### MaBloWriMo

All right. I am now inclined to switch topics a little (I am looking forward to saying a few words about local cohomology), so I will sketch a few details in the present post. The goal is to compute the sheaf cohomology groups of the canonical line bundles on projective space. The argument will follow EGA III.2; Hartshorne does essentially the same thing (namely, analysis of the Cech complex) but without the Koszul machinery, so his approach seems more opaque to me.

Now, let us compute the cohomology of projective space ${X = \mathop{\mathbb P}^n_A}$ over a ring ${A}$. Note that ${X}$ is quasi-compact and separated, so we can compute the Cech cohomology by the above machinery. That is, we will use the Koszul-Cech connection discussed two days ago. In particular, we will consider the quasi-coherent sheaf

$\displaystyle \mathcal{H}=\bigoplus_{m \in \mathbb{Z}} \mathcal{O}(m)$

We shall now approach the proof of the Cartan vanishing theorem. First, however, it will be necessary to describe a spectral sequence between Cech cohomology and derived functor cohomology. For now, the reason is that there isn’t any obvious way for us to compute derived functor cohomology, because injective sheaves are big and scary, while Cech cohomology is nice and concrete. And indeed, all we’ve done so far is compute various Cech cohomologies.

I should mention that I don’t know a standard reference for the material in this post. I didn’t find Godement’s treatment in Theorie des faisceaux to be terribly enlightening, but after a fair bit of googling I found a sketch in James Milne’s online notes on étale cohomology. Fortunately, enough details are given to enable one to work it out more fully for oneself.

Let ${X}$ be a topological space covered by an open cover ${\mathfrak{A} = \left\{U_i\right\}_{i \in I}}$, and consider the category ${\mathfrak{C}}$ of presheaves of abelian groups on ${X}$. Let ${\mathfrak{C}'}$ be the subcategory of sheaves. The spectral sequence will be the Grothendieck spectral sequence of the composite of functors

$\displaystyle \mathfrak{C}' \stackrel{F}{\rightarrow} \mathfrak{C} \stackrel{G}{\rightarrow} \mathbf{Ab}.$

So last time, when (partially) computing the cohomology of affine space, we used a fact about the Koszul complex. Namely, I claimed that the Koszul complex is acyclic when the elements in question generate the unit ideal. This was swept under the rug, and logically I should have covered that before getting to yesterday’s bit of algebraic geometry. So today, I will backtrack into the elementary properties of the Koszul complex, and prove a more general claim.

0.9. A chain-homotopy on the Koszul complex

Before proceeding, we need to invoke a basic fact about the Koszul complex. If ${K_*(\mathbf{f})}$ is a Koszul complex, then multiplication by anything in ${(\mathbf{f})}$ is chain-homotopic to zero. In particular, if ${\mathbf{f}}$ generates the unit ideal, then ${K_*(\mathbf{f})}$ is homotopically trivial, thus exact. This is one reason we should restrict our definition of “regular sequence” (as we do) to sequences that do not generate the unit ideal, or the connection with the exactness of the Koszul complex wouldn’t work as well.

Proposition 33 Let ${g \in (\mathbf{f})}$. Then the multiplication by ${g}$ map ${K_*(\mathbf{f}) \rightarrow K_*(\mathbf{f})}$ is chain-homotopic to zero.

Proof: Let ${\mathbf{f} = (f_1, \dots, f_r)}$ and let ${g = \sum g_i f_i}$. Then there is a vector ${q_g = (g_1, \dots, g_r) \in R^r}$. We can define a map of degree one

$\displaystyle H: K_*(\mathbf{f}) \rightarrow K_*(\mathbf{f})$

It’s funny that the topics I planned to discuss on this blog in the next month have turned out to largely match the topics covered in my commutative algebra class. As a result, I think I will move the focus closer to algebraic geometry. Today, I shall explain how the connection explained yesterday between Koszul and Cech cohomology lets us compute the cohomology of an affine space. This is the proof that Grothendieck gives in EGA III, and it has the advantage (unlike the argument in Hartshorne, though I’ll probably later end up explaining that too) of applying to non-noetherian rings. It is a nice application of the basic properties of the Koszul complex, though it has the disadvantage of requiring a rather non-elementary result in sheaf theory (which I’ll discuss next).

0.10. The cohomology of affine space

We are now going to prove the first fundamental theorem on the cohomology of quasi-coherent sheaves:

Theorem 35 (Cohomology of an affine) Let ${R}$ be a ring, and let ${\mathcal{F}}$ be a quasi-coherent sheaf on ${X=\mathrm{Spec} R}$. Then$\displaystyle H^k( X, \mathcal{F})=0, \quad k \geq 1.$

I have earlier discussed a proof due to Kempf. What we will now sketch is a much less elementary and significantly more complicated argument. Nonetheless, it has the virtue of being general, and telling us something about projective space too, as we shall see eventually.

