Cohomology of Projective Space

We previously defined the line bundle \(\mathcal{O}(d)\) in §Line Bundles and Vector Bundles, ⁋Example 12, and verified through the computation immediately following §Line Bundles and Vector Bundles, ⁋Example 16 that its global sections \(H^0(\mathbb{P}^n, \mathcal{O}(d))\) are isomorphic to the homogeneous polynomials of degree \(d\). However, the sheaf cohomology of §Sheaf Cohomology, ⁋Definition 1 introduced in our previous post is a richer invariant that includes not only \(H^0\) but also the higher cohomology groups \(H^1, H^2, \ldots\). Thus, we shall now determine all information about \(\mathcal{O}(d)\) using not only \(H^0\) but also the higher cohomology groups.

Bott’s Formula

Since \(\mathcal{O}(d)\) is a line bundle, it is a quasi-coherent sheaf, and therefore to compute its sheaf cohomology it suffices to use Čech cohomology with the standard affine cover \(\mathcal{U}=\{U_0,\ldots, U_n\}\). The result of this computation is as follows.

Proposition 1 (Bott) On \(\mathbb{P}^n\), the cohomology of the line bundle \(\mathcal{O}(d)\) is as follows.

\[H^q(\mathbb{P}^n, \mathcal{O}(d)) = \begin{cases} \mathbb{K}[\x_0, \ldots, \x_n]_d & q = 0, d \geq 0 \\ \mathbb{K}[\x_0^{-1}, \ldots, \x_n^{-1}]_{-d-n-1} & q = n, d \leq -n-1 \\ 0 & \text{otherwise} \end{cases}\]

Proof

As explained above, we use Čech cohomology. First, recall that on each open set the section \(\mathcal{O}(d)(U_i)\) is

\[\x_i^d \cdot \mathbb{K}[\x_0/\x_i, \ldots, \widehat{\x_i/\x_i}, \ldots, \x_n/\x_i]\]

(§Line Bundles and Vector Bundles, ⁋Example 12). Then a Čech cochain \(f \in \check{C}^p(\mathcal{U}, \mathcal{O}(d))\) assigns to each \((p+1)\)-tuple \((i_0, \ldots, i_p)\) a regular section on the open set \(U_{i_0}\cap\cdots\cap U_{i_p}\). Here, for a section to be regular on the intersection \(U_{i_0}\cap\cdots\cap U_{i_p}\), only the coordinates that are not \(0\), namely \(\x_{i_0}, \ldots, \x_{i_p}\), are permitted in the denominator, and the sections are generated by the monomials

\[f_{i_0 \cdots i_p} = \x_0^{a_0} \cdots \x_n^{a_n},\qquad \sum_{j=0}^n a_j=d,\quad a_j\geq 0\text{ for $j\not\in \{i_0, \ldots, i_p\}$}\]

for which the others are not permitted.

As for the coboundary map \(\delta : \check{C}^p \rightarrow \check{C}^{p+1}\), it is given by

\[(\delta f)_{i_0 \cdots i_{p+1}} = \sum_{k=0}^{p+1} (-1)^k f_{i_0 \cdots \hat{i_k} \cdots i_{p+1}}\]

For each \((p+1)\)-tuple, this is the alternating sum of the sections corresponding to the \(p\)-tuples obtained by omitting one index.

Now let us use these data to compute the respective cohomology groups. Looking first at the case of \(\mathbb{P}^1\), the Čech complex is given by

\[0 \longrightarrow \check{C}^0\overset{\delta}{\longrightarrow}\check{C}^1\longrightarrow 0\]

where

\[\check{C}^0=\mathcal{O}(d)(U_0)\oplus \mathcal{O}(d)(U_1),\qquad \check{C}^1=\mathcal{O}(d)(U_0\cap U_1)\]

and the respective section spaces are

\[\mathcal{O}(d)(U_0) = \x_0^d \cdot \mathbb{K}[\x_1/\x_0], \qquad \mathcal{O}(d)(U_1) = \x_1^d \cdot \mathbb{K}[\x_0/\x_1], \qquad \mathcal{O}(d)(U_0 \cap U_1) = \mathbb{K}[\x_0^{\pm 1}, \x_1^{\pm 1}]_d\]

First, to compute the cohomology at \(\check{C}^0\), let us analyze \(\ker\delta\). Since \(H^0(\mathbb{P}^n, \mathcal{O}(d))=\Gamma(\mathbb{P}^n, \mathcal{O}(d))\), this is in fact nothing more than re-verifying the computation immediately following §Line Bundles and Vector Bundles, ⁋Example 16, but rather than treating it as a separate example, let us carry out the Čech cohomology computation in this proof.

By definition, that a cochain \((f_0, f_1) \in \check{C}^0\) belongs to \(\ker \delta\) means that \(f_0 = f_1\) holds in \(\mathcal{O}(d)(U_0 \cap U_1)\). Looking first at the \(U_0\) part, we know that in order for an arbitrary monomial to belong to \(\mathcal{O}(d)(U_0)\), it must be of the form \(\x_0^{d-a}\x_1^a\) for some \(a\geq 0\). Similarly, for any monomial to belong to \(\mathcal{O}(d)(U_1)\), it must be of the form \(\x_0^b\x_1^{d-b}\) for some \(b\geq 0\). Now, for a specific cocycle \((f_0,f_1)\) to belong to \(\ker\delta\), we must have \(f_0=f_1\), and therefore only the monomials satisfying \(a+b=d\) can belong to \(\ker\delta\). That is, the following monomials

\[\x_0^d, \quad\x_0^{d-1}\x_1,\quad\ldots, \quad\x_0\x_1^{d-1},\quad \x_1^d\]

form a basis of \(H^0\), which gives the desired result. If \(d<0\), then \(a,b\geq 0\) cannot satisfy this equation, so \(H^0\) becomes \(0\).

