Here is a list of things we’ve added to the Stacks project since last summer. Newer things are listed first:

1. Artin’s theorem on contractions, see Tag 0GH7. The exposition follows Artin’s proof very closely. It was added recently, so improvements can be made and suggestions as to how to do so are welcomed.
2. Rachel Webb pointed out a serious error in the proof of Lemma 0A9Q in the chapter on duality for schemes and the corresponding Lemma 0E58 for algebraic spaces. See Example 0GEU for a counter example to the original formulation.
3. Noah Olander added some material on detecting boundedness of quasi-coherent complexes using a generator. See Section 0GEI and the material at the end of Section 0CSI. We also have the analogous material for algebraic spaces, see Section 0GFE and Lemma 0GFJ and Lemma 0GFL.
4. We upgraded some of the discussion in Pushouts of Spaces because it was needed for the proof of Artin’s theorem on contractions.
5. We discussed various “descent of \’etale sheaves” issues, e.g., if you have a proper surjective morphism f : X —> Y and an \’etale sheaf on X which is constant on the fibres of f, then it comes from an \’etale sheaf on Y. For a precise statement, see discussion in Section 0GEX. There is an analogous section for algebraic spaces somewhere.
6. Thanks to prompting by Tuomas Tajakka, we added the algebraic spaces case of the discussion of ample invertible modules and cohomology. See Section 0GF9
7. We added a rather large amount of material on formal algebraic spaces in Chapter 0AM7. In particular, given an adic, finite type morphism f : X —> Y of locally Noetherian formal algebraic spaces, we introduce carefully a number of “rig-properties” of f and prove some initial lemmas on these. A “rig-property” of f is a property of the restriction of f to the “generic fibres” of X and Y, except that the Stacks project doesn’t contain enough theory to make this precise. Anyway, I want to point something out here: the notion “rig-flat” is a rather tricky one! See Section 0GGK for the corresponding algebra discussion.
8. Yet another application local criterion flatness: Lemma 0GEB is the lemma you always wanted to know about, but you didn’t know it! No, really!
9. Thanks to discussions with Jarod Alper around his lectures on moduli theory on hikes here in WA, we much improved the discussion of the \’etale local structure of morphisms of schemes in Section 0CAT.
10. We explicitly formulated Artin’s axioms in the Noetherian setting for algebraic spaces, see Section 0GE6.
11. We revamped the discussion on algebraization of rig-etale and rig-smooth algebras as discussed in Elkik’s Theorem 7. You can read this in Section 0ALU, Section 0GAU, and Section 0AK5.
12. We fixed several errors pointed out by 李一笑
13. We added relative Poincare duality for de Rham cohomology, see Section 0G8F. Thanks to Shizhang Li for helping me with this. Let f : X —> S be a smooth proper morphism of relative dimension n. The key is to prove that the map d : Rⁿf_*Ω^{n – 1}_X/S ——> Rⁿf_*Ωⁿ_X/S is zero. The hard case is when S is the spectrum of a (nonreduced) Artinian local ring. After trying a *lot* of things that didn’t work, we found a proof using in some sense that the construction of this map is compatible with kunneth and the gysin map for the diagonal of X/S. I would appreciate references to places where relative Poincare is discussed in the literature.
14. We added some additional Kunneth formulas, see Section 0FLN, Section 0G4A, and Section 0FXX.
15. Pullbacks of K-flats with flat terms are K-flat with flat terms. Somehow we missed this the first time around. See Lemma 0G7E
16. We added a bunch of stuff on gysin maps in Hodge cohomology and related lemmas on cohomology with supports of quasi-coherent modules.
17. Lichtenbaum’s theorem Section 0G5D.
18. Duality for compactly supported cohomology coherent modules, see Section 0G59.
19. Bertini a la Jouanolou: just an amazing argument, no idea how you would come up with this. Read the original or see Section 0G4C.

Enjoy!

This week I learned an interesting fact about uniform annihilators of high degree Ext modules from a paper by Iyengar and Takahashi “The Jacobian ideal…”. I dare say there are many other places in the literature to read about it. In fact, I wouldn’t mind at all if you emailed me references where I could learn more about it.

The general gist of the results is that given a “good” Noetherian ring S and an ideal I ⊂ S cutting out the singular locus, then there exist integers m and i_0 such that for all i > i_0 the modules Ext^i_S(M, N) are annihilated by I^m. The key here is that m does not depend on M, N.

