Sheaf Cohomology of Schemes

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

We defined sheaf cohomology on (quasi-projective) varieties via derived functors in [Algebraic Varieties] §Sheaf Cohomology, and examined methods for computing cohomology of quasi-coherent sheaves through comparison with Čech cohomology and Leray’s theorem. Now that we have reformulated quasi-coherent sheaves in the language of schemes, we can do the same thing on schemes. The key point is that the quasi-projective hypothesis in the previous post was excessive; once we establish separatedness and the vanishing theorem for quasi-coherent sheaves on affine schemes, we can prove the same results as in that post.

As with the previous few posts, the goal of this post is to lift the material already covered in [Algebraic Varieties] §Sheaf Cohomology to the setting of schemes. While we will carry out some calculations directly, most of the work will be translating into this language, leaving the bulk of the computations to those earlier posts.

Cohomology as a Derived Functor

On a scheme \(X\), the category \(\Sh(X)\) of sheaves of abelian groups is also an abelian category with enough injectives. Thus we can define the derived functor of the global section functor in exactly the same way as in [Algebraic Varieties] §Sheaf Cohomology. As before, our primary interest is always in quasi-coherent sheaves, but the definition and the resolution take place inside \(\Sh(X)\).

Definition 1 For a sheaf \(\mathcal{F}\) on a scheme \(X\), we define the \(i\)-th sheaf cohomology by taking the right derived functor of the global section functor \(\Gamma(X, -):\Sh(X) \rightarrow \Ab\) ([Homological Algebra] §Derived Functors, ⁋Definition 9) as

\[H^i(X, \mathcal{F})=R^i\Gamma(X, -)(\mathcal{F})=\frac{\ker\bigl(\Gamma(X, \mathcal{I}^i) \rightarrow \Gamma(X, \mathcal{I}^{i+1})\bigr)}{\im\bigl(\Gamma(X, \mathcal{I}^{i-1}) \rightarrow \Gamma(X, \mathcal{I}^i)\bigr)}\]

where \(0 \rightarrow \mathcal{F} \rightarrow \mathcal{I}^\bullet\) is an injective resolution in \(\Sh(X)\).

This definition is nothing more than lifting [Algebraic Varieties] §Sheaf Cohomology, ⁋Definition 1 to an arbitrary scheme, and the formal properties, including the fact that this definition is independent of the choice of \(\mathcal{I}^\bullet\), all follow from standard arguments in homological algebra. In particular, \(H^0(X, \mathcal{F})=\Gamma(X, \mathcal{F})\), and a short exact sequence of sheaves induces a long exact sequence.

Meanwhile, since \(\Gamma(X, -)\) is a functor on \(\Sh(X)\), the \(H^i(X, \mathcal{F})\) are a priori only abelian groups; however, when \(\mathcal{F}\) is an \(\mathcal{O}_X\)-module, they acquire a \(\Gamma(X, \mathcal{O}_X)\)-module structure. For each \(a\in\Gamma(X, \mathcal{O}_X)\), multiplication by \(a\) is an endomorphism of \(\mathcal{F}\) in \(\Sh(X)\), so functoriality of cohomology gives an endomorphism of \(H^i(X, \mathcal{F})\), and this correspondence preserves multiplication. Adding the fact that \(R^i\Gamma(X, -)\) is an additive functor, the map \(a\mapsto H^i(m_a)\) also preserves addition, yielding a ring homomorphism \(\Gamma(X, \mathcal{O}_X) \rightarrow \End_\mathbb{Z}\bigl(H^i(X, \mathcal{F})\bigr)\), which gives the desired module structure.

Proposition 2 For a short exact sequence of sheaves

\[0 \rightarrow \mathcal{F}' \rightarrow \mathcal{F} \rightarrow \mathcal{F}'' \rightarrow 0\]

there exists a long exact sequence

\[0 \rightarrow H^0(X, \mathcal{F}') \rightarrow H^0(X, \mathcal{F}) \rightarrow H^0(X, \mathcal{F}'') \xrightarrow{\delta} H^1(X, \mathcal{F}') \rightarrow H^1(X, \mathcal{F}) \rightarrow \cdots\]

Proof

Since \(\Gamma(X, -)\) is a left exact functor and \(\Sh(X)\) has enough injectives, the long exact sequence of the \(\delta\)-functor defined by right derived functors holds directly.

Cohomology of Affine Schemes

On the other hand, we also introduced Čech cohomology in [Algebraic Varieties] §Sheaf Cohomology; this is in fact a definition at the level of topological spaces, so it works on schemes without any modification. As in that post, the definition of sheaf cohomology in Definition 1 captures its essence precisely, but in practice one usually translates to Čech cohomology. The result that made this possible was [Algebraic Varieties] §Sheaf Cohomology, ⁋Proposition 12, which stated that higher cohomology of quasi-coherent sheaves vanishes on affine varieties. The scheme version of this proposition is as follows.

Theorem 3 (Serre) For the affine scheme \(X=\Spec A\) defined by a Noetherian ring \(A\) and a quasi-coherent sheaf \(\mathcal{F}=\widetilde M\) on it,

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

holds.

Proof

By §Quasi-coherent Sheaves, ⁋Theorem 9, \(\QCoh(\Spec A)\) is equivalent to \(\rMod{A}\), so there exists an \(A\)-module \(M\) with \(\mathcal{F}=\widetilde M\). Since \(\rMod{A}\) has enough injectives, take an injective resolution of \(M\)

\[0 \rightarrow M \rightarrow I^0 \rightarrow I^1 \rightarrow \cdots\]

The associated sheaf functor \(\widetilde{(-)}\) is exact (§Quasi-coherent Sheaves, ⁋Proposition 6), so

\[0 \rightarrow \widetilde M \rightarrow \widetilde{I^0} \rightarrow \widetilde{I^1} \rightarrow \cdots\]

is a resolution of sheaves on \(\Spec A\). Our claim is that each \(\widetilde{I^k}\) is \(\Gamma(\Spec A, -)\)-acyclic; if this holds, then by [Algebraic Varieties] §Sheaf Cohomology, ⁋Proposition 17 we obtain

\[H^i(\Spec A, \widetilde M)\cong H^i\bigl(\Gamma(\Spec A, \widetilde{I^\bullet})\bigr)=H^i(I^\bullet)\]

The second equality follows from the fact that the global sections of the associated sheaf are the original module (§Quasi-coherent Sheaves, ⁋Definition 4), and since \(M \rightarrow I^\bullet\) is a quasi-isomorphism, the cohomology on the right vanishes for all \(i>0\). Thus \(H^i(\Spec A, \widetilde M)=0\) (\(i>0\)).

It remains to show that the associated sheaf \(\widetilde I\) of an injective \(A\)-module \(I\) is acyclic. For this we show that \(\widetilde I\) is flasque. ([Algebraic Varieties] §Sheaf Cohomology, ⁋Proposition 16) Since every open subset of \(\Spec A\) is of the form \(U=\Spec A\setminus V(\mathfrak{a})\), it suffices to show that the restriction \(\widetilde I(\Spec A)=I\rightarrow\widetilde I(U)\) is surjective for each such \(U\). There is an exact sequence connecting the sections of a quasi-coherent sheaf with local cohomology

\[I\longrightarrow\widetilde I(U)\longrightarrow H^1_{\mathfrak{a}}(I)\longrightarrow 0\]

where \(H^i_{\mathfrak{a}}(M)=\varinjlim_n\Ext^i_A(A/\mathfrak{a}^n,M)\). Since \(I\) is injective, \(\Ext^1_A(A/\mathfrak{a}^n,I)=0\) for all \(n\), so \(H^1_{\mathfrak{a}}(I)=0\), and therefore the above restriction is surjective. Then for any two open subsets \(V\subseteq U\), the map \(I \rightarrow \widetilde I(V)\) factors through \(\widetilde I(U)\), so \(\widetilde I(U) \rightarrow \widetilde I(V)\) is also surjective, and hence \(\widetilde I\) is flasque.

The Noetherian hypothesis in Theorem 3 (Serre) is for convenience of proof; in fact the result holds for an arbitrary ring \(A\). However, this goes beyond the scope of this post, so we omit the proof, and use this general form only in Corollary 4 below, which is stated without the Noetherian hypothesis, and in Theorem 6 (Bott) dealing with projective space over an arbitrary ring.

At any rate, the heart of this theorem is that affine schemes are simple spaces from the viewpoint of cohomology. That is, on an affine scheme all information about a quasi-coherent sheaf is contained in \(H^0\), i.e., its global section module, and higher cohomology gives no new information. This is the algebraic-geometric phenomenon corresponding to a topological space being contractible from the viewpoint of Čech cohomology.

From this we immediately obtain the Leray theorem for affine coverings at the level of schemes. [Algebraic Varieties] §Sheaf Cohomology, ⁋Theorem 11 states that if \(\mathcal{F}\) is acyclic on all finite intersections of the cover \(\mathcal{U}\), then \(\check H^p(\mathcal{U}, \mathcal{F})\cong H^p(X, \mathcal{F})\); since this is a theorem at the level of topological spaces, it applies directly on schemes. The only additional hypothesis needed here is separatedness, which ensures that intersections of affines are again affine. (§Valuation Rings, ⁋Definition 3)

Corollary 4 For a separated scheme \(X\), a quasi-coherent sheaf \(\mathcal{F}\) on it, and an affine open cover \(\mathcal{U}=\{U_i\}\), we have

\[\check H^p(\mathcal{U}, \mathcal{F})\cong H^p(X, \mathcal{F})\]

for all \(p\).