Now let us compute \(H^1\); that is, we must compute \(\coker\delta\). From the computation above, we know that the image of \(\delta\) is, for suitable constants \(a_i,b_j\),

\[f_1-f_0=\sum_{i\geq 0}a_i \x_0^{d-i}\x_1^i-\sum_{j\geq 0}b_j\x_0^j\x_1^{d-j}\tag{$\ast$}\]

On the other hand, \(\check{C}^1=\mathcal{O}(d)(U_0\cap U_1)=\mathbb{K}[\x_0^{\pm 1}, \x_1^{\pm 1}]_d\) is an infinite-dimensional space with basis consisting of the degree-\(d\) monomials with no restriction on the exponents,

\[\x_0^a\x_1^{d-a},\qquad a\in\mathbb{Z}\tag{$\ast\ast$}\]

Since the two sums in (\(\ast\)) give respectively the monomials with \(a\leq d\) and the monomials with \(a\geq 0\), the image of \(\delta\) is the subspace generated by those among (\(\ast\ast\)) with \(a\geq 0\) or \(d-a\geq 0\). Therefore, a basis of \(\coker\delta\) consists of those with both exponents negative, that is, for \(d+1\leq a\leq -1\), the monomials corresponding to \(a\). If \(d\geq -1\), no such \(a\) exists, so \(\coker\delta=0\); and if \(d\leq -2\), the elements

\[\x_0^{-1}\x_1^{d+1}, \quad \x_0^{-2}\x_1^{d+2},\quad\ldots,\quad \x_0^{d+1}\x_1^{-1}\]

numbering \(-d-1\), form a basis of \(\coker \delta\). We shall explain the notation in the statement separately after the proof.

We now complete the proof for general \(n\). In the \(\mathbb{P}^1\) computation above, we drew our conclusion by examining for each monomial which belongs to \(\ker\) and which belongs to \(\im\); this is no accident. This is because the coboundary map sends monomials to monomials anyway, so the entire Čech complex splits as a direct sum of subcomplexes for each Laurent monomial \(\x^a=\x_0^{a_0}\cdots\x_n^{a_n}\) (\(a\in\mathbb{Z}^{n+1}\), \(\sum_ja_j=d\)), and therefore computing only the subcomplex for each individual \(a\) yields all \(q\) at once.

Fix a multi-index \(a\), and partition the indices according to the signs of the exponents into two sets, \(N_{<0}(a)=\{j\mid a_j<0\}\) and \(N_{\geq 0}(a)=\{j\mid a_j\geq 0\}\). By the monomial condition seen above, that \(\x^a\) is regular on \(U_{i_0}\cap\cdots\cap U_{i_p}\) is equivalent to

\[N_{<0}(a)\subseteq\{i_0,\ldots, i_p\}\]

Therefore, for the subcomplex corresponding to \(a\), the \(p\)-th term is the vector space formed by the \((p+1)\)-element index sets \(I=\{i_0,\ldots, i_p\}\) containing \(N_{<0}(a)\), and its differential is the alternating sum of those obtained by removing one index from \(I\). Here, since \(I\supseteq N_{<0}(a)\), we can write uniquely \(I=N_{<0}(a)\sqcup J\), where \(J\) is a subset of \(N_{\geq 0}(a)\), and \(\lvert I\rvert=p+1\) is equivalent to \(\lvert J\rvert=p+1-\lvert N_{<0}(a)\rvert\). That is, among \(I\), the \(N_{<0}(a)\) part is forced and only \(J\) is free, so this subcomplex is the complex formed by the subsets of \(N_{\geq 0}(a)\),

\[K^q=\bigoplus_{\substack{J\subseteq N_{\geq 0}(a) \\ \lvert J\rvert=q}}\mathbb{K}\cdot e_J, \qquad \delta(e_J)=\sum_{v\in N_{\geq 0}(a)\setminus J}\pm e_{J\cup\{v\}}\]

shifted in degree by \(\lvert N_{<0}(a)\rvert-1\).

For the complex above, observe that if \(N_{\geq 0}(a)\) is nonempty, then \(K^\bullet\) is exact. This is because, fixing an element \(v\in N_{\geq 0}(a)\) and setting

\[h(e_J)=\begin{cases} e_{J\setminus\{v\}} & v\in J \\ 0 & v\notin J\end{cases}\]

we have \(\delta h+h\delta=\mathrm{id}\) when the signs are adjusted. Then the cohomology of the complex with \(K^0\) removed,

\[0 \rightarrow K^1 \rightarrow K^2 \rightarrow \cdots \rightarrow 0\]

is, at \(q=1\),

\[\ker(K^1 \rightarrow K^2)=\im(K^0 \rightarrow K^1)\cong K^0=\mathbb{K}\]

and is \(0\) in all other degrees. That is, if \(N_{\geq 0}(a)\) is nonempty and the above subcomplex is the whole of \(K^\bullet\), the cohomology vanishes in all degrees; thus, for something to remain, one of these two conditions must fail. As seen above, the latter condition is a question of whether the term corresponding to the empty set \(J=\emptyset\), namely \(K^0\), is included: if \(N_{<0}(a)\neq\emptyset\), then \(I=N_{<0}(a)\) itself satisfies the condition at \(p=\lvert N_{<0}(a)\rvert-1\geq 0\), so that term is included, whereas if \(N_{<0}(a)=\emptyset\), then \(I\) cannot be empty, so it is omitted.

