Intersection Theory

Today I finished the first complete version of a chapter on intersection theory. As usual comments and suggestion are very welcome.

The chapter uses Serre’s Tor formula and moving lemmas to define an intersection product on the Chow groups of nonsingular projective varieties over an algebraically closed ground field and that is all it does. You can read the introduction for a little bit more information.

There are some improvements that can be made to this chapter. The first is that some of the material on Serre’s Tor formula belongs properly in one of the chapters on commutative algebra. Of course, there is a lot more one can say about regular local rings and the Tor formula, leading up to recent work on homological conjectures in commutative algebra. Also, some of the arguments in the moving lemmas use geometric arguments on varieties over algebraically closed fields and we need to write more of the API to easily translate these into scheme theoretic language. Finally, the chapter is missing examples and more references to the literature.

What often happens with new chapters is that a few years down the road, we take a second look and make substantial improvements.

One aspect of the material in the new chapter is that it was not as straightforward to write as the material on constructible sheaves which was like butter. The conclusion must therefore be that intersection theory is not like butter!

Nonetheless: Enjoy!

Updated again

As you may have noticed I do not always immediately respond to the comments on the stacks project. It turns out to be more efficient to wait till there are quite a few of them and then take out a block of time to deal with them. I just finished with this once more and this time it took a bit more time. Thanks especially to Juan Pablo Acosta López.

Chow homology

Last week I started looking at the chapter on chow homology. When I first wrote it in 2009, it was a bit of an experiment. As a consequence, various proofs used three different approaches: one using K-groups, one using blow-up lemmas as in Fulton, and one via the key lemma (see below). Today I rearranged the whole thing to simplify the exposition and stick with the approach using the key lemma.

How much intersection theory does the chapter cover? We work consistently with schemes locally of finite type over a fixed universally catenary, locally Noetherian base scheme. We introduce cycle groups, flat pullback, proper pushforward, and rational equivalence. After proving some basic properties, we introduce the operation c_1(L) ∩ – where L is an invertible sheaf and the Gysin map for an effective Cartier divisor. Having proved the basic properties of these operations, we have enough to introduce chern classes of locally free sheaves and prove their basic properties. The chapter ends with stating the Grothendieck-Riemann-Roch theorem (without proof).

The key lemma is a statement about tame symbols over a Noetherian local domain of dimension 2. I think of it as the statement that the secondary ramifications add up to zero. I’d love it if you could tell me a reference for this lemma in the literature (I assume there is some paper on K-theory that contains this result).

The key lemma implies our key formula. I am sure I discussed this statement with somebody in my office at some point, but I cannot remember who; if it was you, please email me! Anyway, the key formula quickly implies that c_1(L) ∩ – passes through rational equivalence (Section Tag 02TG), the fact that c_1(L) ∩ c_1(N) ∩ – = c_1(N) ∩ c_1(L) ∩ -, and that the Gysin map for an effective Cartier divisor passes through rational equivalence (Section Tag 02TK).


Up to date again

With a very stuffy nose and lot’s of sneezing, today I worked through the comments that were left on the Stacks project. Thanks to all of you!

Here is what is very helpful (especially on those days were my head is all stuffed up): when you find a mistake, please also point out how to fix it (if you know how). Even just making a guess at what went wrong is helpful. Thanks!

Up to date with comments

Hello again! This is just to let you know that I’ve just finished working through the comments on the Stacks project and all (known) mistakes, typos etc are fixed. Thanks for all your help! Please go back there and find more.

Also, please let me know if you or somebody else have helped out in the past and we’ve forgotten to add your or their name to the contributors list. It seems idiotic, but it turns out it is actually kind of hard to keep track of everybody who is helping out (and in some sense we still do not have a really good way to do this). Any suggestions on how to manage this are very welcome too.


So I’ve been playing around with tracing the history of lemmas, etc in the Stacks project over time. I’ve written a bunch of scripts to extract this information out of the git logs. Here a pdf file showing what happened to Nakayama’s lemma over time. Enjoy!

PS: Feel free to improve on the current version of Nakayama’s lemma in the Stacks project and to submit it for inclusion, so we can add more stages to its history!


We clarify the discussion in this post resulting in a generalization of a result of Mike Artin.