Iyengar and Takahashi show that this an essential ingredient if you want to prove strong generation for the category of modules and the derived category D^b_{Coh}(S). I would guess that conversely strong generation of D^b_{Coh}(S) will imply some uniform vanishing of Ext’s but I didn’t try to prove it; have you?

Let me explain a strategy to get a result like this. It only works if you can write S as the quotient of a regular ring and you have plenty of derivations. For example for finite type algebras over perfect fields the proposition below proves what I said above (but you can also find it in the literature of course). If you are still reading, the strategy is given in the proof of the proposition; I suggest skipping the details.

Let R be a regular ring. Let R —> S = R/J be a quotient. Assume we have f_1, …, f_c in J and derivations D_1, …, D_c : R —> R as well as an element z’ in R such that z’ J is contained in (f_1, …, f_c) + J^2. Let

z = det(D_i(f_j))

be as in the previous blog post. Finally, let n be the integer found in the first lemma of the previous post (this integer was found in a nonconstructive manner, but I hope somebody can tell me how to make it effective in some way).

Proposition: Let d = dim(R) < ∞. For any finite S-modules M, N we have z^{2n + 1} (z’)^{2n} annihilates Ext^{d + 1}_S(M, N).

Proof. Denote i_* : D(S) —> D(R) the pushforward and denote i* : D(R) —> D(S) the pullback. We have Ext^{d + 1}_R(i_*M, i_*N) = 0 because R is regular of dimension d. Thus we have Ext^{d + 1}_S(i*i_*M, N) = 0. But in the previous post we have seen that up to multiplication by z^{2n + 1} (z’)^{2n} the module M is a summand of i*i_*M. This concludes the proof. EndProof.

Remark If S is a finite type k algebra for some perfect field k and we choose a surjection R = k[x_1, …, x_t] —> S with kernel J then for choices f_1, …, f_c in J and derivations D_1, …, D_c on R, the elements z^{2n}(z’)^{2n + 1} we obtain in S generate an ideal whose vanishing locus is the singular set of Spec(S).

This blog post will be used in a later one. Please skip ahead to the next one.

Let R be a regular ring. Let R —> S = R/J be a quotient. Assume we have f_1, …, f_c in J and derivations D_1, …, D_c : R —> R as well as an element z’ in R such that z’ J is contained in (f_1, …, f_c) + J^2. Let B be the Koszul algebra on f_1, …, f_c over R and let

z = det(D_i(f_j))

be as in the previous blog post.

We can extend B —> S to a Tate resolution. Thus we may assume we have

R —> B —> A —-> S

where A is gotten from B by adjoining variables and extending the differential. In particular A —> S is a quasi-isomorphism and A is free over R and over B as a graded module and B —> A is the inclusion of a direct summand (as a graded B-module).

Lemma: There exists an n >= 1 such that (zz’)^n annihilates Cone(B ⊗ S —> A ⊗ S) in D(S). (Tensor products over R.)

Proof: After inverting zz’ the immersion is regular by Tag 0GEE. This uses that R is regular! For J = (f_1, …, f_c) and f_1, …, f_c a regular sequence the map is a quasi-isomorphism as both sides compute Tor^R_*(S, S). EndProof

Remark: For a while I tried to see if n = 1 is sufficient. I haven’t yet found a counter example. I think stuff in the literature may say that this is true if S is CM.

Lemma: Let M’ in D(S). Let M be a dg B-module which is graded free and which comes with a quasi-isomorphism M —> M’ of dg B-modules. The cone of the map M ⊗_B S —> M’ is annihilated by (zz’)^n

Proof. By standard things the module M ⊗_B S is quasi-isomorphic to M’ ⊗_B A. Then we can replace B by B’ = B ⊗_R S and A by A’ = A ⊗_R S. Thus we have to show that the cone on M’ ⊗_{B’} A’ —> M’ is annihilated by (zz’)^n. By the lemma above we know that the cone C’ of the map B’ —> A’ is annihilated by (zz’)^n. Now we have the short exact sequence of B’-modules

0 —> M’ —> M’ ⊗_{B’} A’ —> M’ ⊗_{B’} C’ —> 0

whose first arrow is a splitting to the arrow M’ ⊗_{B’} A’ —> M’. Anyway, this shows that the cone we are looking at is isomorphic to a shift of M’ ⊗_{B’} C’ which proves what we want. EndProof

Denote i_* : D(S) —> D(R) the pushforward and denote i* : D(R) —> D(S) the pullback. Let M be an object of D(S). We have a counit map

e : i*i_*M —> M

Lemma: For any S-module M’ there is a map s : M’ —> i*i_*M’ whose composition with e is equal to multiplication by z^{2n + 1} (z’)^{2n}.