Proof

By [Algebraic Varieties] §Sheaf Cohomology, ⁋Theorem 11, it suffices to show that \(\mathcal{F}\) is acyclic on any finite intersection \(U_{i_0}\cap\cdots\cap U_{i_p}\) of \(\mathcal{U}\). Since \(X\) is separated, the diagonal morphism \(\Delta:X \rightarrow X\times_{\Spec \mathbb{Z}}X\) is a closed immersion, and therefore the intersection \(U_i\cap U_j\) of any two affine open subsets \(U_i, U_j\) is again affine. Indeed, \(U_i\cap U_j\) is the fiber product \(U_i\times_X U_j\), which is isomorphic to the closed subscheme \(\Delta^{-1}(U_i\times U_j)\) of the affine scheme \(U_i\times_{\Spec \mathbb{Z}}U_j\), hence affine. Repeating the same argument, any finite intersection \(U_{i_0}\cap\cdots\cap U_{i_p}\) is also an affine scheme. Then the restriction of \(\mathcal{F}\) to this is a quasi-coherent sheaf on an affine scheme, so it is acyclic by Theorem 3 (Serre), and thus the hypothesis of [Algebraic Varieties] §Sheaf Cohomology, ⁋Theorem 11 is satisfied.

Therefore, on a separated scheme, derived functor cohomology is obtained directly by computing the Čech complex with respect to a single affine covering. A morphism between affine schemes is always separated (§Valuation Rings, ⁋Lemma 5), and projective schemes including \(\mathbb{P}^n\) are also separated, so this corollary works for most schemes we actually deal with.

Line Bundles on Projective Space

We now treat the cohomology of the line bundle \(\mathcal{O}(d)\) on projective space at the scheme level, using Čech calculations for an affine covering. Just as we defined projective space over a ring \(A\) as \(\mathbb{P}^n_A=\Proj A[\x_0,\ldots, \x_n]\) using the graded ring \(A[\x_0,\ldots, \x_n]\) (§Projective Schemes, ⁋Definition 1), we must first define \(\mathcal{O}(d)\) in the language of graded modules.

Definition 5 Given the standard grading on \(S_\bullet=A[\x_0,\ldots, \x_n]\), let \(S(d)\) be the degree \(d\)-shift of \(S_\bullet\), i.e., the graded \(S_\bullet\)-module with degree given by

\[S(d)_m=S_{d+m}\]

Then on each chart of the standard affine cover

\[\mathcal{U}=\{D_+(\x_i)=\Spec S_{(\x_i)}\}\]

of \(\mathbb{P}_A^n=\Proj S_\bullet\), we define the associated sheaf \(\widetilde{M_i}\) from the degree \(0\) part

\[M_i=\bigl(S(d)_{\x_i}\bigr)_0=\x_i^d\cdot S_{(\x_i)}\]

of the localization \(S(d)_{\x_i}\), viewed as an \(S_{(\x_i)}\)-module, and we call the quasi-coherent sheaf on \(\mathbb{P}^n_A\) obtained by gluing these via natural identifications on overlaps the twisting sheaf \(\mathcal{O}(d)\).

In [Algebraic Varieties] §Line Bundles and Vector Bundles, ⁋Example 12, for each standard open set \(D_+(\x_i)\) with \(\x_i\neq 0\) we specified a trivialization \(\phi_i(s)=s\cdot\x_i^{-d}\), and described \(\mathcal{O}(d)\) by taking the transition functions \((\x_i/\x_j)^d\) obtained by comparing the two trivializations on overlaps as data. In this description, the space of sections over \(D_+(\x_i)\) was \(\x_i^d\cdot\mathcal{O}(D_+(\x_i))\). Definition 5 is a reformulation using this last space itself, directly employing the module \(M_i=\x_i^d\cdot S_{(\x_i)}\) over the coordinate ring \(S_{(\x_i)}\) of \(D_+(\x_i)\). That these two data are the same can be verified by using §Quasi-coherent Sheaves, ⁋Definition 4 to check that the sections in Definition 5 are exactly \(\mathcal{O}(d)(D_+(\x_i))=M_i\). The shift by \(S(d)\) is notation to move the degree \(d\) part of \(S_{\x_i}\) to the degree \(0\) part that each chart adopts as functions.

As we saw in [Algebraic Varieties] §Line Bundles and Vector Bundles, ⁋Example 12, our basic motivation is that to represent closed subschemes of projective space we need degree \(d\) homogeneous polynomials, but these are basically only well-defined as their zero sets, and the function values themselves are not well-defined, so they cannot be regarded as actual functions. One way to resolve this is to choose the open subset \(D_+(\x_i)\) where each coordinate \(\x_i\) is nonzero, and then divide the polynomial by \(\x_i^d\) to make it degree \(0\), treating it as a function on this open subset. However, this amounts to choosing a non-canonical trivialization for each chart \(D_+(\x_i)\), and since these choices differ from chart to chart they are not compatible on overlaps, so they still cannot become global sections of \(\mathcal{O}_{\mathbb{P}^n_A}\). Yet if we define \(\mathcal{O}(d)\) as above, gathering these functions chart by chart gives a global section of this sheaf, and that section is the original polynomial itself. Also, since each \(M_i\) is a rank \(1\) free \(S_{(\x_i)}\)-module with generator \(\x_i^d\) and the transition functions \((\x_i/\x_j)^d\) are invertible, this sheaf is an invertible sheaf (§Quasi-coherent Sheaves, ⁋Definition 12), and hence can also be interpreted as the line bundle \(\mathcal{O}(d)\). On the other hand, this way of describing an object by data on each chart and comparisons on overlaps is precisely the material for a Čech complex. Therefore, the gluing of this data is determined exactly by the (Čech) cohomology of \(\mathcal{O}(d)\).

Theorem 6 (Bott) The cohomology of the line bundle \(\mathcal{O}(d)\) on the projective space \(\mathbb{P}^n_A\) over a ring \(A\) is given by

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

In particular, it vanishes for all \(d\) when \(0<q<n\).

Proof

Since \(\mathbb{P}^n_A\) is a separated scheme, by Corollary 4 the Čech cohomology with respect to the standard affine cover \(\mathcal{U}=\{D_+(\x_i)\}\) coincides with the derived functor cohomology. But this Čech complex is literally the same as the one appearing in the proof of [Algebraic Varieties] §Cohomology of Projective Space, ⁋Proposition 1 (Bott). That is, on each intersection \(D_+(\x_{i_0}\cdots\x_{i_p})\), the sections of \(\mathcal{O}(d)\) are generated over \(A\) by degree-\(d\) monomials allowing only \(\x_{i_0},\ldots, \x_{i_p}\) in the denominator,

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

and the coboundary map is also given by the same alternating sum formula. The proof there starts from the observation that the coboundary map does not change the exponent vector \(a=(a_0,\ldots, a_n)\) of a monomial, and decomposes the Čech complex into subcomplexes for each \(a\). Letting \(N_{<0}(a)\) denote the set of positions with negative exponents, the above condition says that \(\x^a\) being regular on \(D_+(\x_{i_0}\cdots\x_{i_p})\) is equivalent to \(N_{<0}(a)\subseteq\{i_0,\ldots, i_p\}\), so the subcomplex corresponding to \(a\) becomes the cochain complex of the simplex formed by index subsets containing \(N_{<0}(a)\). Then \(N_{<0}(a)\) contributes only to \(q=0\) when empty, only to \(q=n\) when it is the whole set, and nothing in between, giving the three cases at once. This argument uses nothing about the coefficients being a field, only that each subcomplex consists of free \(A\)-modules, so reading that proof with \(A\)-coefficients yields the result above.

Finally, the reason the space obtained for \(q=n\), \(d\leq -n-1\) is written as \(A[\x_0^{-1},\ldots, \x_n^{-1}]_{-d-n-1}\) is the same as the explanation given right after [Algebraic Varieties] §Cohomology of Projective Space, ⁋Proposition 1 (Bott). That is, substituting \(\y_j=\x_j^{-1}\) for all degree-\(d\) monomials with every exponent at most \(-1\) gives degree-\(\lvert d\rvert\) monomials with every exponent at least \(1\), and removing the common factor \(\y_0\cdots\y_n\) yields a one-to-one correspondence with the space of “negative degree” monomials of degree \(\lvert d\rvert-(n+1)=-d-n-1\).

Coherent Sheaves on Noetherian Projective Schemes

We now examine the sheaf cohomology of a Noetherian projective scheme \(X\) and a coherent sheaf \(\mathcal{F}\) on it more generally. The key results are twofold: each \(H^i(X, \mathcal{F})\) is finite-dimensional, and sufficiently twisting \(\mathcal{F}\) kills higher cohomology, which is Serre vanishing.

