Let ${(A, \mathfrak{m})}$ be a regular local (noetherian) ring of dimension ${d}$. In the previous post, we described loosely the local cohomology functors

$\displaystyle H^i_{\mathfrak{m}}: \mathrm{Mod}(A) \rightarrow \mathrm{Mod}(A)$

(in fact, described them in three different ways), and proved a fundamental duality theorem

$\displaystyle H^i_{\mathfrak{m}}(M) \simeq \mathrm{Ext}^{d-i}(M, A)^{\vee}.$

Here ${\vee}$ is the Matlis duality functor ${\hom(\cdot, Q)}$, for ${Q}$ an injective envelope of the residue field ${A/\mathfrak{m}}$. This was stated initially as a result in the derived category, but we are going to use the above form.

The duality can be rewritten in a manner analogous to Serre duality. We have that ${H^d_{\mathfrak{m}}(A) \simeq Q}$ (in fact, this could be taken as a definition of ${Q}$). For any ${M}$, there is a Yoneda pairing

$\displaystyle H^i_{\mathfrak{m}}(M) \times \mathrm{Ext}^{d-i}(M, A) \rightarrow H^d_{\mathfrak{m}}(A) \simeq Q,$

and the local duality theorem states that it is a perfect pairing.

Example 1 Let ${k}$ be an algebraically closed field, and suppose that ${(A, \mathfrak{m})}$ is the local ring of a closed point ${p}$ on a smooth ${k}$-variety ${X}$. Then we can take for ${Q}$ the module

$\displaystyle \hom^{\mathrm{top}}_k(A, k) = \varinjlim \hom_k(A/\mathfrak{m}^i, k):$

in other words, the module of ${k}$-linear distributions (supported at that point). To see this, note that ${\hom_k(\cdot, k)}$ defines a duality functor on the category ${\mathrm{Mod}_{\mathrm{sm}}(A)}$ of finite length ${A}$-modules, and any such duality functor is unique. The associated representing object for this duality functor is precisely ${\hom^{\mathrm{top}}_k(A, k)}$.

In this case, we can think intuitively of ${H^i_{\mathfrak{m}}(A)}$ as the cohomology

$\displaystyle H^i(X, X \setminus \left\{p\right\}).$

These can be represented by meromorphic ${d}$-forms defined near ${p}$; any such ${\omega}$ defines a distribution by sending a function ${f}$ defined near ${p}$ to ${\mathrm{Res}_p(f \omega)}$. I’m not sure to what extent one can write an actual comparison theorem with the complex case.

1. Grothendieck’s theorem

Our goal is to apply local duality  to commutative algebra. Recall the result from last time.

Theorem 8 (Grothendieck) Let ${(A, \mathfrak{m})}$ be a local ring (not necessarily regular) and let ${M}$ be a finitely generated ${A}$-module. Then

$\displaystyle H^i_{\mathfrak{m}}(M) = 0 \quad \text{for} \ i \notin [\mathrm{depth} M, \dim M],$

and the values at the two endpoints are non-zero.

In the previous post, we gave three different proofs of the vanishing beyond ${\dim M}$. Our next goal is to show that

$\displaystyle H^{\dim M}_{\mathfrak{m}}(M) \neq 0.$

Here the local duality result will be essential.

Let’s start by assuming that ${A}$ is regular. Then we saw that ${H^{\dim M}_{\mathfrak{m}}(M) = \mathrm{Ext}^{d - \dim M}(M, A)^{\vee}}$. So, we are reduced to showing that

$\displaystyle \mathrm{Ext}^{d - \dim M}(M, A) \neq 0 .$

In fact, the Matlis dual of any nonzero module is never zero, so this will suffice. We can prove this non-vanishing using the story that connects depth to ${\mathrm{Ext}}$. I’ll only state the main result:

Proposition 9 (Depth and Ext) Let ${R}$ be a noetherian ring and ${M}$ a finitely generated module. Then, for ${N}$ finitely generated, the first non-vanishing group

$\displaystyle \mathrm{Ext}^i(M, N)$

occurs when ${i}$ is equal to the length of a maximal regular sequence on ${N}$ in ${\mathrm{ann}(M)}$.

For example, given an ideal ${I \subset R}$, we can test the depth of ${I}$ on ${N}$ by figuring out the first place at which ${\mathrm{Ext}^i(R/I, M) \neq 0}$; we could also replace ${R/I}$ with ${R/I^2}$, or anything supported exactly on the set cut out by ${I}$.