Now, the conclusion falls into three cases according to \(N_{<0}(a)\). First, in the case where \(N_{<0}(a)=\emptyset\), that is, all \(a_j\geq 0\), as seen above \(N_{\geq 0}(a)\) is the whole set and \(K^0\) is omitted, so the cohomology is \(\mathbb{K}\) only at \(q=1\), that is, \(p=0\), and these \(\x^a\) form a basis of \(H^0(\mathbb{P}^n, \mathcal{O}(d))\). Since these are degree-\(d\) monomials with all exponents nonnegative, they exist only when \(d\geq 0\) and yield \(\mathbb{K}[\x_0,\ldots, \x_n]_d\). Next, in the case where \(N_{<0}(a)=\{0,\ldots, n\}\), that is, all \(a_j<0\), we have \(N_{\geq 0}(a)=\emptyset\), so the only index set \(I\) satisfying the condition is the whole set, and at \(p=n\) the complex consists of a single \(\mathbb{K}\); these \(\x^a\) form a basis of \(H^n(\mathbb{P}^n, \mathcal{O}(d))\). This time, since all exponents are at most \(-1\), they exist only when their sum is at most \(-n-1\), that is, when \(d\leq -n-1\). Finally, if \(N_{<0}(a)\) is neither the empty set nor the whole set, \(N_{\geq 0}(a)\) is nonempty while \(K^0\) is included, so \(K^\bullet\) itself is exact and does not contribute to any degree. That is, the cohomology vanishes for \(0<q<n\).

In the proof above, we showed that for the variables \(\x_0,\cdots, \x_n\) and for \(d\leq -n-1\), the group \(H^n(\mathbb{P}^n, \mathcal{O}(d))\) is generated by the monomials

\[\x_0^{a_0} \cdots \x_n^{a_n},\qquad a_i \leq -1, \quad \sum a_i=d\]

(Note that \(d\) is negative.) When we think of each \(\x_i^{-1}\) as a new variable \(\y_i=\x_i^{-1}\), this can also be regarded as the space generated by the expressions

\[\y_0^{\lvert a_0\rvert}\cdots \y_n^{\lvert a_n\rvert}\qquad \lvert a_i\rvert\geq 1,\quad \sum \lvert a_i\rvert=\lvert d\rvert\]

Here, since each \(a_i\) and \(d\) are negative, we have \(\lvert a_i\rvert=-a_i\) and \(\lvert d\rvert=-d\). The space above is very similar to the space of homogeneous polynomials of degree \(\lvert d\rvert\), with the difference that the \(\lvert a_i\rvert\) cannot be \(0\). Therefore, substituting \(b_i=\lvert a_i\rvert-1\), we can think of this space as the space of

\[\y_0^{b_0}\cdots \y_n^{b_n},\qquad b_i\geq 0,\quad \sum b_i=\lvert d\rvert-(n+1)\]

That is, this space can be thought of as the space of “negative degree” monomials of degree \(-d-n-1\), and for this reason, we denote this space by

\[\mathbb{K}[\x_0^{-1}, \ldots, \x_n^{-1}]_{-d-n-1}\]

Meanwhile, for later use, we define the Euler characteristic.

Definition 2 For an \(n\)-dimensional projective variety \(X\) and a coherent sheaf \(\mathcal{F}\) defined on it, the Euler characteristic of \(\mathcal{F}\) is defined by the formula

\[\rchi(X, \mathcal{F}) = \sum_{i=0}^{n} (-1)^i \dim H^i(X, \mathcal{F})\]

In the special case where \(X=\mathbb{P}^n\) and \(\mathcal{F}=\mathcal{O}(d)\), since in any case the intermediate cohomology groups all vanish and we need only consider the cohomology at the two ends, the following corollary is easily proved.

Corollary 3 On \(\mathbb{P}^n\), the Euler characteristic of \(\mathcal{O}(d)\) is given by the formula

\[\rchi(\mathbb{P}^n, \mathcal{O}(d)) = \binom{n+d}{n}\]

Proof

By Proposition 1 (Bott), the cohomology falls into three cases.

First, if \(d \geq 0\), then only \(H^0\) is non-zero, so we see that

\[\rchi(\mathcal{O}(d)) = \dim H^0(\mathbb{P}^n, \mathcal{O}(d)) = \dim \mathbb{K}[\x_0, \ldots, \x_n]_d = \binom{n+d}{n}\]

holds.

Second, if \(-n \leq d \leq -1\), then all cohomology vanishes, so \(\rchi(\mathcal{O}(d)) = 0\), and in this case we usually define \(\binom{n+d}{n}=0\), which agrees with the convention.

Finally, consider the case \(d \leq -n-1\). In this case, since only \(H^n\) is non-zero, we have

\[\rchi(\mathcal{O}(d)) = (-1)^n \dim \mathbb{K}[\x_0^{-1}, \ldots, \x_n^{-1}]_{-d-n-1}\]

By the explanation immediately following Proposition 1 (Bott), we know that the dimension of this space is

\[\binom{-d-1}{n}=(-1)^n\binom{n+d}{n}\]

Here \(\binom{n+d}{n}\) follows the usual convention for binomial coefficient notation, just as in the case above.

The Euler characteristic has the important property of additivity with respect to short exact sequences. That is, for a short exact sequence

\[0 \rightarrow \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{H} \rightarrow 0\]

we have \(\rchi(\mathcal{G}) = \rchi(\mathcal{F}) + \rchi(\mathcal{H})\). Thus the Euler characteristic becomes an invariant that is much easier to compute and manipulate, at the cost of losing information about the individual cohomology groups.