One of the tools we use for this is the existence of the line bundle \(\mathcal{O}_X(1)\) defined on a projective scheme. By definition, a projective scheme \(X\) admits a closed embedding \(\iota: X\hookrightarrow \mathbb{P}^n\), and using this we can pull back the line bundle \(\mathcal{O}_{\mathbb{P}^n}(1)\) from \(\mathbb{P}^n\) to define

\[\mathcal{O}_X(1)=\mathcal{O}_{\mathbb{P}^n}(1)\vert_X=\iota^\ast \mathcal{O}_{\mathbb{P}^n}(1)\]

At this point, the restrictions of coordinates \(\x_0\vert_X,\ldots, \x_n\vert_X\) globally generate \(\mathcal{O}_X(1)\), and the morphism they define is precisely the inclusion \(X\hookrightarrow\mathbb{P}^n_\mathbb{K}\), so we may just as well view this line bundle as the embedding itself. That is, \(\mathcal{O}_X(1)\) is a very ample invertible sheaf (§Divisors and Linear Systems, ⁋Definition 17), and for any coherent sheaf \(\mathcal{F}\) we write \(\mathcal{F}(d)=\mathcal{F}\otimes_{\mathcal{O}_X}\mathcal{O}_X(d)\).

The key observation is that cohomology is preserved along the closed embedding \(\iota:X\hookrightarrow\mathbb{P}^n\), that is, for any quasi-coherent sheaf \(\mathcal{F}\) on \(X\) we have

\[H^i(X, \mathcal{F})\cong H^i(\mathbb{P}^n, \iota_\ast\mathcal{F})\tag{$\ast$}\]

This is because the Čech complexes computing both sides are the same complex. Indeed, since a closed embedding is an affine morphism (§Closed Subschemes, ⁋Proposition 3), each \(U_i=\iota^{-1}(D_+(\x_i))\) is an affine scheme and \(\{U_i\}\) is an affine open cover of \(X\). On the other hand, by the definition of pushforward,

\[(\iota_\ast\mathcal{F})(D_+(\x_{i_0}\cdots\x_{i_p}))=\mathcal{F}\bigl(\iota^{-1}(D_+(\x_{i_0})\cap\cdots\cap D_+(\x_{i_p}))\bigr)=\mathcal{F}(U_{i_0}\cap\cdots\cap U_{i_p})\]

and the restriction maps correspond as well, so the Čech complex of \(\mathcal{F}\) with respect to \(\{U_i\}\) and the Čech complex of \(\iota_\ast\mathcal{F}\) with respect to \(\mathcal{U}\) are identical. But since both \(X\) and \(\mathbb{P}^n\) are projective schemes they are separated, and \(\iota_\ast\mathcal{F}\) is also quasi-coherent (§Quasi-coherent Sheaves, ⁋Theorem 16), so by Corollary 4 the cohomology of this single complex computes both sides simultaneously, yielding the isomorphism above.

The two theorems below both start from the fact that a coherent sheaf, when sufficiently twisted, is generated by its global sections, so we establish this first.

Lemma 7 For a coherent sheaf \(\mathcal{F}\) on the projective space \(\mathbb{P}^n_\mathbb{K}\) over a field \(\mathbb{K}\), the sheaf \(\mathcal{F}(d)\) is globally generated for sufficiently large \(d\gg 0\).

Proof

Let \(S_\bullet=\mathbb{K}[\x_0,\ldots, \x_n]\) and consider the graded \(S_\bullet\)-module

\[\Gamma_\ast(\mathcal{F})=\bigoplus_{m\in\mathbb{Z}}\Gamma(\mathbb{P}^n, \mathcal{F}(m))\]

We first show the equality on each standard affine chart:

\[\Gamma(D_+(\x_j), \mathcal{F})=\Gamma_\ast(\mathcal{F})_{(\x_j)}\]

That is, we must show that every section on \(D_+(\x_j)\) comes from a global twisted section, which means that given any section on \(D_+(\x_j)\), multiplying by a sufficiently high power of \(\x_j\) makes it come from a global section of \(\mathbb{P}^n\).

This is obtained by checking chart by chart and then gluing. To shorten notation, write \(U_i=D_+(\x_i)\). Fix \(s\in\Gamma(U_j, \mathcal{F})\) and look at each \(i\). Since \(U_i\cap U_j\) is the principal open set in \(U_i=\Spec S_{(\x_i)}\) obtained by inverting \(\x_j/\x_i\), we obtain from §Quasi-coherent Sheaves, ⁋Proposition 5 that

\[\Gamma(U_i\cap U_j, \mathcal{F})=\Gamma(U_i, \mathcal{F})_{\x_j/\x_i}\]

That is, \(s\vert_{U_i\cap U_j}\) is \(t_i\vert_{U_i\cap U_j}\) divided by \((\x_j/\x_i)^{e_i}\) for suitable \(e_i\geq0\) and \(t_i\in\Gamma(U_i, \mathcal{F})\), and therefore multiplying both sides by \(\x_i^{e_i}(\x_j/\x_i)^{e_i}=\x_j^{e_i}\) gives

\[\x_j^{e_i}\cdot s\vert_{U_i\cap U_j}=(\x_i^{e_i}t_i)\vert_{U_i\cap U_j}\]

Then both sides of this equation are sections of \(\mathcal{F}(e_i)\), the right side is the restriction of the section \(\x_i^{e_i}t_i\in\Gamma(U_i, \mathcal{F}(e_i))\) defined on all of \(U_i\), and thus \(\x_j^{e_i}s\) extends to \(U_i\). Since there are finitely many such charts, letting \(e=\max e_i\) and setting

\[u_i=\x_j^{e-e_i}\x_i^{e_i}t_i\in\Gamma(U_i, \mathcal{F}(e))\]

we have \(u_i\vert_{U_i\cap U_j}=\x_j^es\vert_{U_i\cap U_j}\) for all \(i\). It remains to check whether the \(u_i\) and \(u_{i'}\) obtained from two different charts agree on \(U_i\cap U_{i'}\), that is, the cocycle condition; their difference is \(0\) on \(U_i\cap U_{i'}\cap U_j\), and this open set is also obtained by inverting the ratio of \(\x_j\) on \(U_i\cap U_{i'}\), so by the same localization equality as above a suitable power of \(\x_j\) kills the difference. Again there are finitely many such pairs, so taking a sufficiently large common integer \(f\), the \(\x_j^fu_i\) agree and glue to a single \(t\in\Gamma(\mathbb{P}^n, \mathcal{F}(e+f))\), with \(t\vert_{U_j}=\x_j^{e+f}s\).

Conversely, suppose \(t/\x_j^m\) is \(0\) on \(U_j\) for \(t\in\Gamma(\mathbb{P}^n, \mathcal{F}(m))\). Then for each \(i\) the restriction of \(t\vert_{U_i}\) to \(U_i\cap U_j\) is \(0\), so by the localization equality above there exists \(q_i\) with \((\x_j/\x_i)^{q_i}t\vert_{U_i}=0\). Since there are finitely many charts, letting \(q=\max q_i\) gives \(\x_j^qt=0\), and therefore \(t/\x_j^m\) is already \(0\) in \(\Gamma_\ast(\mathcal{F})_{(\x_j)}\). This establishes the equality above.

Now the right-hand side \(\Gamma_\ast(\mathcal{F})_{(\x_j)}\) is a finitely generated module over \(S_{(\x_j)}\), so on each chart its generators can be written in the form

\[m_{jk}/\x_j^{e_{jk}},\qquad m_{jk}\in\Gamma(\mathbb{P}^n, \mathcal{F}(e_{jk}))\]

Since there are finitely many charts and finitely many generators, \(d_0=\max_{j,k}e_{jk}\) is well defined, and then multiplying each generator \(m_{jk}\) by \(\x_j^{d_0-e_{jk}}\) gives elements \(m_{jk}\x_j^{d_0-e_{jk}}\in\Gamma(\mathbb{P}^n, \mathcal{F}(d_0))\) that generate the stalks of \(\mathcal{F}(d_0)\) on each \(D_+(\x_j)\). Since these charts cover \(\mathbb{P}^n\), the sheaf \(\mathcal{F}(d_0)\) is globally generated, and for \(d\geq d_0\) we have \(\mathcal{F}(d)=\mathcal{F}(d_0)\otimes\mathcal{O}(d-d_0)\), which is also globally generated.

Theorem 8 For a Noetherian projective scheme \(X\) over a field \(\mathbb{K}\) and a coherent sheaf \(\mathcal{F}\) on it, each \(H^i(X, \mathcal{F})\) is a finite-dimensional \(\mathbb{K}\)-vector space, and is \(0\) for sufficiently large \(i\).

Proof

By the isomorphism \((\ast)\) above, we only need to check that the quasi-coherent sheaf \(\iota_\ast\mathcal{F}\) is (§Quasi-coherent Sheaves, ⁋Theorem 16) a coherent sheaf on \(\mathbb{P}^n\) to reduce to the case \(X=\mathbb{P}^n_{\mathbb{K}}\). For this, it suffices to check the finite type condition affine-locally: since \(\iota\) is a closed immersion, on each chart \(\iota_\ast\mathcal{F}\) is the finitely generated module over the \(A\)-algebra \(A/I\) viewed as an \(A\)-module, and because \(A \rightarrow A/I\) is surjective, lifting the generators over \(A/I\) to \(A\) shows that the same elements generate over \(A\) as well.

