before we get to deciding which school to go to, let’s think about what you will do when you get there: you will write a thesis with a thesis advisor. (To me, as an advisor, this is the only thing that matters.) In the first year or so, besides learning new material, you will choose(!?) your advisor. How will this happen? If you think about what math you understand best, then it is probably the material from the math lectures you liked most. Very likely you will end up working with the professor whose lectures you enjoy the most. I say there is no way of predicting how this will end up and I claim that it is best to go into grad school with no preconceived notion of what will happen.

Having said this, here is the Carpe Diem method of choosing a grad school:

1. Go somewhere else; try something new! Don’t become a graduate student at the institution you are an undergrad at.
2. If at all possible, visit the schools you got into. Talk to the graduate students there, attend a random lecture, and generally just soak in the atmosphere.
3. Try not to worry about extraneous issues like: stipend, teaching, housing, etc. (Of course you may have to for some reason.)
4. Don’t worry about availability of professors. You’ll find somebody to work with, but as I said above there is no telling how, when, why this will happen. If an institution has a certain track record of excellence, you can be sure this will continue in the near future. Moreover, once you are in grad school, the institution you are at has a certain responsibility to get you a PhD (provided you work hard, pass your general exam, etc, etc).
5. Finally, make your choice based on where you think you will enjoy living and working the most.

Just today I finally managed to fix the proof of “formally smooth + locally of finite presentation <=> smooth” for morphisms of algebraic spaces, see Lemma Tag 04AM. In fact, the implication “=>” isn’t hard, and is the result that is used in practice. In the current implementation, the proof of “<=” uses infinitesimal deformation of maps, and in particular a topos theoretic description of first order thickenings of algebraic spaces which we alluded to in this post, see Lemma Tag 05ZN and Lemma Tag 05ZN.

Here is a related fact:

Suppose that X —> Y —> Z are morphisms of algebraic spaces or schemes, that X —> Y is etale and that X —> Z is formally smooth. Then Y —> Z is formally smooth too. 

In other words, being formally smooth is etale local on the source and target. See Lemma Tag 061K for a more precise statement.

If X, Y, Z are schemes, then one can prove this by reducing to the affine case, using that formal smoothness is equivalent to the cotangent complex being a projective module in degree 0 [Edit 5/18/2011: Wrong! See here.], and using the distinguished triangle of cotangent complexes associated to a pair of compose-able ring maps.

If X, Y, Z are algebraic spaces, then one has to do a bit more work (I think). The proof of the reference above uses the material mentioned in the first paragraph and that Ω_{X/Z} is a locally projective, quasi-coherent O_X-module (see Lemma Tag 061I), which is fun in and of itself.

Here is a technically straightforward manner in which to introduce various categories of sheaves on algebraic stacks, and it is my intention to introduce sheaves on algebraic stacks in the stacks project along these lines. Please take a look and leave a comment if you see a problem with this approach.

Suppose that C is a site. Using conventions as in the stacks project:

1. If p : X —> C is a stack in groupoids over C, then we declare a family of morphisms {x_i —> x}_{i ∈ I} in X to be a covering if and only if {p(x_i) —> p(x)}_{i ∈ I} is a covering of the site C. In this way X becomes a site.
2. If f : X —> Y is a 1-morphism of stacks in groupoids over C, then f is a continuous and cocontinuous as a functor of sites. Hence f induces a morphism of topoi f : Sh(X) —> Sh(Y) with the property that the pull back of a sheaf G on Y is defined by the simple rule (f^{-1}G)(x) = G(f(x)). This construction is compatible with composition of 1-morphisms of stacks in groupoids.
3. Finally, if a : f —> g is a 2-morphism in the 2-category of stacks in groupoids over C, then a induces a 2-morphism a : f —> g in the 2-category of topoi.

In other words, this is a perfectly reasonable way to associate a site to each and every stack over C.

Next, let C = (Sch/S)_{fppf} be the category of schemes with the fppf topology as in the stacks project. An algebraic stack X is a category fibred in groupoids over C. Hence the construction above gives us a site X_{fppf} which we will call the fppf site of X. According to the remarks above this has a suitable 2-functoriality with regards to morphisms of algebraic stacks.