Serre Vanishing

According to Proposition 1 (Bott), on \(\mathbb{P}^n\), for sufficiently large \(d\), the higher cohomology of \(\mathcal{O}(d)\) all vanishes. Since every line bundle on \(\mathbb{P}^n\) is, for some \(d\), of the form \(\mathcal{O}(d)\), this means that on \(\mathbb{P}^n\), for any line bundle \(\mathcal{L}\), and for sufficiently large \(d\gg 0\), the twisted line bundle using \(\mathcal{O}(d)\),

\[\mathcal{L}\otimes \mathcal{O}(d)\]

necessarily has higher cohomology equal to \(0\).

More generally, we can extend this to arbitrary projective varieties and arbitrary coherent sheaves defined on them. For this we first need something to play the role of \(\mathcal{O}(1)\); in our definition, a projective variety \(X\) is always given by an embedding \(X\hookrightarrow\mathbb{P}^N\), so it suffices to pull back from \(\mathbb{P}^N\) the \(\mathcal{O}(1)\).

Meanwhile, the key ingredient in proving this is the fact that a coherent sheaf becomes globally generated upon sufficient twisting. To gain intuition for this concept, let us first consider the case of line bundles. That a line bundle \(\mathcal{L}\) is basepoint-free, as defined in §Linear Systems, ⁋Definition 5, means that for every point \(p \in X\), we have \(s(p) \neq 0\) for some global section \(s \in H^0(X, \mathcal{L})\). That is, the base locus is empty, so the linear system \(\lvert \mathcal{L} \rvert\) necessarily provides a nonzero value at each point. This is equivalent to the evaluation map

\[H^0(X, \mathcal{L}) \otimes \mathcal{O}_X \rightarrow \mathcal{L}\]

being surjective, and the following definition generalizes this condition to arbitrary coherent sheaves.

Definition 4 A coherent sheaf \(\mathcal{F}\) is said to be globally generated if the evaluation map

\[H^0(X, \mathcal{F}) \otimes \mathcal{O}_X \rightarrow \mathcal{F}\]

is surjective. That is, the stalks can all be generated by global sections.

In particular, for line bundles, being globally generated is equivalent to being basepoint-free. We first verify that twisting on \(\mathbb{P}^N\) actually produces enough global sections.

Lemma 5 On \(\mathbb{P}^N\), for a coherent sheaf \(\mathcal{G}\), for sufficiently large \(n\), \(\mathcal{G}(n)\) is globally generated.

Proof

Let \(S = \mathbb{K}[\x_0, \ldots, \x_N]\), and let \(M = \bigoplus_{n \in \mathbb{Z}} \Gamma(\mathbb{P}^N, \mathcal{G}(n))\) be a graded \(S\)-module. On each standard affine open set \(D_+(\x_j)\), \(\Gamma(D_+(\x_j), \mathcal{G})\) is the degree-0 localisation \(M_{(\x_j)}\), which is a finitely generated \(S_{(\x_j)}\)-module. If we let its generators be \(\overline{m}_1, \ldots, \overline{m}_{r_j} \in M_{(\x_j)}\), then for homogeneous elements \(m_k \in M\) we can write \(\overline{m}_k = m_k / \x_j^{d_k}\). Setting \(d_0 = \max_{j,k} d_k\), multiplying each generator by \(\x_j^{d_0 - d_k}\) yields the homogeneous element \(m_k \cdot \x_j^{d_0 - d_k} \in M_{d_0}\). This is an element of \(\Gamma(\mathbb{P}^N, \mathcal{G}(d_0))\), and since multiplication by \(\x_j^{d_0}\) is, on \(D_+(\x_j)\), an isomorphism between \(\mathcal{G}\) and \(\mathcal{G}(d_0)\), the global sections obtained in this way generate on \(D_+(\x_j)\) the stalks of \(\mathcal{G}(d_0)\). Since the \(D_+(\x_j)\) cover \(\mathbb{P}^N\), we obtain that \(\mathcal{G}(d_0)\) is globally generated. On the other hand, for \(k \geq 0\), the monomials \(\x_0^k, \ldots, \x_N^k\) generate the stalks of \(\mathcal{O}(k)\), so \(\mathcal{O}(k)\) is also globally generated, and taking the tensor product of the two evaluation maps shows that for \(n \geq d_0\), for all such \(n\), \(\mathcal{G}(n) = \mathcal{G}(d_0) \otimes \mathcal{O}(n - d_0)\) is globally generated.

The usefulness of Lemma 5 is that it allows us to write an arbitrary coherent sheaf as a quotient of a direct sum of line bundles, and from this we can extend the computation of Proposition 1 (Bott) to arbitrary coherent sheaves.

Lemma 6 For a coherent sheaf \(\mathcal{G}\) on \(\mathbb{P}^N\), for all sufficiently large \(n\),

\[H^i(\mathbb{P}^N, \mathcal{G}(n)) = 0 \quad (i > 0)\]

holds.

Proof

By Lemma 5, we can choose \(a \gg 0\) such that \(\mathcal{G}(a)\) is globally generated, and tensoring the resulting surjection \(\mathcal{O}^{\oplus r} \twoheadrightarrow \mathcal{G}(a)\) with \(\mathcal{O}(-a)\), we obtain

\[\mathcal{E} := \mathcal{O}(-a)^{\oplus r} \twoheadrightarrow \mathcal{G}\]

That is, any coherent sheaf is a quotient of a direct sum of line bundles, and the kernel \(\mathcal{K}\) of this surjection is also coherent.

