Fix a noetherian local ring ${(A, \mathfrak{m})}$.

Let ${C \in D(A)}$ (for ${D(A)}$ the derived category of ${A}$, or preferably its higher-categorical analog). Let us define the local cohomology functor

$\displaystyle \Gamma_{\mathfrak{m}}: D(A) \rightarrow D(A), \quad \Gamma_{\mathfrak{m}}(C) = \varinjlim \mathbf{Hom}({A}/\mathfrak{m}^i, C).$

We can think of this at a number of levels: for instance, it is the (derived) functor of the ordinary functor on ${A}$-modules which sends an ${A}$-module ${M}$ to its submodule

$\displaystyle \Gamma_{\mathfrak{m}}(M) = \varinjlim\hom(A/\mathfrak{m}^i, M)$

of ${\mathfrak{m}}$-power torsion elements. From this point of view, we can think of the cohomology groups

$\displaystyle H^i_{\mathfrak{m}}(M) \stackrel{\mathrm{def}}{=} H^i (\Gamma_{\mathfrak{m}}(M))$

as defining “cohomology with supports” for the pair ${(\mathrm{Spec} A, \mathrm{Spec} A \setminus \left\{\mathfrak{m}\right\})}$ with coefficients in the sheaf ${M}$. I’ll try to elaborate more on this point of view later.

Notation: The derived categories in this post will use cohomological grading conventions, for simplicity.

Our first goal here is to describe the calculation (which is now quite formal) of ${\Gamma_{\mathfrak{m}}}$ in the regular case, in terms of dualizing objects. So, let’s suppose ${A}$ is regular local on, of dimension ${d = \dim A}$. In this case, each ${A/\mathfrak{m}^i}$ lives in the smaller perfect derived category ${\mathrm{D}_{\mathrm{perf}}(A)}$, and we will use the duality in that category.

Namely, recall that we have a functor ${D: \mathrm{D}_{\mathrm{perf}}(A) \rightarrow \mathrm{D}_{\mathrm{perf}}(A)^{op} }$ given by ${\mathbf{Hom}(\cdot, A)}$, which induces a duality on the perfect derived category of ${A}$, as we saw yesterday.

Let ${K = \varinjlim DA/\mathfrak{m}^i}$. We saw in the previous post that ${K}$ is cohomologically concentrated in the degree ${d}$, and it is a shift of the module ${Q = \varinjlim \mathrm{Ext}^d(A/\mathfrak{m}^i, A)}$: we saw that ${Q}$ was the injective envelope of ${k}$. The next result will reduce the computation of ${\Gamma_{\mathfrak{m}}}$ to an ${\mathrm{Ext}}$ computation.

Theorem 5 (Local duality) If ${C \in \mathrm{D}_{\mathrm{perf}}(A)}$ and ${A}$ is regular, then we have a canonical isomorphism in ${D(A)}$,

$\displaystyle \Gamma_{\mathfrak{m}}(C) \simeq \mathbf{Hom}( D C, K).$ (more…)

I’ve been trying to re-understand some of the proofs in commutative and homological algebra. I never really had a good feeling for spectral sequences, but they seemed to crop up in purely theoretical proofs quite frequently. (Of course, they crop up in computations quite frequently, too.) After learning about derived categories it became possible to re-interpret many of these proofs. That’s what I’d like to do in this post.

Here is a toy example of a result, which does not use spectral sequences in its usual proof, but which can be interpreted in terms of the derived category.

Proposition 1 Let ${(A, \mathfrak{m})}$ be a local noetherian ring with residue field ${k}$. Then a finitely generated ${A}$-module ${M}$ such that ${\mathrm{Tor}_i(M, k) = 0, i > 0}$ is free.

Let’s try to understand the usual proof in terms of the derived category. Throughout, this will mean the bounded-below derived category ${D^-(A)}$ of ${A}$-modules: in other words, this is the category of bounded-below complexes of projectives and homotopy classes of maps. Any module ${M}$ can be identified with an object of ${D^-(A)}$ by choosing a projective resolution.

So, suppose ${M}$ satisfies ${\mathrm{Tor}_i(M, k) = 0, i > 0}$. Another way of saying this is that the derived tensor product

$\displaystyle M \stackrel{\mathbb{L}}{\otimes} k$

has no homology in negative degrees (it is ${M \otimes k}$ in degree zero). Choose a free ${A}$-module ${P}$ with a map ${P \rightarrow M}$ which induces an isomorphism ${P \otimes k \simeq M \otimes k}$. Then we have that

$\displaystyle P \stackrel{\mathbb{L}}{\otimes} k \simeq M \stackrel{\mathbb{L}}{\otimes} k$

by hypothesis. In particular, if ${C}$ is the cofiber (in ${D^-(A)}$) of ${P \rightarrow M}$, then ${C \stackrel{\mathbb{L}}{\otimes} k = 0}$.

We’d like to conclude from this that ${C}$ is actually zero, or that ${P \simeq M}$: this will imply the desired freeness. Here, we have:

Lemma 2 (Derived Nakayama) Let ${C \in D^-(A)}$ have finitely generated homology. Suppose ${C \stackrel{\mathbb{L}}{\otimes} k = 0}$. Then ${C = 0}$. (more…)

The next big application of the Koszul complex and this general machinery that I have in mind is to projective space. Namely, consider a ring ${A}$, and an integer ${n \in \mathbb{Z}_{\geq 0}}$. We have the ${A}$-scheme ${\mathop{\mathbb P}^n_A = \mathrm{Proj} A[x_0, \dots, x_n]}$. Recall that on it, we have canonical line bundles ${\mathcal{O}(m)}$ for each ${m \in \mathbb{Z}}$, which come from homogeneous localization of the ${A[x_0, \dots, x_n]}$-modules obtained from ${A[x_0, \dots, x_n]}$ itself by twisting the degrees by ${m}$. When ${A}$ is a field, the only line bundles on it are of this form. (I am not sure if this is true in general. I think it will be true, but perhaps someone can confirm.)

It will be useful to compute the cohomology of these line bundles. For one thing, this will lead to Serre duality, from a very convenient isomorphism that will spring up. For another, we will see that they are finitely generated over ${A}$. This is far from obvious. The scheme ${\mathop{\mathbb P}^n_A}$ is not finite over ${A}$, and a priori this is not expected.

But to start, let’s think more abstractly. Let ${X}$ be any quasi-compact, quasi-separated scheme; we’ll assume this for reasons below. Let ${\mathcal{L}}$ be a line bundle on ${X}$, and ${\mathcal{F}}$ an arbitrary quasi-coherent sheaf. We can consider the twists ${\mathcal{F} \otimes \mathcal{L}^{\otimes m}}$ for any ${m \in \mathbb{Z}}$. This is a bunch of sheaves, but it is something more.

Let us package these sheaves together. Namely, let us consider the sheaves:

$\displaystyle \bigoplus \mathcal{L}^{\otimes m}, \quad \mathcal{H}=\bigoplus \mathcal{F} \otimes \mathcal{L}^{\otimes m}$

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…)