Variants: If X is an algebraic stack, then p : X —> C is also a stack fibred in groupoids over C endowed with the Zariski, smooth=etale (see this post), or syntomic topology. Hence we obtain variants X_{Zar}, X_{smooth}, and X_{syntomic} satisfying functorialities as above. Note that the underlying category is X in each case.

Here are some (I think) properties of these definitions:

1. if x is an object of X with U = p(x), then X_{fppf}/x is equivalent (as a site) to (Sch/U)_{fppf}. Hence given a sheaf F on X_{fppf} the cohomology groups H^p(x, F) are just fppf cohomology groups of some sheaf on (Sch/U)_{fppf}. This also works with the other topologies.
2. when the topology is etale=smooth or Zariski, then H^p(x, F) can be computed on the small etale or Zariski site of U.
3. In general X does not have a final object and does not have fibre products. If the diagonal of X is representable (by schemes) then X has all fibre products.
4. Assume the diagonal of X is representable. Let x_0 be an object of X such that U_0 = p(x_0) is a scheme surjective, flat, locally of finite presentation over X. The representable sheaf h_{x_0} surjects onto the singleton sheaf * in Sh(X_{fppf}). Moreover, the fibre products h_{x_0} \times_{*} h_{x_0}, h_{x_0} \times_{*} h_{x_0} \times_{*} h_{x_0}, etc are representable by x_1, x_2, etc with p(x_1) = U \times_X U, p(x_2) = U \times_X U \times_X U, etc. It follows formally from this (compare with Lemma Tag 01GC and Lemma Tag 01GY) that there is a spectral sequence E_1^{p, q} = H^q(x_p, F) => H^{p + q}(X_{fppf}, F) and by the above H^q(x_p, F) corresponds to fppf cohomology of F over the scheme U_p.
5. There is a similar spectral sequence for the smooth=etale topology if the morphism U_0 –> X is surjective and smooth and the diagonal of X is representable.
6. If X is general there is still a spectral sequence with E_1^{p, q} = H^q(U_p, F), but then the U_p are algebraic spaces.

The sheaf of differentials Ω_{X/S} of one scheme X over another scheme S is the target of the universal O_S-derivation d_{X/S} : O_X —> Ω_{X/S}. I remember being surprised to learn that people habitually define this sheaf using the conormal sheaf C_{X/Xx_SX} of the diagonal morphism of X over S^[1].

Why is it not the “right thing” to do? The reason is that both the conormal sheaf and the sheaf of differentials have a natural functoriality, and that the identification of C_{X/Xx_SX} with Ω_{X/S} is not compatible with this! Namely, consider the morphism that flips the factors on Xx_SX. This should clearly act by -1 on the conormal sheaf C_{X/Xx_SX} and by +1 on Ω_{X/S}. So there you go!

When X —> S is a morphism of algebraic spaces, then the diagonal morphism isn’t an immersion in general so the conormal sheaf is harder to define. In this case defining Ω_{X/S} as the target of the universal O_S-derivation d_{X/S} : O_X —> Ω_{X/S} on the small etale site of X works fine, see Tag 04CR.

Finally, suppose that X —> S is a morphism of algebraic stacks. We have yet to choose (in the stacks project) which site to use to define quasi-coherent sheaves on X. But in order to study differentials the only reasonable choice seems to be the lisse-etale site X_{lisse, etale}. Again there is a universal O_{S_{lisse, etale}}-derivation d : O_{X_{lisse, etale}} —> Ω. Now, (I think) Ω is not a quasi-coherent O_{X_{lisse, etale}}-module, and it is not what authors on algebraic stacks define as Ω_{X/S}, but for some purposes it might be the right thing to look at (e.g., deformation theory?).

Footnote 1: Yes, currently the stacks project also introduces sheaves of differentials for morphisms of schemes using this method. The first result is then that d_{X/S} is a universal derivation, see Lemma Tag 01UR. Having proven this, maps involving Ω_{X/S} are defined using the universal property.