The given claim can now be proved by descending induction on \(i\). First, for the base step, for \(i\) greater than \(N\), the standard affine open cover of \(\mathbb{P}^N\) consists of \(N+1\) open sets, so the cohomology of any quasi-coherent sheaf vanishes even without twisting. Now assume \(0 < i \leq N\) and that the claim holds for \(i+1\) for all coherent sheaves. The long exact sequence obtained by tensoring the short exact sequence \(0 \rightarrow \mathcal{K} \rightarrow \mathcal{E} \rightarrow \mathcal{G} \rightarrow 0\) with \(\mathcal{O}(n)\) gives

\[H^i(\mathcal{E}(n)) \rightarrow H^i(\mathcal{G}(n)) \rightarrow H^{i+1}(\mathcal{K}(n))\]

Since \(\mathcal{E}(n) = \mathcal{O}(n-a)^{\oplus r}\), by Proposition 1 (Bott) the left term vanishes for \(n - a \geq -N\), and by the inductive hypothesis on \(\mathcal{K}\) the right term vanishes for all sufficiently large \(n\). Therefore, we obtain \(H^i(\mathcal{G}(n)) = 0\) for all sufficiently large \(n\). Since there are only finitely many values of \(i\) from \(1\) to \(N\), taking the maximum of the lower bounds obtained for each yields an \(n_0\) such that vanishing holds simultaneously for all \(i > 0\).

Then all that remains is to embed an arbitrary projective variety into \(\mathbb{P}^N\) and apply Lemma 6.

Proposition 7 (Serre Vanishing) Let \(X\) be a projective variety, \(\mathcal{L}\) an ample line bundle, and \(\mathcal{F}\) a coherent sheaf. Then for sufficiently large \(m\),

\[H^i(X, \mathcal{F} \otimes \mathcal{L}^{\otimes m}) = 0 \quad (i > 0)\]

holds.

Proof

Since \(\mathcal{L}\) is ample, for sufficiently large \(m_0\), \(\mathcal{L}^{\otimes m_0}\) is very ample. That is, there exists a suitable embedding \(i \colon X \hookrightarrow \mathbb{P}^N\) such that \(\mathcal{L}^{\otimes m_0} = i^\ast\mathcal{O}(1)\). Restricting the standard affine cover \(\{U_i\}\) of \(\mathbb{P}^N\) to \(X\) yields an affine open cover \(\{X \cap U_i\}\). Since a finite intersection \(U_{i_0} \cap \cdots \cap U_{i_p}\) is affine, \((X \cap U_{i_0}) \cap \cdots \cap (X \cap U_{i_p}) = X \cap (U_{i_0} \cap \cdots \cap U_{i_p})\) is also affine. Therefore the two Čech complexes are literally the same, so

\[\check{H}^i(\{X \cap U_j\}, \mathcal{F}) = \check{H}^i(\{U_j\}, i_\ast\mathcal{F})\]

holds. On the other hand, by §Sheaf Cohomology, ⁋Theorem 11 (Leray), the cohomology of a quasi-coherent sheaf on a variety is computed by Čech cohomology with respect to an affine open cover, so the following identity

\[H^i(X, \mathcal{F}) = \check{H}^i(\{X \cap U_j\}, \mathcal{F}) = \check{H}^i(\{U_j\}, i_\ast\mathcal{F}) = H^i(\mathbb{P}^N, i_\ast\mathcal{F})\]

holds. We apply this identity with \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) in place of \(\mathcal{F}\). Writing a given \(m\) as \(m = m_0 n + r\) (\(0 \leq r < m_0\)), we have \(\mathcal{L}^{\otimes m} = \mathcal{L}^{\otimes r} \otimes i^\ast\mathcal{O}(n)\), so by the projection formula

\[i_\ast(\mathcal{F} \otimes \mathcal{L}^{\otimes m}) = i_\ast(\mathcal{F} \otimes \mathcal{L}^{\otimes r}) \otimes \mathcal{O}(n) = \mathcal{G}_r(n), \qquad \mathcal{G}_r := i_\ast(\mathcal{F} \otimes \mathcal{L}^{\otimes r})\]

and therefore \(H^i(X, \mathcal{F} \otimes \mathcal{L}^{\otimes m}) = H^i(\mathbb{P}^N, \mathcal{G}_r(n))\). Here the \(\mathcal{G}_r\) are finitely many coherent sheaves on \(\mathbb{P}^N\) determined by \(r = 0, \ldots, m_0-1\), and if \(m\) is sufficiently large then \(n = (m-r)/m_0\) is also sufficiently large, so applying Lemma 6 to each \(\mathcal{G}_r\) and taking the maximum of the finitely many lower bounds obtained yields the desired conclusion for all sufficiently large \(m\).

Generally, vanishing theorems of this kind play a major role in cohomology computations. If Proposition 7 (Serre Vanishing) said that sufficient twisting can make all higher cohomology \(0\), the next proposition says that cohomology in sufficiently high degree is always \(0\) even without twisting.

Proposition 8 (Grothendieck Vanishing) For an \(n\)-dimensional projective variety \(X\) and a coherent sheaf \(\mathcal{F}\) on it, if \(i > n\), then

\[H^i(X, \mathcal{F}) = 0\]

holds.