Proof: This proof proceeds first by analyzing the Cech cohomology. We will show that this is zero. Then, we shall appeal to some general sheaf-theoretic business to prove the result for standard cohomology. In particular, we are going to prove:

Theorem 36 Let ${\mathcal{F}}$ be a quasi-coherent sheaf on ${X=\mathrm{Spec} R}$. Let ${\left\{f_i\right\} \subset R}$ be a finite set of elements generating the unit ideal. Then the higher Cech cohomology of ${\mathcal{F}}$ with respect to the open cover ${D(f_i)}$ vanishes.

Up until now, I have been talking primarily about the commutative algebra purely. I think I now want to go in a more algebro-geometric direction, partially because I find it easier to understand that way. Today, I will explain how the Koszul complex lets you compute certain types of Cech cohomology.

0.8. The Koszul complex and Cech cohomology

What we now want to show is that on a reasonable scheme, Cech cohomology of a quasi-coherent sheaf is really a type of Koszul cohomology. Namely, let’s start with a scheme ${X}$, which I will take to be quasi-compact and quasi-separated. (If you are what Ravi Vakil calls a noetherian person, then you can ignore the previous remark.)

Let ${\mathcal{F}}$ be a quasi-coherent sheaf on ${X}$. Let ${f_1, \dots, f_r \in \Gamma(X, \mathcal{O}_X)}$ be global regular functions on ${X}$. Then we can define the sets ${X_{f_i}}$ where the functions ${f_i}$ “don’t vanish” (more precisely, are units in the local ring). One of the basic results one proves is that taking sections over these basic open sets corresponds to localization:

Proposition 29 ${\Gamma(X_f, \mathcal{F}) = \Gamma(X, \mathcal{F})_f}$ if ${\mathcal{F}}$ is quasi-coherent.

Proof: This is a general fact about quasi-coherent sheaves, and one way to see it is to use the fact that if ${A = \Gamma(X, \mathcal{O}_X)}$ is the ring of global functions, there is a morphism ${g: X \rightarrow \mathrm{Spec} A}$. This is a quasi-separated, quasi-compact morphism by hypothesis. Thus the direct image ${g_*(\mathcal{F})}$ is quasi-coherent. In particular, this means that

$\displaystyle \Gamma(\mathrm{Spec} A, g_*(\mathcal{F}))_f = \Gamma(D(f), g_*(\mathcal{F}))$

where ${D(f) \subset \mathrm{Spec} A}$ is the basic open set. When one translates this back via the definition of ${f_*}$, one gets the proposition.

We now continue with the original question. So let ${M = \Gamma(X, \mathcal{F})}$ be the global sections of the sheaf ${\mathcal{F}}$. We have seen that ${M_{f_i}}$ is ${\Gamma(X_{f_i}, \mathcal{F})}$ for each ${i}$. Similarly, ${M_{f_{i_1} \dots f_{i_k}}}$ is ${\Gamma(X_{f_{i_1}} \cap \dots \cap X_{f_{i_k}}, \mathcal{F})}$ for any ${k}$-tuple of the ${f_i}$. To avoid triple subscripts, let us write ${U_{i_k}}$ instead of ${X_{f_{i_k}}}$. This is precisely what we need to consider the Cech cohomology with respect to the open sets ${\mathfrak{A} = \left\{X_{f_i}\right\}}$. (more…)

I’ve not been a very good MaBloWriMo participant this time around. Nonetheless, coursework does tend to sap the time and energy I have for blogging. I have been independently looking as of late at the formal function theorem in algebraic geometry, which can be phrased loosely by saying that the higher direct images under a proper morphism of schemes commute with formal completions. This is proved in Hartshorne for projective morphisms by first verifying it for the standard line bundles and then using a (subtle) exactness argument, but EGA III.4 presents an argument for general proper morphisms. The result is quite powerful, with applications for instance to Zariski’s main theorem (or at least a weak version thereof), and I would like to say a few words about it at some point, at least after I have a fuller understanding of it than I do now. So I confess to having been distracted by algebraic geometry.

For today, I shall continue with the story on the Koszul complex, and barely begin the connection between Koszul homology and regular sequences. Last time, we were trying to prove:

Proposition 24 Let ${\lambda: L \rightarrow R, \lambda': L' \rightarrow R}$ be linear functionals. Then the Koszul complex ${K_*(\lambda \oplus \lambda')}$ is the tensor product ${K_*(\lambda) \otimes K_*(\lambda')}$ as differential graded algebras.