Thus it suffices to prove the statement for a coherent sheaf \(\mathcal{F}\) on \(\mathbb{P}^n\). First, \(H^i=0\) for sufficiently large cohomological dimension \(i>n\), because \(\mathbb{P}^n\) is covered by \(n+1\) open sets so these terms are already \(0\) at the level of the Čech complex.

We now prove finiteness of the remaining terms by descending induction on \(i\). We have already shown this is \(0\) in large dimension, so it suffices to give the inductive step. For any coherent sheaf \(\mathcal{F}\), by Lemma 7 there is some \(d\gg 0\) such that \(\mathcal{F}(d)\) is globally generated, so finitely many global sections give a surjection

\[\mathcal{O}_{\mathbb{P}^n}^{\oplus r} \twoheadrightarrow \mathcal{F}(d)\]

and twisting this by \(\mathcal{O}(-d)\) yields \(\mathcal{O}(-d)^{\oplus r}\twoheadrightarrow\mathcal{F}\). Since \(\mathbb{P}^n\) is Noetherian, on each affine chart a submodule of a finitely generated module is again finitely generated ([Commutative Algebra] §Basic Notions, ⁋Theorem 3), and therefore the kernel \(\mathcal{K}\) is also of finite type, hence a coherent sheaf. Then from the long exact sequence of the short exact sequence of coherent sheaves

\[0 \rightarrow \mathcal{K} \rightarrow \mathcal{O}(-d)^{\oplus r} \rightarrow \mathcal{F} \rightarrow 0\]

we look at

\[H^i(\mathbb{P}^n, \mathcal{O}(-d)^{\oplus r}) \rightarrow H^i(\mathbb{P}^n, \mathcal{F}) \rightarrow H^{i+1}(\mathbb{P}^n, \mathcal{K})\]

The left term is finite-dimensional by Theorem 6 (Bott), and the right term is finite-dimensional by the induction hypothesis, so the middle term \(H^i(\mathbb{P}^n, \mathcal{F})\) is also finite-dimensional.

The key argument in the proof above is to twist a coherent sheaf sufficiently to make it globally generated, and then cover it by a free sheaf. From this, the finiteness of cohomology on \(\mathbb{P}^n\) propagated along the long exact sequence to give the theorem’s claim. But Theorem 6 (Bott) shows not only that cohomology is finite-dimensional in every degree, but also that higher-degree cohomology vanishes entirely, so developing the argument in this direction yields the following result.

Theorem 9 (Serre Vanishing) For a Noetherian projective scheme \(X\) over a field \(\mathbb{K}\) and a coherent sheaf \(\mathcal{F}\) on it, for all sufficiently large \(d\gg 0\),

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

holds. Moreover, for such \(d\), the sheaf \(\mathcal{F}(d)\) is globally generated.

Proof

As in Theorem 8, by \((\ast)\) we may reduce to the case \(X=\mathbb{P}^n_{\mathbb{K}}\) and \(\mathcal{O}_X(1)=\mathcal{O}(1)\). Since the proposition involves twisting, the additional equality we need for this is

\[\iota_\ast(\mathcal{F}(d))\cong(\iota_\ast\mathcal{F})(d)\]

and this follows immediately from \(\mathcal{O}_X(1)=\iota^\ast\mathcal{O}(1)\) by §Quasi-coherent Sheaves, ⁋Proposition 17 (Projection formula).

Under this reduction, global generation is also preserved. Since \((\iota_\ast\mathcal{G})_{\iota(x)}=\mathcal{G}_x\) at \(x\in X\) and \(\Gamma(\mathbb{P}^n, \iota_\ast\mathcal{G})=\Gamma(X, \mathcal{G})\) along a closed immersion, if \((\iota_\ast\mathcal{F})(d)\) is globally generated then so is \(\mathcal{F}(d)\). Then Lemma 7 gives a \(d_0\) making \(\mathcal{F}(d)\) globally generated for every \(d\geq d_0\), so only the vanishing remains.

Now the remaining task is the vanishing, which we treat by descending induction on \(i\), just as in Theorem 8. Since \(H^i=0\) for \(i>n\) by the number of charts, it suffices to show the inductive step. For any \(i\geq1\), using the globally generated property we pick a surjection \(\mathcal{O}^{\oplus r}\twoheadrightarrow\mathcal{F}(d_0)\) and obtain the following short exact sequence

\[0 \rightarrow \mathcal{K} \rightarrow \mathcal{O}^{\oplus r} \rightarrow \mathcal{F}(d_0) \rightarrow 0\]

where \(\mathcal{K}\) is the kernel of this surjection and is a coherent sheaf for the same reason as in the proof of Theorem 8. Twisting this by \(\mathcal{O}(d-d_0)\) yields

\[0 \rightarrow \mathcal{K}(d-d_0) \rightarrow \mathcal{O}(d-d_0)^{\oplus r} \rightarrow \mathcal{F}(d) \rightarrow 0\]

and consider the long exact sequence it gives

\[H^i(\mathbb{P}^n, \mathcal{O}(d-d_0)^{\oplus r}) \rightarrow H^i(\mathbb{P}^n, \mathcal{F}(d)) \rightarrow H^{i+1}(\mathbb{P}^n, \mathcal{K}(d-d_0)).\]

The left term is \(0\) for \(d-d_0\gg0\) and \(i>0\) by Theorem 6 (Bott). The right term is the inductive hypothesis applied to \(\mathcal{K}\), so vanishing at \(i+1\) holds for sufficiently large twists. Hence if \(d\) is large enough to make both of these \(0\), the middle term \(H^i(\mathbb{P}^n, \mathcal{F}(d))\) vanishes. Since \(i\) ranges only from \(1\) to \(n\), we can choose a common \(d_1\) guaranteeing vanishing for all \(i>0\) simultaneously, and for \(d\geq d_1\) we have \(H^i(\mathbb{P}^n, \mathcal{F}(d))=0\) (\(i>0\)).

This is the scheme-level translation of the Serre vanishing from [Algebraic Varieties] §Cohomology of Projective Space, ⁋Proposition 4 (Serre Vanishing). The global generation obtained along the way allows us to write any coherent sheaf as a quotient of a finite direct sum of \(\mathcal{O}(-d)\)’s, and repeating this yields a resolution by such sheaves.

Cohomological Criterion for Ampleness

On the other hand, the crucial fact used to prove the results in the preceding section was that \(\mathcal{O}_X(1)\) comes from an embedding of \(X\) into projective space, that is, it is very ample. We already know from [Algebraic Varieties] §Linear Systems, ⁋Definition 10 that very ampleness and ampleness are closely related, and hence the (scheme version of) ampleness defined in §Divisors and Linear Systems should also be related. First, the following holds.

Corollary 10 A very ample invertible sheaf \(\mathcal{L}\) on a Noetherian projective scheme \(X\) over a field \(\mathbb{K}\) is ample. (§Divisors and Linear Systems, ⁋Definition 18)

Proof

Since \(\mathcal{L}\) is very ample, finitely many sections globally generating it define a locally closed embedding \(\iota:X \rightarrow \mathbb{P}^N_\mathbb{K}\) and satisfy \(\mathcal{L}\cong\iota^\ast\mathcal{O}(1)\). (§Divisors and Linear Systems) Since \(X\) is projective over \(\mathbb{K}\), the image \(\iota(X)\) is a closed set (§Valuation Rings, ⁋Corollary 16), and hence \(\iota\) is a closed immersion. Then \(\mathcal{L}\) can play exactly the same role as \(\mathcal{O}_X(1)\) in Theorem 9 (Serre Vanishing), so for any coherent sheaf \(\mathcal{F}\) and all sufficiently large \(d\), the sheaf \(\mathcal{F}\otimes\mathcal{L}^{\otimes d}\) is globally generated. This is the definition of ampleness.

However, the converse of this corollary does not hold. An ample invertible sheaf may lack enough sections to give an embedding, and to remedy this one must tensor it repeatedly to increase the number of sections. The following theorem tells us that this is always possible.

Theorem 11 For an invertible sheaf \(\mathcal{L}\) on a Noetherian projective scheme \(X\) over a field \(\mathbb{K}\), the following are equivalent: \(\mathcal{L}\) is ample, and \(\mathcal{L}^{\otimes m}\) is very ample for some \(m>0\).

Proof

Suppose \(\mathcal{L}^{\otimes m}\) is very ample. By Corollary 10, \(\mathcal{L}^{\otimes m}\) is ample, and by the second result of §Divisors and Linear Systems, ⁋Proposition 19, \(\mathcal{L}\) is also ample.

Conversely, suppose \(\mathcal{L}\) is ample. We must produce a locally closed embedding defined by global sections of a suitable power of \(\mathcal{L}\). By definition, some power of \(\mathcal{L}\) is globally generated, and such sections always give a morphism \(X \rightarrow \mathbb{P}^M\), so what we actually need to show is that this morphism is an embedding.