Let’s return to the assertion that we were trying to prove: for ${A}$ regular of dimension ${d}$, and ${M}$ any finitely generated ${A}$-module, we had

$\displaystyle \mathrm{Ext}^{d- \dim M}(M, A) \neq 0.$

This now follows because ${M}$ is supported on the subscheme of ${\mathrm{Spec} A}$ cut out by ${\mathrm{ann}(M)}$, which has height ${\dim M}$. But the depth of ${\mathrm{ann} M}$ on the module ${A}$ is precisely ${d - \dim M}$, because ${A}$ is a regular (in particular, Cohen-Macaulay) ring, which means that its depth with respect to any ideal is precisely the codimension of that ideal. So not only do we get that ${ \mathrm{Ext}^{d- \dim M}(M, A) \neq 0}$, but that this is the smallest such ${\mathrm{Ext}}$ group which is nonzero: that corresponds to the fact that the local cohomology is nonzero at ${\dim M}$ and zero beyond that.

We have now proved our goal, that ${H^{\dim M}_{\mathfrak{m}}(M) \neq 0}$, when ${A}$ is regular. The general case can now be handled by a trick described in the previous post: we can replace ${A}$ by its completion (which doesn’t change either local cohomology or dimensions), and then replace ${\hat{A}}$ by a complete regular local ring which surjects onto it. Again, we have to use the fact that a complete local ring is a quotient of a complete regular local ring.

2. The other half of Grothendieck’s theorem

We’ve now handled the difficult parts of Grothendieck’s vanishing/non-vanishing theorem. We’ve shown that

$\displaystyle H^i_{\mathfrak{m}}(M) = 0 \quad \text{for } i> \dim M,$

while ${H^{\dim M}_{\mathfrak{m}}(M) \neq 0}$. The other half states that

$\displaystyle H^i_{\mathfrak{m}}(M) = 0 \quad \text{for } i < \mathrm{depth} M,$

and there is a similar non-vanishing at ${\mathrm{depth} M}$. This doesn’t require anything as fancy as local duality to prove: in fact, we use the interpretation of ${H^i_{\mathfrak{m}}}$ as a colimit of ${\mathrm{Ext}}$ groups. Our original definition, in fact, was

$\displaystyle H^i_{\mathfrak{m}}(M) = \varinjlim \mathrm{Ext}^i(A/\mathfrak{m}^k, M).$

Using the general result (mentioned above) connecting ${\mathrm{Ext}}$ groups to depth, one can see that these groups vanish for ${i < \mathrm{depth} M}$. Moreover, all the groups in the colimit are nonzero for ${i = \mathrm{depth} M}$, and the connecting maps are injections (by the long exact sequence). It follows that we get both the vanishing and the non-vanishing.

3. An application

Let’s apply the abstract theory developed in the past couple of posts to the following concrete problem: How many equations does it take to define a variety?

Given a variety ${X \subset \mathbb{A}^n_k}$, it takes at least ${\mathrm{codim} X}$ equations to cut out ${X}$: this follows from the fact that cutting with a hyperplane can drop the dimension by at most one. That is, if ${Y}$ is an affine variety and ${H }$ a hypersurface, then ${Y \cap H}$ is either empty or has dimension one less than ${Y}$. This is a geometric version of Krull’s principal ideal theorem.

However, in general, a variety may need more equations to cut it out. Local cohomology is a relevant tool here. We are going to see that the union of two skew lines in ${\mathbb{P}^3}$, for instance, is not cut out set-theoretically by two equations. Alternatively, the ideal

$\displaystyle I = (u, v) \cap (x, y) \subset k[u, v, x,y ]$

is not generated up to radical by two polynomials. How can we prove this? We will show that

$\displaystyle H^3_I( k[u, v, x, y]) \neq 0,$

whereas if ${I}$ were generated up to radical by two elements, the Koszul-type complex which defines ${H_I^\bullet}$ would show that this group would have to vanish. Our goal is thus to compute the local cohomology of this ideal ${I}$, which geometrically corresponds to two planes in 4-space meeting at a point. We will use a Mayer-Vietoris sequence to reduce this to a simpler computation.