Proof

The homogeneous coordinate ring of \(X \subseteq \mathbb{P}^M\), \(S(X)\), has Krull dimension \(n+1\), so by [Commutative Algebra] §Noether Normalization, ⁋Theorem 1 (Noether normalization lemma) there exists a polynomial subring over which \(S(X)\) is a finitely generated module, and since \(\mathbb{K}\) is algebraically closed, it is an infinite field, so by the second case of [Commutative Algebra] §Noether Normalization, ⁋Lemma 2 we can choose its generators to be linear forms \(\y_0, \ldots, \y_n\). These have no common zero on \(X\), so they define a morphism

\[\pi \colon X \rightarrow \mathbb{P}^n, \qquad p \mapsto [\y_0(p) : \cdots : \y_n(p)]\]

and on each \(D_+(\y_j)\) the coordinate ring of \(\pi^{-1}(D_+(\y_j))\) is \(S(X)_{(\y_j)}\), which is a finitely generated module over \(\mathbb{K}[D_+(\y_j)]\), so \(\pi\) is finite in the sense of §Dimension, ⁋Definition 11, and surjective by lying over for integral extensions. Geometrically, this is the projection centered at the linear subspace disjoint from \(X\), \(V(\y_0, \ldots, \y_n)\).

Since for a finite morphism the preimage of an affine open set is again affine, the standard affine cover of \(\mathbb{P}^n\), \(\{U_j\}\), pulled back to \(\{\pi^{-1}(U_j)\}\), is an affine open cover of \(X\), and its finite intersections are also affine. Then, just as in the proof of Proposition 7 (Serre Vanishing), the two Čech complexes agree termwise as

\[\check{C}^p(\{\pi^{-1}(U_j)\}, \mathcal{F}) = \prod_{j_0 < \cdots < j_p} \mathcal{F}(\pi^{-1}(U_{j_0} \cap \cdots \cap U_{j_p})) = \check{C}^p(\{U_j\}, \pi_\ast\mathcal{F})\]

and by §Sheaf Cohomology, ⁋Theorem 11 (Leray) the Čech cohomology on each side is \(H^i(X, \mathcal{F})\) and \(H^i(\mathbb{P}^n, \pi_\ast\mathcal{F})\) respectively, so these are equal. Now \(\{U_j\}\) consists of \(n+1\) open sets, so for \(p > n\) we have \(\check{C}^p(\{U_j\}, \pi_\ast\mathcal{F}) = 0\), and therefore if \(i > n\), the right-hand side vanishes.

Regularity

Proposition 7 (Serre Vanishing) gave a qualitative result that higher cohomology vanishes after sufficiently large twisting. Regularity quantifies this, measuring specifically how much twisting is needed.

Intuitively, higher cohomology arises from failures in lower-degree cohomology, so this twisting is “less” necessary in higher degrees. With this in mind, the following definition is natural.

Definition 9 Fix a projective variety \(X\) and a very ample line bundle \(\mathcal{L}\) on it. Then we say that a coherent sheaf \(\mathcal{F}\) on \(X\) is \(m\)-regular if for all \(i>0\),

\[H^i(X, \mathcal{F} \otimes \mathcal{L}^{\otimes m - i}) = 0\]

holds.

In general it is almost impossible to compute all cohomology of a coherent sheaf, but our basic idea is that higher cohomology vanishes after sufficient twisting. Regularity goes further: it is a concept that measures specifically how much twisting is needed.

To define regularity in general, we first need the notion of a twist. On \(\mathbb{P}^n\) we use \(\mathcal{O}(1)\) as the standard, so we write \(\mathcal{F}(d) := \mathcal{F} \otimes \mathcal{O}(d)\). On an arbitrary projective variety \(X\) we choose an ample line bundle \(\mathcal{L}\) and define \(\mathcal{F}(d) := \mathcal{F} \otimes \mathcal{L}^{\otimes d}\). Twisting satisfies the following properties. By associativity of the tensor product, \(\mathcal{F}(d)(e) = \mathcal{F}(d+e)\) holds. Moreover, the tensor product functor \(- \otimes \mathcal{L}^{\otimes d}\) is exact because \(\mathcal{L}^{\otimes d}\) is a line bundle, so for a short exact sequence

\[0 \rightarrow \mathcal{F} \rightarrow \mathcal{G} \rightarrow \mathcal{H} \rightarrow 0\]

the sequence

\[0 \rightarrow \mathcal{F}(d) \rightarrow \mathcal{G}(d) \rightarrow \mathcal{H}(d) \rightarrow 0\]

is also a short exact sequence.

Proposition 10 (Castelnuovo-Mumford Regularity) Let \(X\) be a projective variety, \(\mathcal{L}\) a very ample line bundle, and \(\mathcal{F}\) a coherent sheaf. If \(\mathcal{F}\) is \(m\)-regular with respect to \(\mathcal{L}\), then the following hold.

1. \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated.
2. \(\mathcal{F}\) is \((m+p)\)-regular with respect to \(\mathcal{L}\) for all \(p \geq 0\).

Proof

Set \(n = \dim X\) and proceed by induction on the dimension of \(X\). If \(\dim X = 0\), then \(X\) is a point and the coherent sheaf \(\mathcal{F}\) is a finite-dimensional vector space, so all cohomology except \(H^0\) vanishes automatically. Now assume \(\dim X \geq 1\).