Consider the morphism \(\varphi:X \rightarrow \mathbb{P}^M\) given by sections \(s_0,\ldots, s_M\in\Gamma(X, \mathcal{L}^{\otimes k})\) globally generating \(\mathcal{L}^{\otimes k}\). This is the morphism pulling back the coordinates \(\x_i\) of \(\mathbb{P}^M\) to \(s_i\), and from this the preimage of the standard chart \(D_+(\x_i)\) is exactly the open subset \(X_{s_i}\) where \(s_i\) does not vanish. Since being a closed immersion is a local property on the target, it suffices to show that each of the morphisms

\[\varphi\vert_{X_{s_i}}: X_{s_i} \rightarrow D_+(\x_i)\]

is a closed immersion. Our claim is that if we choose the above sections very well, then the \(X_{s_i}\) become affine schemes, so the above closed immersion is a morphism between affine schemes, and hence this criterion reduces to the surjectivity of the corresponding ring homomorphism. Here, the coordinate ring of \(D_+(\x_i)\) is the polynomial ring generated by the ratios \(\x_j/\x_i\), and the ring homomorphism corresponding to \(\varphi\vert_{X_{s_i}}\) sends these to \(s_j/s_i\in\Gamma(X_{s_i}, \mathcal{O}_X)\), so surjectivity is equivalent to these ratios generating \(\Gamma(X_{s_i}, \mathcal{O}_X)\) as a \(\mathbb{K}\)-algebra.

Therefore our first goal is to find sections globally generating \(\mathcal{L}^{\otimes k}\) such that all the \(X_{s_i}\) are affine. For this, for each closed point \(x\in X\) we will construct an affine \(X_s\) containing \(x\). First, choose a trivializing open affine neighborhood \(U\) of \(x\) in \(\mathcal{L}\). If a suitable section \(s\) satisfies \(X_s\subseteq U\), then \(X_s\) is affine because it is the principal open set defined by the function \(s\) inside \(U\), so the section \(s\) we need is one that is non-vanishing at \(x\) and identically zero on \(Y=X\setminus U\).

To achieve this, give \(Y\) the reduced closed subscheme structure and consider its ideal sheaf \(\mathcal{I}_Y\). This is the kernel of \(\mathcal{O}_X \rightarrow \iota_\ast\mathcal{O}_Y\), hence quasi-coherent, and on each affine chart \(\Spec A\) it corresponds to an ideal of \(A\). Since \(X\) is Noetherian, so is \(A\), and thus this ideal is finitely generated ([Commutative Algebra] §Basic Notions, ⁋Theorem 3), so \(\mathcal{I}_Y\) is a coherent sheaf. Therefore, by the ampleness of \(\mathcal{L}\), we know that \(\mathcal{I}_Y\otimes\mathcal{L}^{\otimes n}\) is globally generated for some \(n>0\). Since \(x\not\in Y\), we have \((\mathcal{I}_Y)_x=\mathcal{O}_{X,x}\), and hence there exists a section \(s\in\Gamma(X, \mathcal{I}_Y\otimes\mathcal{L}^{\otimes n})\subseteq\Gamma(X, \mathcal{L}^{\otimes n})\) not vanishing at \(x\), from which we can choose an affine open set \(X_s\) containing each closed point. Moreover, since \(X\) is of finite type over \(\mathbb{K}\), the coordinate ring of each affine chart is a Jacobson ring ([Commutative Algebra] §Nullstellensatz, ⁋Theorem 4), and hence every non-empty closed subset of \(X\) always contains a closed point of \(X\). That is, the union of the open sets obtained above at closed points is all of \(X\), because its complement is a closed set containing no closed point, and since \(X\) is quasi-compact, finitely many of them \(X_{s_1},\ldots, X_{s_q}\) already cover \(X\). However, these were obtained from sheaves with different exponents \(s_i\in\Gamma(X, \mathcal{L}^{\otimes n_i})\), so to define a single morphism together we must match the exponents to their least common multiple \(m\) and replace the \(s_i\) by \(s_i^{m/n_i}\) so that \(s_i\in \Gamma(X, \mathcal{L}^{\otimes m})\), and this is justified because \(X_{s^e}=X_s\) for any \(e\geq1\).

Now since each \(X_{s_i}\) is affine and \(X\) is of finite type over \(\mathbb{K}\), each \(B_i=\Gamma(X_{s_i}, \mathcal{O}_X)\) is a finitely generated \(\mathbb{K}\)-algebra, and we can choose generators \(b_{i1},\ldots, b_{ir_i}\). Our claim is that there exist a common \(N\) and global sections \(t_{ij}\) such that

\[b_{ij}=t_{ij}/s_i^N, \qquad t_{ij}\in\Gamma(X, \mathcal{L}^{\otimes mN})\]

holds, and that \(s_1^N,\ldots, s_q^N\) together with the \(t_{ij}\) globally generate \(\mathcal{L}^{\otimes mN}\). If this holds, then among the ratios given by these on the chart of \(s_i^N\), we have \(t_{ij}/s_i^N=b_{ij}\), so the desired surjectivity, i.e. the fact that \(\varphi\) becomes a closed immersion when restricted to affine schemes, will be proved. The charts we check correspond only to the \(s_i^N\), but since the \(X_{s_i}\) cover \(X\), this means \(\varphi\) is a closed immersion into the union of these charts, that is, a locally closed embedding.

For this, choose a trivializing affine open cover \(V_1,\ldots, V_p\) of \(\mathcal{L}^{\otimes m}\) and fix a trivialization on each \(V_l=\Spec A_l\); then \(s_i\) corresponds to a function \(g_{il}\in A_l\) and \(X_{s_i}\cap V_l=D(g_{il})\). Then the restriction of the functions \(b_{ij}\) on \(X_{s_i}\) to \(D(g_{il})\) is an element of \((A_l)_{g_{il}}\), so multiplying by a suitable power of \(g_{il}\) makes it an element of \(A_l\), and since there are finitely many indices for \(i\), \(j\), and \(l\), taking the maximum \(N_0\) of the exponents allows us to make \(s_i^{N_0}b_{ij}\) extend to a section of \(\mathcal{L}^{\otimes mN_0}\) on every \(V_l\). It remains to show that these can be glued into a global section \(t_{ij}\).

To see this, consider the extensions obtained from two charts \(V_l\) and \(V_{l'}\); they agree on \(X_{s_i}\cap V_l\cap V_{l'}\) but there is no reason they should agree on all of \(V_l\cap V_{l'}\). To resolve this, take an affine open cover simultaneously trivializing \(\mathcal{L}^{\otimes m}\) and \(\mathcal{L}^{\otimes mN_0}\) and requiring these to cover \(V_l\cap V_{l'}\). Since \(X\) is Noetherian, we can arrange that there are finitely many such affine open sets. Then on such an affine open set \(\Spec C\), \(s_i\) corresponds to a function \(g\in C\), and the difference of the two extensions from \(V_l\) and \(V_{l'}\) becomes an element \(h\in C\) vanishing on \(D(g)\). Then \(h=0\) in \(C_g\), so \(g^ch=0\) for some exponent \(c\), meaning that multiplying both extensions by \(s_i^c\) makes them agree on all of \(V_l\cap V_{l'}\). Since there are finitely many such exponents, taking their maximum \(c_0\) and setting \(N=N_0+c_0\), the chart-wise extensions of \(s_i^{N_0}b_{ij}\) multiplied by \(s_i^{c_0}\) agree and glue to a single \(t_{ij}\in\Gamma(X, \mathcal{L}^{\otimes mN})\), and by construction the restriction of \(t_{ij}\) to \(X_{s_i}\) is \(s_i^Nb_{ij}\), and that these globally generate \(\mathcal{L}^{\otimes mN}\) is already obvious since the \(X_{s_i}\) cover \(X\). According to the discussion above, this was the last step needed to obtain the desired result, so the proof is complete.

By part 2 of §Divisors and Linear Systems, ⁋Proposition 19, \(\mathcal{L}\) being ample is equivalent to \(\mathcal{L}^{\otimes m}\) being ample for every \(m\geq1\), so Theorem 11 tells us that the only difference between the two notions lies in taking a power. Adding Theorem 9 (Serre Vanishing) to this, we can characterize ampleness solely by cohomology vanishing.

Theorem 12 (Serre criterion) For an invertible sheaf \(\mathcal{L}\) on a Noetherian projective scheme \(X\) over a field \(\mathbb{K}\), the following two conditions are equivalent.

1. \(\mathcal{L}\) is ample.
2. For any coherent sheaf \(\mathcal{F}\), there exists \(n_0\) such that for all \(i>0\) and \(n\geq n_0\), \(H^i(X, \mathcal{F}\otimes_{\mathcal{O}_X}\mathcal{L}^{\otimes n})=0\).

Proof

The direction that the first condition implies the second is easy, so let us look at it first. If \(\mathcal{L}\) is ample, then by Theorem 11 there exists \(m>0\) such that \(\mathcal{L}^{\otimes m}\) is very ample. Then, as in the proof of Corollary 10, the immersion defined by \(\mathcal{L}^{\otimes m}\) is a closed immersion, so we may apply Theorem 9 (Serre Vanishing) with \(\mathcal{L}^{\otimes m}\) as the twisting sheaf. In particular, applying this to each of the finitely many coherent sheaves

\[\mathcal{F}\otimes\mathcal{L}^{\otimes q},\qquad q=0,1,\ldots, m-1\]

we obtain, for each \(q\), integers \(p_q\) such that for all \(p>p_q\),

\[H^i\bigl(X, \mathcal{F}\otimes\mathcal{L}^{\otimes q}\otimes(\mathcal{L}^{\otimes m})^{\otimes p}\bigr)=0 \qquad (i>0).\]

Now setting \(n_0=m(\max p_q+1)\), any \(n\geq n_0\) satisfies

\[n\geq n_0>m p_q+ q\]

for all \(q\), and we obtain the desired result.

