The cohomology of projective space

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

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

(So here ${\mathcal{L} = \mathcal{O}(1)}$ , to use the notation of two days ago.) We have the sections ${x_i \in \mathcal{O}(1)}$ for ${ 0 \leq i \leq n}$ . It is a basic fact that the sets ${X_{x_i}}$ are affine subsets of ${\mathop{\mathbb P}^n_A}$ ; in fact they are sometimes called basic open affines. Each is isomorphic to ${ \mathbb{A}^n_A}$ .

In order to apply the machinery, we will need to first find ${\Gamma(X, \mathcal{H})}$ .

Lemma 45 The global sections of ${\mathcal{O}(m)}$ are precisely the degree ${m}$ polynomials. (When ${m <0}$ , there are none.)

Proof: ${\mathcal{O}(m)}$ is the sheaf associated to the graded ${S=A[x_0, \dots, x_n]}$ -module ${S(m)}$ whose graded part is defined by ${S(m)_k = S_{k+m}}$ . The sections over ${X_{x_i}}$ are isomorphic to ${S(m)_{(x_i)}}$ . But ${S}$ is the intersection of the localizations,

$\displaystyle S = \bigcap S_{x_i} \subset A[x_0, \dots, x_n, (x_0 \dots x_n)^{-1}]$

This identity makes sense as graded rings. Thus if something lies in ${S(m)_{(x_i)}}$ for all ${i}$ , then it must lie in ${S}$ , and is obviously of degree ${m}$ . In particular, the lemma tells us that

$\displaystyle \Gamma(X, \mathcal{H}) = S$

as, in fact, a ring!

Proposition 46 The Cech cohomology ${H^k(\mathfrak{A}, \mathcal{H})}$ is as follows. It is zero if ${k \neq 0, n}$ . For ${k=0}$ , it is isomorphic as a graded ${A}$ -module to ${S}$ . For ${k=n}$ , it is a free ${A}$ -module on rational functions ${x_0^{-a_0} \dots x_n^{-a_n}}$ where ${a_0, \dots, a_n>0}$ .

Proof: Indeed, we know that Cech cohomology is a direct limit of Koszul cohomologies. In particular, we have for ${k \geq 1}$ ,

$\displaystyle H^k(\mathfrak{A}, \mathcal{H}) =\varinjlim_m H^{k+1}( \mathbf{x}^m, S)$

where ${\mathbf{x} = (x_0, \dots, x_n)}$ . However, the sequence ${\mathbf{x}}$ , and consequently all powers ${\mathbf{x}^m}$ , are regular sequences on ${S}$ of length ${n+1}$ . This implies that in Koszul cohomology, by a lemma below, that

$\displaystyle H^i( \mathbf{x}^m, S) = 0$

for ${i \neq 0,n+1}$ . This implies the vanishing of ${H^k(\mathfrak{A}, \mathcal{H})}$ for ${k \neq 0,n}$ .

Last, but not least, we have to do ${H^{n}(\mathfrak{A}, \mathcal{H})}$ . Here again we can use Koszul cohomology. We know that this is the direct limit ${\varinjlim H^{n+1}( \mathbf{x}^m, S)}$ . As we will see below, this is the direct limit of the ${S/(x_1^m, \dots, x_n^m)}$ over ${m}$ getting larger, where the maps ${S/(x_1^m, \dots, x_n^m) \rightarrow S/(x_1^{m+1}, \dots, x_1^{m+1})}$ are multiplication by ${x_1 \dots x_n}$ .

To compute the colimit of this, note that at any element in the colimit is measured by “its distance from the end.” In particular, the elements are spanned by the negative monomials ${x_1^{-a_1} \dots x_n^{-a_n}}$ for ${a_1, \dots, a_n>0}$ . For each ${m}$ large, this negative monomial corresponds to the element

$\displaystyle x_1^{m-a_1} \dots x_n^{m-a_n} \in S/(x_1^m, \dots, x_n^m)$

in a way that is manifestly compatible with the maps in the colimit diagram.

Theorem 47 Let ${A}$ be a ring. The cohomology of the line bundle ${\mathcal{O}(m)}$ on projective space ${X=\mathop{\mathbb P}^n_A}$ is as follows. If ${k = 0}$ , then ${H^0(X,\mathcal{O}(m))}$ is the collection of polynomials of degree ${m}$ if ${m \geq 0}$ , and ${0}$ otherwise. If ${ 0< k < n-1}$ , then ${H^k(X, \mathcal{O}(m)) = 0}$ . If ${k=n-1}$ , then ${H^{n-1}(X, \mathcal{O}(m))}$ is zero for ${m > -n-1}$ , but for other ${m}$ is free on the set of negative monomials ${x_1^{-a_1} \dots x_n^{-a_n}}$ for ${a_1 + \dots + a_n = m}$ .

Proof: This now follows from the previous result, if we split the gradings. Note that we are using the fact that derived functor cohomology coincides with Cech cohomology for an affine covering on a separated scheme.

It is an interesting and important fact that all the cohomology groups here are finitely generated, which is a special property of projective space. It is also a curious fact that if ${\mathcal{L}}$ is any line bundle of this form, then for ${m \gg 0}$ , we have

$\displaystyle H^i(X, \mathcal{L}(m))=0, \quad \forall i >0.$

These will figure in the future.

2.7. A loose end: self-duality of the Koszul complex

Earlier, we saw that if ${x_1, \dots, x_n}$ is an ${M}$ -sequence, then the complex ${K_*(\mathbf{x}, M)}$ is acyclic in dimension not zero. But in the above computation, we used a dual analog:

Proposition 48 Let ${\mathbf{x}=(x_1, \dots, x_n)}$ be an ${M}$ -sequence. Then the complex ${K^*(\mathbf{x},M)}$ is acyclic in dimension not ${n}$ .

Proof: This will now follow from a dualization of the argument. Namely, the claim is that the Koszul chain complexes and cochain complexes are dual to each other. That is, ${K^*(\mathbf{x}) }$ is the cochain complex ${\hom(K_*(\mathbf{x}, R)}$ with the degrees reversed. In fact, let us state this more carefully.

Let ${F}$ be a free module of rank ${n}$ . Let ${\mathbf{x} = (x_1, \dots,x_n) \in R^n = F}$ . Then ${\wedge^p F}$ is isomorphic to ${\wedge^{n-p} F}$ . In fact, if ${e_1, \dots, e_n}$ is a basis for ${F}$ , then the map can be taken to be

$\displaystyle e_{i_1} \wedge \dots \wedge e_{i_p} \rightarrow e_{j_1} \wedge \dots \wedge e_{j_{n-p}}$

where the ${\left\{j_k\right\}}$ are the complementary set to the ${i_k}$ and the ${j_k,i_k}$ are ordered appropriately. Anyway, the point is that we have an isomorphism