4. The Mayer-Vietoris sequence

If ${X}$ is any topological space and ${Z \subset X}$ a closed subset, we can define functors

$\displaystyle H^i_Z: \mathbf{Sh}(X) \rightarrow \mathbf{Ab}$

which are the derived functors of “sections with supports on ${Z}$.” When ${X = \mathrm{Spec} A}$ and ${Z }$ is cut out by an ideal ${\mathfrak{a} \subset A}$, then for quasi-coherent sheaves,

$\displaystyle H^i_Z(\cdot) = \varinjlim \mathrm{Ext}^i( A/\mathfrak{a}^k, \cdot) ,$

and we can write these functors as ${H^i_{\mathfrak{a}}}$.

The relevant observation here, which is easier to see from the geometric viewpoint than the algebraic one, is that there is a natural Mayer-Vietoris sequence. Namely, let ${X}$ be a space as above, and ${Z , Z' \subset X}$ closed subspaces. There is a natural long exact sequence

$\displaystyle 0 \rightarrow H^0_{Z \cap Z'}(\mathcal{F}) \rightarrow H^0_Z(\mathcal{F}) \oplus H^0_{Z'}(\mathcal{F}) \rightarrow H^0_{Z \cup Z'}(\mathcal{F}) \rightarrow H^1_{Z \cap Z'}(\mathcal{F}) \rightarrow \dots,$

precisely analogous to the Mayer-Vietoris sequence in topology. The sequence arises because a section supported on ${Z \cap Z'}$ certainly defines a section supported on ${Z}$, and a section supported on ${Z'}$.

If we translate this into algebraic language, we find that if ${\mathfrak{a}, \mathfrak{a}' \subset A}$, then there is a long exact sequence

$\displaystyle 0 \rightarrow H^0_{\mathfrak{a} + \mathfrak{a}'}(M) \rightarrow H^0_{\mathfrak{a}}(M) \oplus H^0_{\mathfrak{a}'}(M) \rightarrow H^0_{\mathfrak{a} \cap \mathfrak{a}'}(M) \rightarrow \dots,$

natural in the ${A}$-module ${M}$.

5. Hartshorne’s example

Let’s now apply the Mayer-Vietoris sequence to show that a certain local cohomology group doesn’t vanish, and that the union of two skew-lines in $\mathbb{P}^3$ is not cut out by two equations.

Example 2 (Hartshorne) Let ${A = k[u, v, x, y]}$ and take the two ideals ${\mathfrak{a} = (u,v), \mathfrak{a}' = (x,y)}$. In other words, take affine 4-space, and take for subvarieties two planes ${P_1, P_2}$ meeting only at the origin. Then we have a sequence

$\displaystyle H^3_{P_1}(A) \oplus H^3_{P_2}(A) \rightarrow H^3_{P_1 \cup P_2}(A) \rightarrow H^4_{P_1 \cap P_2}(A) \rightarrow H^4_{P_1}(A) \oplus H^4_{P_2}(A),$

corresponding to the cohomologies of the individual planes, their intersection, and their union. Since ${P_1, P_2}$ are cut out by two equations, the Koszul complexes defining ${H^i_{P_1}, H^i_{P_2}}$ exist only in degrees ${0,1,2}$, and the extreme points of the four-term exact sequence vanish. Since ${P_1 \cap P_2}$ has codimension zero, Grothendieck’s non-vanishing theorem implies that ${H^4_{P_1 \cap P_2}(A) \neq 0}$—one localizes at the origin. It follows that

$\displaystyle H^3_{P_1 \cup P_2}(A) \neq 0,$

which means that ${P_1 \cup P_2}$ cannot be cut out by two equations.

To be fair, we didn’t need Grothendieck’s non-vanishing theorem to assert that ${H^4_{P_1 \cap P_2}(A) \neq 0}$: it is given by a Koszul complex (more or less the same Koszul complex which computes the cohomology of projective space, and by no coincidence), which can be evaluated explicitly. The main point of the above argument is the following. For an ideal ${\mathfrak{a} \subset A}$, we can define the arithmetic rank

$\displaystyle \mathrm{ara} \mathfrak{a}$

to be the number of elements generating ${\mathfrak{a}}$ up to radical (i.e., the number of equations needed to cut out ${V(\mathfrak{a})}$ set-theoretically). We have then that

$\displaystyle H^i_{\mathfrak{a}}(M) = 0 , \quad i > \mathrm{ara} \mathfrak{a},$

and we can use this to get a lower bound on ${\mathrm{ara} \mathfrak{a}}$. That was precisely the approach of the above argument. More generally, one can apply it to the union of two set-theoretic complete intersections meeting at a single point.