Thus the heart of this theorem is to prove the first condition from the second. That is, fixing an arbitrary coherent sheaf \(\mathcal{F}\), we must show that \(\mathcal{F}\otimes\mathcal{L}^{\otimes n}\) is globally generated for all sufficiently large \(n\).

First, fix a closed point \(x\in X\), and let \(\mathcal{I}_x\) be the ideal sheaf of the reduced closed subscheme structure on \(\{x\}\). That is, \(\mathcal{I}_x\) is the kernel of \(\mathcal{O}_X \rightarrow \iota_\ast\mathcal{O}_{\{x\}}\) induced by the closed embedding \(\iota:\{x\} \rightarrow X\) (§Closed Subschemes, ⁋Definition 5); since \(x\) is a closed point, for any open set \(U\) not meeting \(\{x\}\) we have \((\iota_\ast\mathcal{O}_{\{x\}})(U)=\mathcal{O}_{\{x\}}(U\cap\{x\})=0\), so cleanly \(\mathcal{I}_x=\mathcal{O}_X\) on the open set \(X\setminus \{x\}\). Now let \(\mathcal{I}_x\mathcal{F}\) be the image of the multiplication map \(\mathcal{I}_x\otimes\mathcal{F} \rightarrow \mathcal{F}\); then on each affine chart \(\Spec A\), \(\mathcal{I}_x\) and \(\mathcal{F}\) correspond to an ideal \(I\subseteq A\) and a finitely generated \(A\)-module \(M\), and since the associated sheaf functor is exact (§Quasi-coherent Sheaves, ⁋Proposition 6), \(\mathcal{I}_x\mathcal{F}\) is the associated sheaf of the submodule \(IM\subseteq M\) on each chart. Since \(X\) is Noetherian, \(IM\) is also finitely generated ([Commutative Algebra] §Basic Notions, ⁋Theorem 3), so \(\mathcal{I}_x\mathcal{F}\) is a coherent sheaf. We thus obtain a short exact sequence

\[0 \rightarrow \mathcal{I}_x\mathcal{F} \rightarrow \mathcal{F} \rightarrow \mathcal{F}/\mathcal{I}_x\mathcal{F} \rightarrow 0.\]

But from the computation above, we have already seen that \(\mathcal{I}_x\) becomes all of \(\mathcal{O}_X\) outside \(\{x\}\), so the stalk of the last term \(\mathcal{F}/\mathcal{I}_x\mathcal{F}\) vanishes away from \(x\), and at \(x\), since \(\mathcal{I}_x\) is the ideal sheaf of the reduced structure, it is exactly the skyscraper sheaf at \(x\) with value

\[(\mathcal{F}/\mathcal{I}_x\mathcal{F})_x=\mathcal{F}_x/\mathfrak{m}_x\mathcal{F}_x=\mathcal{F}_x\otimes_{\mathcal{O}_{X,x}}\kappa(x).\]

Now tensoring the above exact sequence with \(\mathcal{L}^{\otimes n}\), since this is invertible it is locally isomorphic to \(\mathcal{O}_X\), and thus exactness is preserved, so

\[0 \rightarrow \mathcal{I}_x\mathcal{F}\otimes\mathcal{L}^{\otimes n} \rightarrow \mathcal{F}\otimes\mathcal{L}^{\otimes n} \rightarrow (\mathcal{F}/\mathcal{I}_x\mathcal{F})\otimes\mathcal{L}^{\otimes n} \rightarrow 0\]

is also exact, and the last term becomes the skyscraper sheaf at \(x\) with value

\[(\mathcal{F}\otimes\mathcal{L}^{\otimes n})_x\otimes_{\mathcal{O}_{X,x}}\kappa(x).\]

Applying the given hypothesis to the coherent sheaf \(\mathcal{I}_x\mathcal{F}\), there exists \(n_1\) such that for all \(n\geq n_1\), \(H^1(X, \mathcal{I}_x\mathcal{F}\otimes\mathcal{L}^{\otimes n})=0\), so a portion of the long exact sequence

\[\Gamma(X, \mathcal{F}\otimes\mathcal{L}^{\otimes n}) \longrightarrow (\mathcal{F}\otimes\mathcal{L}^{\otimes n})_x\otimes_{\mathcal{O}_{X,x}}\kappa(x) \longrightarrow 0\]

is exact. That is, the global sections of \(\mathcal{F}\otimes\mathcal{L}^{\otimes n}\) surject onto the fiber at \(x\). From this we obtain global generation in a neighborhood of \(x\).

Indeed, the stalk \(M=(\mathcal{F}\otimes\mathcal{L}^{\otimes n})_x\) is a finitely generated module over the Noetherian local ring \(\mathcal{O}_{X,x}\), and the fiber of the above skyscraper sheaf is \(M/\mathfrak{m}_xM\), so we can choose finitely many global sections \(s_1,\ldots, s_c\) whose germs generate \(M/\mathfrak{m}_xM\), and by the second result of [Commutative Algebra] §Integral Extensions, ⁋Lemma 8 (Nakayama) they generate \(M\) itself. Then the cokernel \(\mathcal{Q}\) of the map \(\mathcal{O}_X^{\oplus c} \rightarrow \mathcal{F}\otimes\mathcal{L}^{\otimes n}\) defined by \(s_1,\ldots, s_c\) is a coherent sheaf satisfying \(\mathcal{Q}_x=0\). Taking an affine open neighborhood \(\Spec A\) containing \(x\), letting \(N\) be the finitely generated \(A\)-module corresponding to \(\mathcal{Q}\) on it and \(\mathfrak{p}\) the prime ideal corresponding to \(x\), we have \(N_\mathfrak{p}=0\), so for each generator of \(N\) there exists an element outside \(\mathfrak{p}\) annihilating it; since \(N\) has finitely many generators, letting \(f\) be the product of the elements thus obtained, we have \(N_f=0\). Hence \(\mathcal{Q}\) vanishes on the open neighborhood \(D(f)\) of \(x\), which means that \(\mathcal{F}\otimes\mathcal{L}^{\otimes n}\) is generated by \(s_1,\ldots, s_c\) on \(D(f)\).

That is, we have verified that for each closed point \(x\) there exists an open neighborhood \(D(f)\) such that \(\mathcal{F}\otimes\mathcal{L}^{\otimes n}\) is globally generated at every point of \(D(f)\). The somewhat subtle point is that the definition of this \(D(f)\) varies with \(n\), and to finish the proof of the theorem we need a single neighborhood that works simultaneously for all sufficiently large \(n\), so before the main argument we must find a way to increase the exponent of \(\mathcal{L}\) on a fixed neighborhood. For this, applying the argument so far to \(\mathcal{F}=\mathcal{O}_X\), there exists \(e\geq1\) such that \(\mathcal{L}^{\otimes e}\) and \(\mathcal{L}^{\otimes(e+1)}\) are each globally generated on some open neighborhood of \(x\), and hence both are so on the intersection \(W\) of the two neighborhoods. Since the tensor product of two globally generated sheaves is again globally generated by the tensors of stalk generators, \(\mathcal{L}^{\otimes(ae+b(e+1))}\) is globally generated on \(W\) for \(a,b\geq0\). But writing any integer \(k\geq e^2\) as \(k=se+b\) with quotient \(s\) and remainder \(b\) upon division by \(e\), we have \(s\geq e>b\geq0\), so \(a=s-b\) is nonnegative and \(ae+b(e+1)=se+b=k\); thus on \(W\), \(\mathcal{L}^{\otimes k}\) is globally generated for all \(k\geq e^2\). Returning now to \(\mathcal{F}\) itself, choosing some \(n_2\geq n_1\), we have that \(\mathcal{F}\otimes\mathcal{L}^{\otimes n_2}\) is globally generated on some open neighborhood \(V\) of \(x\), so for all \(n\geq n_2+e^2\),

\[\mathcal{F}\otimes\mathcal{L}^{\otimes n}\cong(\mathcal{F}\otimes\mathcal{L}^{\otimes n_2})\otimes\mathcal{L}^{\otimes(n-n_2)}\]

is globally generated on \(W\cap V\). That is, for each closed point \(x\) we obtain an open neighborhood \(U_x=W\cap V\) and a lower bound \(n_x=n_2+e^2\) such that \(\mathcal{F}\otimes\mathcal{L}^{\otimes n}\) is globally generated on \(U_x\) whenever \(n\geq n_x\).