The key is to use the restriction exact sequence for an effective divisor \(D\) defined by a global section \(s \in H^0(X, \mathcal{L})\). Since \(\mathcal{L}\) is very ample, \(D\) is a hyperplane section of \(X\) embedded in projective space, and hence by choosing a general \(s\), §Linear Systems, ⁋Proposition 11 (Bertini’s theorem) implies that \(D\) is smooth away from the singular points of \(X\). We then obtain the following short exact sequence.

\[0 \rightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes k-1} \xrightarrow{\cdot s} \mathcal{F} \otimes \mathcal{L}^{\otimes k} \rightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes k}\vert_D \rightarrow 0\]

The long exact sequence in cohomology of this sequence yields

\[\cdots \rightarrow H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes k-1}) \rightarrow H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes k}) \rightarrow H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes k}\vert_D) \rightarrow H^{i+1}(\mathcal{F} \otimes \mathcal{L}^{\otimes k-1}) \rightarrow \cdots\]

In the special case of \(\mathbb{P}^n\), we have \(\mathcal{L} = \mathcal{O}(1)\), \(s\) is a general linear form, and \(D\) becomes a hyperplane \(H\) isomorphic to \(\mathbb{P}^{n-1}\).

First, we show that \(\mathcal{F}\vert_D\) is \(m\)-regular with respect to \(\mathcal{L}\vert_D\). Since \(\mathcal{F}\) is \(m\)-regular with respect to \(\mathcal{L}\), we have \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i}) = 0\) for \(i > 0\). Substituting \(k = m - i\) into the restriction sequence (\(0 < i \leq n-1\)) gives

\[0 \rightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes m-i-1} \rightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes m-i} \rightarrow \mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m-i} \rightarrow 0\]

and from its long exact sequence we have

\[H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i}) \rightarrow H^i(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m-i}) \rightarrow H^{i+1}(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i-1})\]

By \(m\)-regularity, \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i}) = 0\) and \(H^{i+1}(\mathcal{F} \otimes \mathcal{L}^{\otimes m-i-1}) = 0\) (\(i+1 > 0\)), so we obtain

\[H^i(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m-i}) = 0\]

for \(0 < i \leq n-1\). This means that \(\mathcal{F}\vert_D\) is \(m\)-regular with respect to \(\mathcal{L}\vert_D\).

Now apply the inductive hypothesis to \(D\). The divisor \(D\) is a projective variety with \(\dim D < \dim X\), and \(\mathcal{L}\vert_D\) is an ample line bundle. Since \(\mathcal{F}\vert_D\) is \(m\)-regular, the inductive hypothesis implies that \(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m}\) is globally generated on \(D\).

We now show that \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated. It suffices to verify that for any point \(x \in X\), the fiber \((\mathcal{F} \otimes \mathcal{L}^{\otimes m})_x\) is generated by the images of global sections. Choose a general divisor \(D\) passing through \(x\), and substitute \(k = m\) in the restriction sequence:

\[0 \rightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes m-1} \rightarrow \mathcal{F} \otimes \mathcal{L}^{\otimes m} \rightarrow \mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m} \rightarrow 0\]

From \(m\)-regularity with \(i = 1\), we have \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m-1}) = 0\), so

\[H^0(\mathcal{F} \otimes \mathcal{L}^{\otimes m}) \rightarrow H^0(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m})\]

is surjective. By the inductive hypothesis, \(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m}\) is globally generated on \(D\), so its fiber at \(x\) is generated by the image of \(H^0(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m})\). Since the restriction map is surjective, the global sections of \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) also generate the fiber at \(x\). Therefore \(\mathcal{F} \otimes \mathcal{L}^{\otimes m}\) is globally generated.

Finally, we show that \(\mathcal{F}\) is \((m+p)\)-regular by induction on \(p\). When \(p = 0\), that \(\mathcal{F}\) is \(m\)-regular holds by definition. Assume \(p \geq 1\), and let us show that \(\mathcal{F}\) is \((m+p)\)-regular, that is, \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i}) = 0\) (\(i > 0\)). For \(i = 1\), substituting \(k = m + p - 1\) into the restriction sequence gives

\[H^0(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-1}) \rightarrow H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-2}) \rightarrow H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-1}) \rightarrow H^1(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-1})\]

By the inductive hypothesis (for \(p-1\)), \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-2}) = 0\). Also, as shown above, \(\mathcal{F}\vert_D\) is \(m\)-regular, so by the inductive hypothesis on dimension, \(\mathcal{F}\vert_D\) is \((m+p)\)-regular, and hence \(H^1(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-1}) = 0\). From the exact sequence, \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-1})\) embeds into \(H^1(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-1})\), so we obtain \(H^1(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-1}) = 0\). For \(i \geq 2\), from the same restriction sequence

\[H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i-1}) \rightarrow H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i}) \rightarrow H^i(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-i})\]

the left term vanishes by the inductive hypothesis (for \(p-1\)). For the right term, if \(i \leq n-1\), then by the inductive hypothesis on dimension, \(\mathcal{F}\vert_D\) is \((m+p)\)-regular, so \(H^i(\mathcal{F}\vert_D \otimes (\mathcal{L}\vert_D)^{\otimes m+p-i}) = 0\); if \(i \geq n\), then \(\dim D = n-1\), so this term vanishes automatically by Proposition 8 (Grothendieck Vanishing). In either case we obtain \(H^i(\mathcal{F} \otimes \mathcal{L}^{\otimes m+p-i}) = 0\).