So in other words, not only is the algebra structure preserved by taking the tensor product, but when you think of them as chain complexes, ${K_*(\lambda \oplus \lambda') \simeq K_*(\lambda) \oplus K_*(\lambda')}$. This is a condition on the differentials. Here ${\lambda \oplus \lambda'}$ is the functional ${L \oplus L' \stackrel{\lambda \oplus \lambda'}{\rightarrow} R \oplus R \rightarrow R}$ where the last map is addition.

So for instance this implies that ${K_*(\mathbf{f}) \otimes K_*(\mathbf{f}') \simeq K_*(\mathbf{f}, \mathbf{f}')}$ for two tuples ${\mathbf{f} = (f_1, \dots, f_i), \mathbf{f}' = (f'_1, \dots, f'_j)}$. This implies that in the case we care about most, catenation of lists of elements corresponds to the tensor product.

Before starting the proof, let us talk about differential graded algebras. This is not really necessary, but the Koszul complex is a special case of a differential graded algebra.

Definition 25 A differential graded algebra is a graded unital associative algebra ${A}$ together with a derivation ${d: A \rightarrow A}$ of degree one (i.e. increasing the degree by one). This derivation is required to satisfy a graded version of the usual Leibnitz rule: ${d(ab) = (da)b + (-1)^{\mathrm{deg} a} a (db) }$. Moreover, ${A}$ is required to be a complex: ${d^2=0}$. So the derivation is a differential.

So the basic example to keep in mind here is the case of the Koszul complex. This is an algebra (it’s the exterior algebra). The derivation ${d}$ was immediately checked to be a differential. There is apparently a category-theoretic interpretation of DGAs, but I have not studied this.

Proof: As already stated, the graded algebra structures on ${K_*(\lambda), K_*(\lambda')}$ are the same. This is, I suppose, a piece of linear algebra, about exterior products, and I won’t prove it here. The point is that the differentials coincide. The differential on ${K_*(\lambda \oplus \lambda')}$ is given by extending the homomorphism ${L \oplus L' \stackrel{\lambda \oplus \lambda'}{\rightarrow} R}$ to a derivation. This extension is unique. (more…)

We are now going to discuss another mechanism for determining the length of maximal ${M}$-sequences, namely the Koszul complex. This is going to be super-useful in a whole bunch of ways. For one thing, it is integral in the proof that regular local rings are of finite global dimension, because the Koszul complex becomes a free resolution of the residue field.

Another one, which has excited me as of late, is that if you have a suitable scheme (say, quasi-compact and quasi-separated) and a quasi-coherent sheaf on it, then its Cech cohomology is in fact a direct limit of Koszul cohomologies! So properties of Koszul cohomology can be used to compute the cohomology of projective space as in Hartshorne, and thus to prove the fundamental theorem that higher direct images by projective (and, eventually, proper) morphisms preserve coherence. But this is getting rather far afield of what I want to talk about today.

Let ${L}$ be a finitely generated ${R}$-module. Consider the graded commutative algebra ${K = \bigwedge L = \bigoplus \wedge^i L}$ with the product given by the wedge product; the graded commutativity is similar to the cup-product in cohomology, and implies that

$\displaystyle x \wedge y = (-1)^{\deg x \deg y} y \wedge x.$

Given ${\lambda: L \rightarrow R}$, we can define a differential on ${K}$ as follows. Namely, we define

$\displaystyle d( x_1 \wedge \dots \wedge x_n) = \sum_i (-1)^i\lambda(x_i) x_1 \wedge \dots \wedge \hat{x_i} \wedge \dots \wedge x_n.$

(More precisely, this clearly defines an alternating map ${L^n \rightarrow \wedge^{n-1} L}$, and this thus factors through the alternating product by the universal property.) It is very easy to see that ${d \circ d = 0}$. Moreover, ${d}$ is an anti-derivation. If ${x,y \in K}$ are homogeneous elements of the graded algebra, then

$\displaystyle d(x\wedge y) = d(x)\wedge y + (-1)^x x \wedge d(y)$

Definition 20 The complex, together with the multiplicative structure, just defined is called the Koszul complex and is denoted ${K_*(\lambda)}$.

So I’ve missed a few days of MaBloWriMo. But I do have a talk topic now (I was mistaken–it’s actually tomorrow)! I’ll be speaking about some applications of Sperner’s lemma. Notes will be up soon.

Today I want to talk about how depth (an “arithmetic” invariant) compares to dimension (a “geometric” invariant). It turns out that the geometric invariant wins out in size. When they turn out to be equal, then the relevant object is called Cohen-Macaulay. This is a condition I’d like to say more about in future posts.

