In this post I claimed that a formally smooth ring map has a cotangent complex which is quasi-isomorphic to a projective module sitting in degree 0. I thought this was in Illusie’s thesis. But when Wansu Kim asked me for a reference, and when I tried to find it today, I couldn’t find it.
Now I think it is simply wrong! I constructed what I think is a counter example and put it in the chapter on examples (search for cotangent complex). Let me know if I made a mistake… again.
The stacks project now contains Artin’s Theorem Tag 06DC:
Let f : X —> Y be a 1-morphism of stacks in groupoids on (Sch/S)_{fppf}. Assume that X is representable by an algebraic space, f is representable by algebraic spaces, surjective, locally of finite presentation, and flat. Then Y is an algebraic stack.
Extremely loosely speaking this means that to verify that a stack in groupoids Y is an algebraic stack, it suffices to find a flat cover by scheme, smoothness not required!
This has some pleasing consequences which we have not yet spelled out in the stacks project (most of these are similar to the consequences to the corresponding result for algebraic spaces, see Theorem Tag 04S6 ff). For example
And so on and so forth. Moreover, in the process of proving the theorem stated above we proved some results on algebraicity of spaces of sections, relative morphisms, restriction of scalars, and finite Hilbert stacks, most of which can now be considerably improved.
Our next goal in the stacks project is to add more basic theory on algebraic stacks, add some material on deformation theory, and (perhaps) something on approximation theory.
A fun fact is that the graph of logical dependencies for Theorem Tag 06DC has depth 68 and has 4006 edges!
This is my second report on the second semester of a yearlong algebraic geometry course for first year graduate students here at Columbia University, based on Ravi Vakil’s lecture notes. The first report is here. Please visit Ravi’s Math 216 blog and find the complete set of lecture notes here.
Besides some minor and unimportant annoyances with the text, I found teaching out of these lecture notes very pleasant. What turned out to be, for me, a key feature of his notes is Ravi’s intent to do things in the correct generality:
We will work with as much generality as we need for most readers, and no more. In particular, we try to have hypotheses that are as general as possible without making proofs harder. The right hypotheses can make a proof easier, not harder, because one can remember how they get used.
As I worked my way through the material I felt that Ravi mostly succeeded in this and it gave me the confidence to be less general! For example, I proved a bunch of results on ample and very ample invertible sheaves in the course working only with morphisms between schemes of finite type over a field. I stuck with Noetherian integral schemes whilst defining the Weil divisor class group. I talked about effective Cartier divisors, but avoided talking about Cartier divisors (a horrrible invention IMHO). Etc, etc.
Doing this allowed me to cover more ground than I usually do in an algebraic geometry course. I was able to do pushforward and pullback of divisors for finite morphisms of regular curves and prove the “n = ∑ e_i f_i” formula if you know what I mean. I was able to introduce cohomology for quasi-coherent sheaves on quasi-compact and separated schemes and actually prove some interesting theorems about it, by only doing Cech cohomology (this is probably the best time saving feature of the notes — it is one of those “why didn’t I think of that” things). Using this I was able to prove q-gr(A) = Coh(X) when X = Proj(A). A trivial consequence of these basic theorems is then the Riemann-Roch theorem in the form χ(X, L) = deg(L) + χ(X, O_X) on a projective regular curve X.
Finally, at the end I diverged from Ravi’s notes. I introduced dualizing sheaves for projective regular curves X by requiring Serre duality to hold for locally free sheaves (since there is only H^0 and H^1 you don’t need the cup product here nor Ext groups). I proved the existence of an invertible dualizing sheaf ω_X on X by first proving it for P^1 and then (using duality for a finite flat morphism) for any X by choosing a nonconstant rational functor. I defined the genus of a projective regular curve X over a field k, assuming H^0(X, O_X) = k, as g(X) = dim H^1(X, O_X). Then Serre duality gives deg(ω_X) = 2g(X) – 2. I proved that every genus 0 regular projective curve X is a conic. I proved that ω_X = Ω_X^1 if X is smooth over k (although here I had to assume something about a trace map on differentials). Finally, I explained how this leads to Riemann-Hurewitz using functoriality of differentials.
It may seem depressing to not be able to get much beyond RR and RH in a yearlong algebraic geometry course. But what really counts is for students to learn a whole language. Working through Ravi’s notes is a great way for students to do this. Thanks Ravi!
Let R be a local ring. Let J ⊂ R be an ideal generated by a Koszul-regular sequence. Let I ⊂ J be an ideal such that R/I is a perfect object of D(R) and such that R/J is a perfect object of D(R/I). Then, is it true that I and J/I are generated by Koszul-regular sequences in R and R/I?
In the Noetherian case you can just say “regular sequence” and the conditions just mean that I has finite projective dimension over R and R/J has finite projective dimension over R/I. But the way the question is formulated makes it believe-able that if the question has answer “yes” in the Noetherian case then the answer is yes in the general case. I have tried to prove this and I have tried to find counter examples, but I failed on both counts. I would appreciate any comments or suggestions.