Example 11 On \(\mathbb{P}^n\), let us compute the regularity of the line bundle \(\mathcal{O}(d)\). Here, since \(\mathcal{L} = \mathcal{O}(1)\), the twist is \(\mathcal{O}(d) \otimes \mathcal{O}(m) = \mathcal{O}(d+m)\). The \(m\)-regularity condition is \(H^i(\mathbb{P}^n, \mathcal{O}(d+m-i)) = 0\) (\(i > 0\)). If \(d \geq 0\) and we choose \(m = 0\), we need to check \(H^i(\mathcal{O}(d-i))\); when \(i = 1\), \(H^1(\mathcal{O}(d-1))\) is, if \(d \geq 1\), \(0\), and if \(d = 0\), \(H^1(\mathcal{O}(-1)) = 0\) (in Bott’s formula, since \(-1 \geq -n\), all cohomology is \(0\)). In general, when \(d \geq 0\) and \(i > 0\), if \(d - i \geq -n\) then \(H^i(\mathcal{O}(d-i)) = 0\), and in the case where \(d - i < -n\), that is, \(i > d + n\), we have \(i > n\), so \(H^i = 0\) anyway. Thus \(\mathcal{O}(d)\), with respect to \(\mathcal{L} = \mathcal{O}(1)\), is \(0\)-regular. On the other hand, in the case where \(d < 0\), \(\mathcal{O}(d)\) is \((-d)\)-regular. By Proposition 10 (Castelnuovo-Mumford Regularity), \(\mathcal{O}(d) \otimes \mathcal{L}^{\otimes 0} = \mathcal{O}(d)\) is globally generated when \(d \geq 0\), which agrees with what was confirmed in the calculation immediately following §Line Bundles and Vector Bundles, ⁋Example 16.

Properties of Very Ample and Ample

The above Proposition 7 (Serre Vanishing) and Proposition 10 (Castelnuovo-Mumford Regularity) are representative results on the properties of ample line bundles. We conclude this post by examining additional properties of ample and very ample line bundles.

Proposition 12 If \(\mathcal{L}\) is very ample and \(\mathcal{M}\) is a globally generated line bundle, then \(\mathcal{L} \otimes \mathcal{M}\) is very ample.

Proof

Since \(\mathcal{L}\) is very ample, there exists a projective embedding \(i: X \hookrightarrow \mathbb{P}^N\) such that \(\mathcal{L} = i^\ast\mathcal{O}_{\mathbb{P}^N}(1)\). On the other hand, since \(\mathcal{M}\) is globally generated, global sections \(s_0, \ldots, s_n \in H^0(X, \mathcal{M})\) generate the stalk at every point, and from these we can define a morphism \(\phi: X \rightarrow \mathbb{P}^n\).

Now consider the closed embedding \((i, \phi): X \rightarrow \mathbb{P}^N \times \mathbb{P}^n\). Then, composing this with the Segre embedding (§Projective Varieties, ⁋Example 16)

\[\sigma: \mathbb{P}^N \times \mathbb{P}^n \hookrightarrow \mathbb{P}^{Nn+N+n}\]

since \(\sigma^\ast\mathcal{O}(1) = \pi_1^\ast\mathcal{O}(1) \otimes \pi_2^\ast\mathcal{O}(1)\), we know that

\[(\sigma \circ (i, \phi))^\ast\mathcal{O}(1) = i^\ast\mathcal{O}(1) \otimes \phi^\ast\mathcal{O}(1) = \mathcal{L} \otimes \mathcal{M}\]

That is, \(\mathcal{L} \otimes \mathcal{M}\) is very ample.

That is, although the explanation was somewhat involved, the key point is that for a globally generated line bundle \(\mathcal{M}\), the morphism \(\phi:X\rightarrow \mathbb{P}^n\) it defines need not be a closed embedding, but by tensoring this with \(\mathcal{L}\) and mapping into projective space in the form \((i,\phi)\), the first component \(i\) makes this map a closed embedding. From this, the following useful result can also be proved.

Proposition 13 For an ample line bundle \(\mathcal{L}\) and an arbitrary line bundle \(\mathcal{M}\) defined on a projective variety \(X\), \(\mathcal{M} \otimes \mathcal{L}^{\otimes n}\) is very ample for sufficiently large \(n\).

Proof

First, since \(\mathcal{L}\) is ample, \(\mathcal{L}^{\otimes m}\) is very ample for some \(m>0\). On the other hand, by Proposition 7 (Serre Vanishing) there exists \(k_0\) such that \(H^i(X, \mathcal{M} \otimes \mathcal{L}^{\otimes k}) = 0\) for all \(k \geq k_0\) and \(i > 0\). Now choose any \(k \geq k_0 + m\dim X\); then for \(1 \leq i \leq \dim X\) we have \(k - mi \geq k_0\), so

\[H^i(\mathcal{M} \otimes \mathcal{L}^{\otimes k} \otimes (\mathcal{L}^{\otimes m})^{\otimes -i}) = H^i(\mathcal{M} \otimes \mathcal{L}^{\otimes k-mi}) = 0\]

and for \(i > \dim X\), this cohomology vanishes automatically by Proposition 8 (Grothendieck Vanishing). That is, \(\mathcal{M} \otimes \mathcal{L}^{\otimes k}\) is \(0\)-regular with respect to the very ample line bundle \(\mathcal{L}^{\otimes m}\), so by Proposition 10 (Castelnuovo-Mumford Regularity) it is globally generated. Now by Proposition 12,

\[(\mathcal{M} \otimes \mathcal{L}^{\otimes k}) \otimes \mathcal{L}^{\otimes m} = \mathcal{M} \otimes \mathcal{L}^{\otimes (k+m)}\]

is very ample, and from this, setting \(n = k + m\) completes the proof.

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References

[Hart] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer, 1977.
[Bot] R. Bott, Homogeneous vector bundles, Annals of Mathematics, 1957.
[Laz] R. Lazarsfeld, Positivity in Algebraic Geometry I, Ergebnisse der Mathematik, Springer, 2004.
[Mum] D. Mumford, Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, Princeton, 1966.