0.5. Depth and dimension

Consider an ${R}$-module ${M}$, which is always assumed to be finitely generated. Let ${I \subset R}$ be an ideal with ${IM \neq M}$. We know that if ${x \in I}$ is a nonzerodivisor on ${M}$, then ${x}$ is part of a maximal ${M}$-sequence in ${I}$, which has length ${\mathrm{depth}_I M}$ necessarily. It follows that ${M/xM}$ has a ${M}$-sequence of length ${\mathrm{depth}_I M - 1}$ (because the initial ${x}$ is thrown out) which can be extended no further. In particular, we find

Proposition 15 Hypotheses as above, let ${x \in I}$ be a nonzerodivisor on ${M}$. Then$\displaystyle \mathrm{depth}_I (M/xM) = \mathrm{depth} M - 1.$

This is strikingly analogous to the dimension of the module ${M}$. Recall that ${\dim M}$ is defined to be the Krull dimension of the topological space ${\mathrm{Supp} M = V( \mathrm{Ann} M)}$ for ${\mathrm{Ann} M}$ the annihilator of ${M}$. But the “generic points” of the topological space ${V(\mathrm{Ann} M)}$, or the smallest primes in ${\mathrm{Supp} M}$, are precisely the associated primes of ${M}$. So if ${x}$ is a nonzerodivisor on ${M}$, we have that ${x}$ is not contained in any associated primes of ${M}$, so that ${\mathrm{Supp}(M/xM)}$ must have smaller dimension than ${\mathrm{Supp} M}$. That is,

$\displaystyle \dim M/xM \leq \dim M - 1.$ (more…)

Thanks to all who responded to the bleg yesterday. I’m still haven’t completely decided on the topic owing to lack of time (I actually wrote this post last weekend), but the suggestions are interesting. My current plan is, following Omar’s comment, to look at Proofs from the Book tomorrow and pick something combinatorial or discrete-ish, like the marriage problem or Arrow’s theorem. I think this will be in the appropriate spirit and will make for a good one-hour talk.

4. ${\mathrm{Ext}}$ and depth

One of the first really nontrivial facts we need to prove is that the lengths of maximal ${M}$-sequences are all the same. This is a highly useful fact, and we shall constantly use it in arguments (we already have, actually). More precisely, let ${I \subset R}$ be an ideal, and ${M}$ a finitely generated module. Assume ${R}$ is noetherian.

Theorem 12 Suppose ${M}$ is a f.g. ${R}$-module and ${IM \neq M}$. All maximal ${M}$-sequences in ${I}$ have the same length. This length is the smallest value of ${r}$ such that ${\mathrm{Ext}^r(R/I, M) \neq 0}$.

I don’t really have time to define the ${\mathrm{Ext}}$ functors in any detail here beyond the fact that they are the derived functors of ${\hom}$. So for instance, ${\mathrm{Ext}(P, M)=0}$ if ${P}$ is projective, and ${\mathrm{Ext}(N, Q) = 0}$ if ${Q}$ is injective. These ${\mathrm{Ext}}$ functors can be defined in any abelian category, and measure the “extensions” in a certain technical sense (irrelevant for the present discussion).

So the goal is to prove this theorem. In the first case, let us suppose ${r = 0}$, that is there is a nontrivial ${R/I \rightarrow M}$. The image of this must be annihilated by ${I}$. Thus no element in ${I}$ can act as a zerodivisor on ${M}$. So when ${r = 0}$, there are no ${M}$-sequences (except the “empty” one of length zero).

Conversely, if all ${M}$-sequences are of length zero, then no element of ${I}$ can act as a nonzerodivisor on ${M}$. It follows that each ${x \in I}$ is contained in an associated prime of ${M}$, and hence by the prime avoidance lemma, that ${I}$ itself is contained in an associated prime ${\mathfrak{p}}$ of ${M}$. This prime avoidance argument will crop up quite frequently.

(I don’t know why these <br >’s appear below. If anyone with better HTML knowledge than I could explain what I’m doing wrong, I’d appreciate it!)

Today, we will continue with our goal of understanding some aspects of commutative algebra, by defining depth.

0.3. Depth

Constructing regular sequences sequences is a useful task. We often want to ask how long we can make them subject to some constraint. For instance,

Definition 8
Suppose ${I}$ is an ideal such that ${IM \neq M}$. Then we define the ${I}$-depth of ${M}$ to be the maximum length of a maximal ${M}$-sequence contained in ${I}$. When ${R}$ is a local ring and ${I}$ the maximal ideal, then that number is simply called the depth of ${M}$.

The depth of a proper ideal ${I \subset R}$ is its depth on ${R}$.

The definition is slightly awkward, but it turns out that all maximal ${M}$-sequences in ${I}$ have the same length. So we can use any of them to compute the depth.

The first thing we can prove using the above machinery is that depth is really a “geometric” invariant, in that it depends only on the radical of ${I}$. (more…)

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