Just an update on what’s been going on since the last update. The following list is roughly in chronological order.
Among some of the properties of these we proved that local complete intersection morphisms are fppf local on the target and syntomic local on the source. Hence it makes sense to say that a morphism of algebraic spaces is a local complete intersection morphism. We should now be in a good position to define the “lci-locus” in the Hilbert stack, which is our next goal.
It turns out that moving a bunch of text inside one of the tex files is a very bad case for git’s diff algorithm. But if you use the patience diff using a switch then it works fine. For example
$ git --no-pager log --oneline --stat -n1 a081d8a
a081d8a Moved sections
more-morphisms.tex |11761 ++++++++++++++++++++++++++--------------------------
1 files changed, 5879 insertions(+), 5882 deletions(-)
and with patience diff I get
$ git --no-pager log --oneline --stat --patience -n1 a081d8a
a081d8a Moved sections
more-morphisms.tex | 1717 ++++++++++++++++++++++++++--------------------------
1 files changed, 857 insertions(+), 860 deletions(-)
I use “git diff” to quickly review what I’ve changed since the last commit. I was annoyed that moving a section or a lemma caused what is clearly the wrong output. Now I know how to avoid it.
The (entirely uninteresting) diff for the commit mentioned above can be found here. This link in particular shows that github doesn’t use patience diff!
Let A be an r x (r – 1) matrix. Set δ_i = (-1)^i times the determinant of the (r – 1) x (r – 1) matrix you get by deleting the ith row. Then we have the equality
(◊) (δ_1, …, δ_r) A = 0.
Let I be an ideal in a Noetherian regular local ring R with dim(R) = dim(R/I) + 2 = depth(R/I) + 2. Then I has a minimal resolution of the form
0 —> R^{r – 1} —> R^r —> I —> 0
where r is the minimal number of generators for I. Denote A the matrix defining the map R^{r – 1} —> R^r. In this situation Burch’s theorem tells us that I is generated by the δ_i, and in fact the map R^r —> I is (up to a unit) given by the row vector (δ_1, …, δ_r).
Why is this useful? Well, suppose you want to deform R/I (see this post). It is often easy to see that there are lots of deformations, but what isn’t so easy is to prove that there are any unobstructed deformations. But in the situation above we can just choose a family of matrices A(t) formally depending on an auxiliary parameter t. Then the minors δ_1(t), …, δ_r(t) of A(t) generate an ideal I(t) in R[[t]]. Then R[[t]]/I(t) is a flat deformation of R/I by the criterion from the post on deformation theory: all the relations lift to R[[t]] because the equation (◊) is universal and hence holds also for our matrix A(t).
As an example consider a fat point in C^2, for example given by the ideal I = (x^n, x^{n – 1}y, x^{n – 2}y^2, …, y^n). The matrix A is the matrix whose ith column look is (0, …, 0, x, -y, 0, …, 0) with x in the ith spot. We can deform this by picking (0, …, 0, x – ta_i, -y + tb_i, 0, …, 0) with a_i, b_i 2n pairwise distinct complex numbers. The deformed scheme for t = 1 has n(n + 1)/2 reduced points, namely the points (a_i, b_j) with j ≥ i.
In fact you can show that any deformation is given by deforming the matrix (by applying the Burch’s theorem which is more general than what I said above to the resolution of the deformed ideal), and hence all deformations are unobstructed and the deformation space of the singularity defined by I is smooth. This in particular shows that the Hilbert scheme of points of a smooth surface is smooth.
The key to the solution to (***) which we formulated here is the construction of a derivation on the Tate resolution of R/I. I am not going to explain this in detail, partly because I do not have a good intuition for why this derivation should exist. So I am only going to give you the flavor of the thing: although the following is correct in spirit it is likely not completely correct in all details (in particular, some of the arguments below should not be done on the level of co/homology but rather on the level of complexes).
Let R = k[[x_1, …, x_n]] and I = (f_1, …, f_r) ⊂ m_R with r minimal for I. Consider as before the sequence
0 — > Rel —> R^{⊕ r} —> I —> 0
Note that if (***) is false, then we obtain a surjective map ξ : Rel —> R/I which annihilates the submodule TrivRel of trivial relations. This in particular implies that Rel/TrivRel = R/I ⊕ (Other part). Because R^{⊕ r} —> R —> R/I —> 0 is the beginning of a minimal resolution we see that we obtain an element e(ξ) ∈ Ext_R^2(R/I, R/I). Cupping with e(ξ) determines an operation
e(ξ) : Tor_n(R/I, M) —> Tor_{n – 2}(R/I, M)
for any module M (in particular M = k or M = R/I). Use Tate’s method to find a free dga with divided powers A which is a resolution of R/I as in the post here. It turns out that the fact that ξ is zero on trivial relations implies that we can represent e(ξ) by a derivation j : A —> A compatible with divided power structure which is homogeneous of degree -2 (this is where Gulliksen whose name I mentioned before comes in). On the other hand, the decomposition Rel/TrivRel = R/I ⊕ (Other part) implies that there exists a nonzero element x in Tor_2(R/I, k) such that δ(x) is nonzero (as an element of k). But this is a contradiction because some divided power of x is zero (as R is regular so finite projective dimension) and on the other hand δ^n(γ_n(x)) = (δ(x))^n is nonzero.