Proof. Denote a_* : D(S) —> D(B), a* : D(B) —> D(S), b_* : D(B) —> D(R), b* : D(R) —> D(B) the usual functors. By the previous lemma we see that there is a map a*M —> M’ whose cone is annihilated by (zz’)^n. Then it suffices to prove the lemma for a*M but with the multiplier being z (in this argument we get the 2n powers of z and z’ in the statement). So we want to construct a map

a*M —> i*i_*a*M = a*b*b_*a_*a*M

For this we can first use the unit for a to get a*b*b_*M —> a*b*b_*a_*a*M. Then we can use the construction of the previous post to get a*M —> a*b*b_*M and this is where z comes in! EndProof

This blog post will be used in a later one. Please skip ahead to the next one.

Let f_1, …, f_c be elements of a ring R. We view the Koszul algebra B = K(R, f_1, …, f_c) as a differential graded R-algebra sitting in cohomological degrees -c, …, 0. So we have R = B^0 and B^{-1} is free over R with a basis x_1, …, x_c such that d(x_i) = f_i.

Of course, if f_1, …, f_c is a regular sequence, then B —> R/(f_1, …, f_c) is a quasi-isomorphism. But we are interested in the general case too.

Denote i_* : D(B) —> D(R) the pushforward and denote i* : D(R) —> D(B) the pullback. Let M be an object of D(B). We have a counit map

e : i*i_*M —> M

We want to find an element z in R such that for every M in D(B) there is a map s : M —> i*i_*M whose composition with e is multiplication by z on M. In other words, we want to split e up to multiplication by z.

Example: suppose that f_1, …, f_r is a regular sequence and that the map R —> R/(f_1, …, f_r) has a section. Then R —> B has a section too and we get what we want with z = 1.

Let M be a (right) dg module over B which is free as a graded module. Any dg module over B is quasi-isomorphic to one of these, so there is no loss of generality. The map e is the map

multiplication : M ⊗ B —> M

where the tensor product is over R. We are going to construct a map s : M —> M ⊗ B using derivations. Before we continue, observe that we do have a map of dg B-modules

ξ : M —> (M ⊗ B)[-c]

Namely, we can send m in M to

(sign)m ⊗ 1 (x_1 ⊗ 1 – 1 ⊗ x_1) … (x_c ⊗ 1 – 1 ⊗ x_c)

where the sign is (-1) to the power cdeg(m). The reason this works is that x_i ⊗ 1 – 1 ⊗ x_i is killed by the differential. There are some sign rules for the multiplication on B ⊗ B: we have (x_i ⊗ 1)(1 ⊗ x_j) = x_i ⊗ x_j and we have (1 ⊗ x_j)(x_i ⊗ 1) = – x_i ⊗ x_j.

Let D : R —> R be a derivation. Then we can extend D to a degree zero map D’ : B —> B which satisfies the Leibniz rule by setting D'(x_i) = 0. Of course D’ does not commute with d in general.

Suppose that M is a (right) dg module over B which is free as a graded module. (Any dg module over B is quasi-isomorphic to one of these.) Then we can similarly find a degree zero map D’ : M —> M which satisfies the Leibniz rule over D’.

In both cases consider the map θ : B —> B and θ : M —> M defined by the formula &theta = D’ o d – d o D’. Then θ has degree 1, defines a map B —> B[1] and M —> M[1] of complexes, θ : B —> B is a derivation, and θ : M —> M satisfies the Leibniz rule θ(mb) = θ(m)b + (sign) m θ(b) where the sign is (-1) to the power the degree of m.

A simple calculation shows that θ(x_i) = D(f_i).

Next, suppose we have c derivations D_1, …, D_c. Then we get c maps θ_1, …, θ_c : M —> M[1]. Then we can consider the composition

M — ξ –> (M ⊗ B)[-c] — θ_1 … θ_c ⊗ 1 –> M ⊗ B

Unless I made a calculation error (which is very possible) the composition of these maps with the map e : M ⊗ B —> M is equal to multiplication by

z = det(D_i(f_j))

Thus we conclude what we want with z as above.

The conclusion of this is a precise version of something we all already know: if we have a closed embedding i of codimension c and we have c tangent fields spanning the normal bundle, then we can split the counit map i*i_*M —> M using those tangent fields.