Let X be a Noetherian algebraic space. Let T ⊂ X be a closed subspace. Let us denote X/T the formal completion of X along T (Tag 0AIX). Let W —> X/T be a rig-etale morphism of formal algebraic spaces, which means that

  1. W —> X/T is representable by algebraic spaces, i.e., it is an adic morphism of formal algebraic spaces Tag 0AQ2),
  2. W —> X/T is locally of finite type, i.e., it is etale locally on affine formal algebraic pieces given by a continuous ring map A —> B which is topologically of finite type Tag 0ALL),
  3. these ring maps A —> B have a completed cotangent complex whose cohomology groups are annihilated by an ideal of definition of A, for more details see Tag 0ALP and Tag 0AQK.

These three conditions correspond to condition (i) of Definition 1.7 of Artin’s paper “Formal Moduli: II”. The first result is that

Given W —> X/T rig-etale there exists a morphism of algebraic spaces Y —> X which is an isomorphism over X – T and whose completion Y/T —> X/T is isomorphic to W —> X/T.

In fact, Theorem 0ARB tells us we obtain an equivalence of categories.

This theorem does not often produce separated morphisms Y —> X if we start with a random W —> X/T. A typical thing that happens can be seen by starting with X equal to the affine line, T = {0} and W two copies of the completion of X at 0. Then the resulting Y is the affine line with 0 doubled.

Thus to get Artin’s theorem on dilatations we need to impose conditions on W —> X/T guaranteeing that Y —> X is separated or even proper. To do this we will use the notion of a rig-surjective morphism W’ —> W of locally Noetherian formal algebraic spaces W, W’ defined by requiring adic morphisms Spf(R) —> W with R a cdvr to lift to Spf(R’) —> W’ for some extension of cdvr R ⊂ R’ (Tag 0AQP). Let’s say an adic morphism W —> W’ of locally Noetherian formal algebraic spaces is a rig-monomorphism if the diagonal morphism is rig-surjective. In this language the conditions (ii) and (iii) from Definition 1.7 of Artin’s paper have the following interpretations:

  1. If W —> X/T as above is separated and a rig-monomorphism, then Y —> X is separated (Tag 0ARW),
  2. If W —> X/T as above is proper, a rig-monomorphism, and rig-surjective, then Y —> X is proper (Tag 0ARX).

The second statement recovers exactly Artin’s theorem on dilatations.

One typically applies the result to construct modifications Y —> X (Tag 0AD7) by taking the complete local ring A of X at a closed point x and setting W equal to the formal completion of the blow up of A at an ideal I ⊂ A which is locally principal on the puctured spectrum of A. Here the funny situation occurs that we can first read Tag 0ARX backwards over Spec(A) to conclude that W —> X/x has the required properties and then forwards to conclude that Y —> X is proper. In other situations it may not be that easy to verify the assumptions needed for the application of the theorem and it would behoove us to prove a few lemmas that help with this task.

Your help with this and other tasks is always welcome!

Spaces are fpqc sheaves

Although already mentioned in the previous post I’d like to point out once again that we have, thanks to Ofer Gabber, now a proof that algebraic spaces satisfy the sheaf condition for fpqc coverings, answering one of the questions in this post. Also thanks go to Bhargav Bhatt who sent me an email (on 9/13/13) explaining Gabber’s argument. With small comments/corrections/clarifications added over the weekend, I hope most of the kinks I introduced have been ironed out.

To read more, start here. As usual comments and suggestions for improvements are welcome. Enjoy!


Since the last update of June 2013 we have added the following material:

  1. Algebraicity of stack of coherent sheaves Tag 09DS
  2. Epp’s Theorem Tag 09F9 and Tag 09II
  3. Reduced fibre theorem Tag 09IL
  4. New chapter on fields Tag 09FA
  5. Perfect generator for DQCoh(X) for schemes Tag 09IS
  6. Perfect generator for DQCoh(X) for algebraic spaces Tag 09IY
  7. New chapter on differential graded algebra Tag 09JD
  8. Section on Noetherian schemes of dimension 1 Tag 09N7
  9. Finite set of codimension 1 points on separated scheme are in an affine Tag 09NN
  10. Section on curves Tag 0A22
  11. Versions of Avramov’s result Tag 09Q7 and Tag 09QF
  12. Obtaining triangulated categories Tag 09P5 written by John Yu and Yifei Zhao
  13. Rickard’s theorem Tag 09S5
  14. Section on compact objects Tag 09SM
  15. Section on generators in triangulated categories Tag 09SI
  16. DQCoh = D(QCoh) for Noetherian schemes Tag 09TI
  17. DQCoh = D(QCoh) for Noetherian spaces Tag 09TH
  18. Final exam commutative algebra Fall 2013 Tag 09TV
  19. Lots of stuff about pseudo-coherence
  20. Cohomology of locally compact spaces Tag 09V0
  21. Proper base change in topology 09V4
  22. New chapter on simplicial spaces Tag 09VI
  23. Cohomological descent for proper hypercoverings in topology Tag 09XA
  24. Section on henselian pairs Tag 09XD
  25. Finite cover by a scheme Tag 0ACX
  26. Partitions and stratifications Tag 09XY
  27. Bunch of stuff about limits of sites and spaces and cohomology
  28. Gabber’s result on Henselian pairs Tag 09Z8
  29. Theorem of formal functiosns via derived completion Tag 0A0H
  30. Lots of edits to the chapter on etale cohomology culminating in a proof of the proper base change theorem Tag 095S
  31. Cohomological dim <= Krull dim for spectral spaces Tag 0A3C
  32. Equivalence beteen torsion and derived complete modules Tag 0A6V
  33. Local duality a la Grothendieck Tag 0A81
  34. Brown representability for triangulated categories Tag 0A8E
  35. Final exam algebraic geometry Spring 2014 Tag 0AAL
  36. Bunch of stuff about twisted inverse image, dualizing complexes, upper shriek functors
  37. New mostly empty chapter on resolution of surfaces Tag 0ADW
  38. Algebraic spaces are schemes in codimension 1 Tag 0ADD
  39. Extremely careful discussion of points on fibres of morphisms of algebraic fibres and when this implies the morphism is quasi-finite at the points Tag 0AC0
  40. Finite groupoids Tag 0AB8
  41. fppf descent data for spaces over spaces are effective Tag 0ADV
  42. New chapter on pushouts of algebraic spaces Tag 0AHT including a discussion of something called formal glueing which should really have another name
  43. Regular local rings are UFDs Tag 0AFW
  44. Approximation for henselian pairs Tag 0AH4

This brings us up to July 1 of this year. At this point I decided to work out what happens with dilatations in Artin’s paper “Formal Moduli II”. In fact, I wrote a blog post trying to figure out what could be true. Anyway, it turns out that Artin’s theorem on dilatations holds without assuming the base Noetherian algebraic space is excellent and in the proof we don’t need to use Artin’s criteria. See Theorem Tag 0ARB and combined this with Lemma Tag 0ARX to obtain Artin’s theorem — I hope to discuss what these results mean in more detail in a later blog post. Here are some related and not so related changes to the Stacks project.

  1. A chapter on formal algebraic spaces. Our approach is somehow a combination of all approaches in the literature: EGA, McQuillan, Yasuda, Beilinson-Drinfeld, Abbes, Fujiwara-Kato, and Knutson. Namely, we define a formal algebraic space as an fppf sheaf on (Sch) which \’etale locally looks like an Ind-scheme whose transition morphisms are thickenings. For the Stacks project this approach turns out to be exactly the right one and many of the earlier results on algebraic spaces can be brought to bear to get ahead fairly quickly.
  2. A followup chapter discussing the notion of rig-etale (homo)morphisms ending with a proof of the algebraization theorem mentioned above. These rig-etale maps are used by Artin in his paper mentioned above to describe formal modifications and these (as well as the similarly defined rig-smooth ring maps) are the ring maps studied in Elkik’s famous paper.
  3. Adjoint functor theorem Tag 0AHM
  4. Remarkable result by Yasuda that every qcqs formal algebraic space is a filtered colimit of algebraic spaces along thickenings Tag 0AJD
  5. Non flat completions Tag 0AL8
  6. QCoh not abelian for formal algebraic spaces Tag 0ALF
  7. Ofer Gabber answered completely this question see Tag 0APW
  8. Algebraic spaces are fpqc sheaves by Ofer Gabber Tag 03W8


150 contributors

Just finished working through the recent comments left on the Stacks project site and because there were a couple of new contributors we are now up to 150 contributors. Go us!

Anyway, please continue helping out by finding mistakes, etc, thank you very much. Also, we are going to have a party(!) when we reach 5000 pages, so please help: submit a new chapter, add an omitted proof, provide an alternative argument, or just randomly submit a piece of material you think may be useful, so we can get there more quickly. Thanks!