It is really a beautiful trick (apparently due to Gulliksen) to play off against each other the derivation pushing things down in degree and the divided power structure to go back up into the area where the Tors are zero.
We continue the discussion. Let R be a regular complete local ring with residue field k. Let S = R[[x_1, …, x_n]]. Let I ⊂ R and J ⊂ S be ideals such that IS &sub J and such that A = R/I —> B = S/J is a flat ring homomorphism. Consider the map
(**) I/m_RI —> J/m_SJ
In the previous post we claimed that the cokernel of this map is (J + m_RS)/(m_SJ + m_RS). To see this choose f’_1, …, f’_r ∈ J whose images f_1, …, f_r in S/m_RS form a minimal system of generators for the ideal (J + m_RS)/m_RS which is the ideal cutting out B/m_AB in S/m_RS. Think of B as a flat deformation of B/m_AB over A. Then by the discussion in this post, the flatness assumption implies that any relation between f_1, …, f_r in S/m_RS lifts to a relation in S/IS. Hence any element h of J ∩ m_RS is an element of IS + m_RJ as desired.
But more is true. Namely, recall from this post that with these choices we obtain an obstruction map
Ob : Rel/TrivRel —> S/(JS + m_RS) \otimes_k I/m_RI
(unfortunately the notation between these two posts isn’t compatible) where Rel is the module of relations between the f_1, …, f_r in S/m_RS. Now because J ⊂ m_S we can compose Ob with the canonical map S/(JS + m_R S) —> k to get a reduced map
Ob_reduced : Rel/TrivRel —> I/m_RI.
At this point an argument along the lines of the argument in the first paragraph shows that (**) is injective if this reduced obstruction map is zero (in fact I think it is equivalent, but I didn’t check this).
Having arrived at this point we see that it suffices to prove the following (changing back to the notation in the post on deformation theory):
(***) Given a proper ideal I in R = k[[x_1, …, x_n]], a minimal set of generators f_1, …, f_r for I with module of relations Rel and submodule of trivial relations TrivRel ⊂ Rel. Then any R-module homomorphism Rel/TrivRel —> R/I has image contained in m_R/I.
It turns out that this result is contained in a paper by Vasconcelos and with a little bit more detail on the proof it is Lemma 2 in this paper by Rodicio. The key appears to be the technique of Tate to find divided power dga resolutions of R/I combined with a technique for constructing derivations on dgas which is due to Gulliksen. We will return to this in a future post.
Let A —> B be a flat local homomorphism of Noetherian local rings. By the local criterion for flatness this also implies that the map on completions A* —> B* is flat. Hence, in order to prove Avramov’s theorem that I mentioned here it suffices to prove it for a flat map of Noetherian complete local rings.
Let A be a complete local Noetherian ring. Write, using the Cohen structure theorem, A = S/I where S is a regular complete local ring. The complete intersection defect of A is the nonnegative integer
cid(A) = dim_k(I/mI) – dim(S) + dim(A)
where k = S/m is the residue field of S. Note that dim_k(I/mI) is the minimal number of generators of the ideal I. Since S is regular we see that A is a complete intersection if and only if cid(A) = 0.
Next, let A —> B be a flat local homomorphism of complete local Noetherian rings. Avramov proved, among other things, that
(*) cid(B) = cid(A) + cid(B/m_AB)
in this situation. It is clear that this proves the result we mentioned in the previous post.
What does (*) mean in more elementary terms? For simplicity, let us assume that the residue fields of A and B are identified by the map A —> B. Write A = R/I. Set S = R[[x_1, …, x_n]] for some large n and choose a surjection S –> B (here we use that the residue fields are equal). Set J ⊂ S equal to the kernel of S —> B so that B = S/J. Consider the induced map
(**) I/m_RI —> J/m_SJ
The equality (*) is equivalent to the injectivity of (**). Namely, flatness of A –> B gives dim(B) = dim(A) + dim(B/m_AB). By construction dim(S) = dim(R) + dim(k[[x_1, …, x_n]]). Finally, the map J/m_SJ —> (J + m_RS)/(m_SJ + m_RS) is surjective with kernel equal to the image of (**). (Proof omitted, but see next post.)
OK, now why is (**) injective? I claim that this is a question about the obstruction space for the deformation theory of the algebra B/m_AB over k. I will discuss this in the next post.