It remains to cover \(X\) by these neighborhoods; as seen in the proof of Theorem 11, since \(X\) is of finite type over \(\mathbb{K}\), the coordinate ring of each affine chart is a Jacobson ring, so every nonempty closed subset of \(X\) contains a closed point. Hence the union of the \(U_x\) is the complement of a closed set with no closed points, which is all of \(X\), and since \(X\) is quasi-compact, finitely many \(U_{x_1},\ldots, U_{x_r}\) suffice to cover \(X\). Then setting \(n_0=\max_jn_{x_j}\), for every \(n\geq n_0\) the sheaf \(\mathcal{F}\otimes\mathcal{L}^{\otimes n}\) is generated by global sections at every point of \(X\), and since \(\mathcal{F}\) was an arbitrary coherent sheaf, \(\mathcal{L}\) is ample.

A noteworthy observation from the proof is that although Theorem 12 (Serre criterion) required vanishing for all \(i>0\), in deriving (1) from (2) only the vanishing of \(H^1\) was actually used; this is because the obstruction to lifting sections of the quotient sheaf to global sections lies in the \(H^1\) term of the long exact sequence.

Euler Characteristic and Hilbert Polynomial

By Theorem 8, a coherent sheaf on a projective scheme has only finitely many finite-dimensional cohomologies, so we may take their alternating sum.

Definition 13 For a Noetherian projective scheme \(X\) over a field \(\mathbb{K}\) and a coherent sheaf \(\mathcal{F}\) on it, we define the Euler characteristic of \(\mathcal{F}\) by the formula

\[\rchi(X, \mathcal{F})=\sum_{i\geq 0}(-1)^i\dim_\mathbb{K}H^i(X, \mathcal{F}).\]

By Theorem 8 the right-hand side is a finite sum and each term is finite, so \(\rchi(X, \mathcal{F})\) is an integer; when \(X\) is clear from context we abbreviate it as \(\rchi(\mathcal{F})\). This is the scheme-theoretic analogue of [Algebraic Varieties] §Cohomology of Projective Space, ⁋Definition 2. While the individual dimensions \(\dim_\mathbb{K}H^i(X, \mathcal{F})\) can jump when the sheaf is perturbed slightly, their alternating sum is far more stable; the source of this is the following additivity.

Proposition 14 For coherent sheaves on a Noetherian projective scheme \(X\) over a field \(\mathbb{K}\), the following hold.

1. For a short exact sequence \(0 \rightarrow \mathcal{F}' \rightarrow \mathcal{F} \rightarrow \mathcal{F}'' \rightarrow 0\), we have \(\rchi(\mathcal{F})=\rchi(\mathcal{F}')+\rchi(\mathcal{F}'')\).
2. For a finite exact sequence \(0 \rightarrow \mathcal{F}_k \rightarrow \cdots \rightarrow \mathcal{F}_1 \rightarrow \mathcal{F}_0 \rightarrow 0\), we have \(\sum_{j=0}^k(-1)^j\rchi(\mathcal{F}_j)=0\).

Proof

First consider an exact sequence \(0 \rightarrow V_0 \rightarrow V_1 \rightarrow \cdots \rightarrow V_t \rightarrow 0\) of finite-dimensional vector spaces. Letting \(r_j\) be the rank of the \(j\)-th linear map (with \(r_{-1}=r_t=0\)), exactness gives \(\dim V_j=r_{j-1}+r_j\), so in the alternating sum \(\sum_j(-1)^j\dim V_j\) adjacent terms cancel and the result is \(0\).

For (1), the long exact sequence given by Proposition 2,

\[0 \rightarrow H^0(X, \mathcal{F}') \rightarrow H^0(X, \mathcal{F}) \rightarrow H^0(X, \mathcal{F}'') \rightarrow H^1(X, \mathcal{F}') \rightarrow \cdots,\]

consists of finite-dimensional vector spaces by Theorem 8 and terminates in sufficiently high degree, so it is finite. Applying the observation above, the alternating sum of the cohomology dimensions of the three sheaves is \(0\), and rearranging signs gives the desired formula.

For (2), set \(\mathcal{Z}_j=\ker(\mathcal{F}_j \rightarrow \mathcal{F}_{j-1})\) for \(j\geq1\) and \(\mathcal{Z}_0=\mathcal{F}_0\). Since the kernel of a morphism between coherent sheaves is again coherent, each \(\mathcal{Z}_j\) is a coherent sheaf; by exactness \(\mathcal{Z}_k=0\), and for each \(j\geq1\) we have a short exact sequence

\[0 \rightarrow \mathcal{Z}_j \rightarrow \mathcal{F}_j \rightarrow \mathcal{Z}_{j-1} \rightarrow 0.\]

Applying (1) to this gives \(\rchi(\mathcal{F}_j)=\rchi(\mathcal{Z}_j)+\rchi(\mathcal{Z}_{j-1})\), and adding these with alternating signs causes the intermediate terms to cancel, yielding \(\sum_{j=1}^k(-1)^j\rchi(\mathcal{F}_j)=-\rchi(\mathcal{Z}_0)=-\rchi(\mathcal{F}_0)\).

In particular, if a coherent sheaf \(\mathcal{F}\) has a finite resolution \(0 \rightarrow \mathcal{E}_k \rightarrow \cdots \rightarrow \mathcal{E}_0 \rightarrow \mathcal{F} \rightarrow 0\), then from (2) of Proposition 14 we obtain \(\rchi(\mathcal{F})=\sum_{j=0}^k(-1)^j\rchi(\mathcal{E}_j)\). This is the standard path for actually computing the Euler characteristic, and its starting point is line bundles on projective space.

Corollary 15 For projective space \(\mathbb{P}^n_\mathbb{K}\) over a field \(\mathbb{K}\) and any integer \(d\),

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

holds. Here \(\binom{n+d}{n}\) is read as the value of the polynomial \(t(t-1)\cdots(t-n+1)/n!\) at \(t=n+d\).

Proof

Setting \(A=\mathbb{K}\) in Theorem 6 (Bott), there are three cases. If \(d\geq0\), only \(H^0\) survives and its dimension is the number of degree \(d\) monomials in \(n+1\) variables, \(\binom{n+d}{n}\). If \(-n\leq d\leq-1\), all cohomology vanishes; in this range \(t=n+d\) is an integer between \(0\) and \(n-1\), so one of the factors in the product \(t(t-1)\cdots(t-n+1)\) is \(0\), giving \(\binom{n+d}{n}=0\). If \(d\leq -n-1\), only \(H^n\) survives and its dimension is the number of degree \(d\) monomials with all exponents negative, \(\binom{-d-1}{n}\), so

\[\rchi(\mathbb{P}^n_\mathbb{K}, \mathcal{O}(d))=(-1)^n\binom{-d-1}{n}=\binom{n+d}{n}.\]

The last equality follows from \(t(t-1)\cdots(t-n+1)=(-1)^n(n-t-1)(n-t-2)\cdots(-t)\) for \(t=n+d\).

This value is a polynomial of degree \(n\) in \(d\), matching the variety version of [Algebraic Varieties] §Cohomology of Projective Space, ⁋Corollary 3. That is, as we twist by \(\mathcal{O}(d)\), the alternating sum of cohomology follows a single polynomial regardless of the three phases that the individual cohomologies undergo. The following theorem states that this also holds for general coherent sheaves; to formulate it we define the support of a coherent sheaf \(\mathcal{F}\) as \(\supp\mathcal{F}=\{x\in X\mid \mathcal{F}_x\neq0\}\). On an affine chart this is the zero set of the annihilator ideal of the corresponding module, hence a closed set.

Theorem 16 (Hilbert) For a closed subscheme \(X\) of the projective space \(\mathbb{P}^n_\mathbb{K}\) over a field \(\mathbb{K}\) and a coherent sheaf \(\mathcal{F}\) on it, there exists a unique numerical polynomial \(P_\mathcal{F}\) such that ([Commutative Algebra] §Hilbert–Samuel Function, ⁋Definition 1) for every integer \(d\),

\[\rchi(\mathcal{F}(d))=P_\mathcal{F}(d)\]

holds. Moreover, if \(\mathcal{F}\neq0\), then the degree of \(P_\mathcal{F}\) equals \(\dim\supp\mathcal{F}\), and for sufficiently large \(d\) we have \(P_\mathcal{F}(d)=\dim_\mathbb{K}\Gamma(X, \mathcal{F}(d))\).

Proof

The last claim follows immediately from Theorem 9 (Serre Vanishing): for sufficiently large \(d\) we have \(H^i(X, \mathcal{F}(d))=0\) (\(i>0\)), so only \(H^0\) remains in the alternating sum. Uniqueness follows from the fact that two distinct polynomials cannot agree at infinitely many integers.

We first reduce to the case \(X=\mathbb{P}^n_\mathbb{K}\). For the closed immersion \(\iota:X\hookrightarrow\mathbb{P}^n_\mathbb{K}\), combining the isomorphism \((\ast)\) just before Theorem 8 with the equality \(\iota_\ast(\mathcal{F}(d))\cong(\iota_\ast\mathcal{F})(d)\) seen in the proof of Theorem 9 (Serre Vanishing), we obtain \(H^i(X, \mathcal{F}(d))\cong H^i(\mathbb{P}^n, (\iota_\ast\mathcal{F})(d))\); and since \(\supp\iota_\ast\mathcal{F}=\iota(\supp\mathcal{F})\), as seen in the proof of Theorem 8 we may replace \(\mathcal{F}\) by the coherent sheaf \(\iota_\ast\mathcal{F}\) without loss of generality.

