Last time we gave the axiomatic description of the Stiefel-Whitney classes. Today, following Milnor-Stasheff, we want to look at what happens in the particular case of real projective space ${\mathbb{RP}^n}$. In particular, we want to compute the Stiefel-Whitney classes of the tangent bundle ${T(\mathbb{RP}^n)}$. The cohomology ring of ${\mathbb{RP}^n}$ with ${\mathbb{Z}/2}$-coefficients is very nice: it’s ${\mathbb{Z}/2[t]/(t^{n+1})}$. We’d like to find what ${w(T(\mathbb{RP}^n)) \in \mathbb{Z}/2[t]/(t^{n+1})}$ is.

On ${\mathbb{RP}^n}$, we have a tautological line bundle ${\mathcal{L}}$ such that the fiber over ${x \in \mathbb{RP}^n}$ is the set of vectors that lie in the line represented by ${x}$. Let’s start by figuring out the Stiefel-Whitney classes of this. I claim that

$\displaystyle w(\mathcal{L}) = 1+t \in H^*(\mathbb{RP}^n, \mathbb{Z}/2).$

The reason is that, if ${\mathbb{RP}^1 \hookrightarrow \mathbb{RP}^n}$ is a linear embedding, then ${\mathcal{L}}$ pulls back to the tautological line bundle ${\mathcal{L}_1}$ on ${\mathbb{RP}^1}$. In particular, by the axioms, we know that ${w(\mathcal{L}_1) \neq 1}$, and in particular has nonzero ${w_1}$. This means that ${w_1(\mathcal{L}) \neq 0}$ by the naturality. As a result, ${w_1(\mathcal{L})}$ is forced to be ${t}$, and there can be nothing in other dimensions since we are working with a 1-dimensional bundle. The claim is thus proved. (more…)

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

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

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:

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

One of the standard facts in algebraic geometry is that a projective scheme is proper. In the language of varieties, one says that the image of a projective variety is closed. The precise statement one proves is that:

Theorem 1 Let ${Y}$ be any variety over the algebraically closed field ${k}$. Let ${Z \subset \mathbb{P}^n_k \times Y}$ be a closed subset. Then the projection of ${Z}$ to ${Y}$ is closed.

This statement is sometimes phrased as saying that ${\mathbb{P}_k^n}$ is “complete.” In many ways, it is a compactness statement. Recall that a compact space ${X}$ has the property that the map ${X \times Y \rightarrow Y}$ for any ${Y}$ is a closed map. The converse is also true if there are reasonable assumptions (I think locally compact Hausdorff on ${X}$ will do it). Of course, these reasonable assumptions don’t apply to a variety with the Zariski topology, but they do if we are working with a variety (say, a quasiprojective variety) over ${\mathbb{C}}$ and we can define the complex topology. And in fact, it turns out that projective varieties are indeed compact in the complex topology. The scheme-theoretic version is a little different. First, the theorem for varieties said that the object ${\mathbb{P}_k^n}$ was “complete” in a certain sense. But the philosophy of Grothendieck is to consider not so much schemes but morphisms of schemes, and to do everything in a relative context. The idea is:

Definition 2 A morphism ${f: X \rightarrow Y}$ is proper if it is separated, of finite type, and universally closed (i.e. any base change is a closed morphism).

I don’t really want to define separated in this post. On the other hand, I should explain what’s going on for quasiprojective varieties over an algebraically closed fields. In this case, the conditions of finite type and separated are redundant. The key condition is universal closedness, which won’t be satisfied for a general morphism. For instance, to say that ${\mathbb{P}^n_k \rightarrow \mathrm{Spec} k}$ is universally closed implies the first lemma about closedness of the projection.
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