Moreover, we may assume that the field \(\mathbb{K}\) is infinite. Since the conclusion of the theorem is stated solely in terms of the dimensions of cohomology and the dimension of \(\supp\mathcal{F}\), to prove this it suffices to show that these two invariants are preserved under an extension to an infinite field \(\mathbb{K}\hookrightarrow \mathbb{L}\). First, for the dimensions of cohomology, the Čech complex for the standard affine cover is \(\check C^\bullet(\mathcal{U}, \mathcal{F})\otimes_\mathbb{K}\mathbb{L}\) with only the coefficients changed, and since \(-\otimes_\mathbb{K}\mathbb{L}\) is exact, Corollary 4 gives \(\dim_\mathbb{L}H^i(\mathbb{P}^n_\mathbb{L}, \mathcal{F}_\mathbb{L}(d))=\dim_\mathbb{K}H^i(\mathbb{P}^n_\mathbb{K}, \mathcal{F}(d))\). For the dimension of the support, we first observe that the support itself is compatible with field extension. On each affine chart \(\Spec A\), letting \(m_1,\ldots, m_r\) be generators of the finitely generated module \(M\) corresponding to \(\mathcal{F}\), the annihilator \(\ann M\) is the kernel of \(A \rightarrow M^{\oplus r}\) given by \(a\mapsto(am_1,\ldots, am_r)\). ([Commutative Algebra] §Basic Notions, ⁋Definition 1) But the \(m_k\otimes1\) generate \(M\otimes_\mathbb{K}\mathbb{L}\) and \(-\otimes_\mathbb{K}\mathbb{L}\) preserves kernels, so \(\ann(M\otimes_\mathbb{K}\mathbb{L})=(\ann M)\otimes_\mathbb{K}\mathbb{L}\); hence if on each chart \(\supp\mathcal{F}\) is \(\Spec(A/\ann M)\), then \(\supp\mathcal{F}_\mathbb{L}\) is given on each chart by \(\Spec\bigl((A/\ann M)\otimes_\mathbb{K}\mathbb{L}\bigr)\). Now the dimension of a finitely generated \(\mathbb{K}\)-algebra does not change under field extension ([Commutative Algebra] §Noether Normalization, ⁋Proposition 5), so these two invariants are preserved and we may assume from the outset that \(\mathbb{K}\) is infinite.

The overall structure of the proof is now the same dévissage used in the proofs of Theorem 8 and Theorem 9 (Serre Vanishing): we construct a short exact sequence

\[0 \rightarrow \mathcal{F}(-1) \rightarrow \mathcal{F} \rightarrow \mathcal{F}'' \rightarrow 0\tag{$\ast\ast$}\]

and transfer the desired property along it. Here \(\mathcal{F}(-1) \rightarrow \mathcal{F}\) is the morphism given by multiplication by an element \(\ell\in S_1\) of degree \(1\) in the homogeneous coordinate ring \(S_\bullet=\mathbb{K}[\x_0,\ldots, \x_n]\), and \(\mathcal{F}''\) is its cokernel; intuitively this is the restriction of \(\mathcal{F}\) to the hyperplane \(V_+(\ell)\), i.e. a hyperplane section of \(\mathcal{F}\). The difference from the previous proofs is that, whereas those ran induction by covering a coherent sheaf with a free sheaf and computing with the long exact sequence, we use the additivity from Proposition 14 to run induction on the dimension \(r=\dim\supp \mathcal{F}\) of the support.

First, such a short exact sequence cannot exist for an arbitrary choice of \(\ell\). The issue is the injectivity of \(\times \ell\): if \(\ell\) belongs to some associated prime of \(\mathcal{F}\), that is, a prime \(\mathfrak{p}=\ann(m)\) arising as the annihilator of a non-zero section \(m\), then \(\ell m=0\) and injectivity fails. Conversely, since the zerodivisors are exactly the union of the associated primes, these are the only obstructions to injectivity. Geometrically, the condition that the point \(\mathfrak{p}\) lies on the hyperplane \(V_+(\ell)\) is exactly \(\ell\in\mathfrak{p}\), so this condition is equivalent to saying that \(V_+(\ell)\) passes through none of the finitely many points given by the associated primes of \(\mathcal{F}\). Among these points are the generic points of the irreducible components of \(\supp\mathcal{F}\) (§Algebraic Structure of Schemes), so such a hyperplane does not contain any component of \(\supp\mathcal{F}\) and meets each only in a proper subset. Thus \(\ell\) must avoid all the points given by the associated primes of \(\mathcal{F}\), and our idea is to use the fact that we have arranged \(\mathbb{K}\) to be infinite, even though there are only finitely many associated primes.

If \(\mathcal{F}=0\) then all cohomology vanishes and the theorem is trivial. Hence assume \(\mathcal{F}\neq0\). On each chart \(D_+(\x_j)=\Spec S_{(\x_j)}\), the sheaf \(\mathcal{F}\) corresponds to a finitely generated module \(M_j\) (§Quasi-coherent Sheaves, ⁋Theorem 10, §Quasi-coherent Sheaves, ⁋Definition 11), and each \(\Ass M_j\) is a finite set. ([Commutative Algebra] §Associated Primes, ⁋Theorem 7, part 1) Collecting all the corresponding points of \(\mathbb{P}^n\) gives finitely many points, each yielding a homogeneous prime ideal \(\mathfrak{p}_1,\ldots, \mathfrak{p}_t\subseteq S\) not containing the irrelevant ideal. Since each \(\mathfrak{p}_k\) is homogeneous, if \(S_1\subseteq\mathfrak{p}_k\) then \(\mathfrak{p}_k\) would contain the irrelevant ideal; hence each \(\mathfrak{p}_k\cap S_1\) is a proper subspace of \(S_1\). On the other hand, since \(\mathbb{K}\) is infinite, the vector space \(S_1\) cannot be the union of finitely many proper subspaces, and therefore there exists \(\ell\in S_1\) not belonging to any \(\mathfrak{p}_k\).

The \(\ell\) chosen in this way avoids the homogeneous primes \(\mathfrak{p}_k\) of \(S\), whereas what obstructs injectivity in the short exact sequence are the associated primes on each chart; so we must transfer this condition to the \(S_{(\x_j)}\) side via the correspondence of §Projective Schemes, ⁋Lemma 8. Viewing the morphism \(\mathcal{F}(-1) \rightarrow \mathcal{F}\) given by multiplication by \(\ell\) on the chart \(D_+(\x_j)\), it acts by multiplying each element of \(M_j\) by \(\ell/\x_j\). Our claim is that this element \(\ell/\x_j\) is not a zerodivisor on \(M_j\), hence injectivity is guaranteed; for this we must show that \(\ell/\x_j\) does not belong to any associated prime of \(M_j\). ([Commutative Algebra] §Associated Primes, ⁋Theorem 7, part 2) Indeed, each associated prime of \(M_j\) is a prime \(\mathfrak{p}_kS_{\x_j}\cap S_{(\x_j)}\) given by some \(\mathfrak{p}_k\) lying in \(D_+(\x_j)\); if \(\ell/\x_j\) belonged to such a prime, then \(\ell=\x_j\cdot(\ell/\x_j)\) would lie in \(\mathfrak{p}_kS_{\x_j}\cap S=\mathfrak{p}_k\), contradicting the choice of \(\ell\). Hence this morphism is injective. Letting \(\mathcal{F}''\) be its cokernel, for each \(d\) we obtain a short exact sequence

\[0 \rightarrow \mathcal{F}(d-1) \rightarrow \mathcal{F}(d) \rightarrow \mathcal{F}''(d) \rightarrow 0\]

and \(\supp\mathcal{F}''=\supp\mathcal{F}\cap V_+(\ell)\). Indeed, looking at the stalk at a point \(\mathfrak{p}\) we have \(\mathcal{F}''_\mathfrak{p}=\mathcal{F}_\mathfrak{p}/\ell\mathcal{F}_\mathfrak{p}\); if \(\ell\notin\mathfrak{p}\) then \(\ell\) is a unit in the local ring so this is \(0\), and if \(\ell\in\mathfrak{p}\) while \(\mathcal{F}_\mathfrak{p}\neq0\) then \(\mathcal{F}_\mathfrak{p}/\ell\mathcal{F}_\mathfrak{p}\neq0\) by part 1 of [Commutative Algebra] §Integral Extensions, ⁋Lemma 8.

As announced, the rest proceeds by applying induction on dimension using the short exact sequence (\(\ast\ast\)) thus obtained. For this we must first show that a hyperplane section actually drops the dimension of the support by exactly one. Since minimal primes are always associated primes, \(\ell\) does not vanish identically on any irreducible component of \(\supp\mathcal{F}\). Giving each component \(Z\) its reduced structure and measuring the dimension of the intersection, since \(Z\) is irreducible and of finite type over \(\mathbb{K}\), for each chart meeting \(Z\) the ring \(A_j\) of \(Z\cap D_+(\x_j)=\Spec A_j\) is a finitely generated \(\mathbb{K}\)-algebra that is an integral domain, and its fraction field is the function field of \(Z\), independent of the chart. That is, by §Dimension, ⁋Proposition 10 the dimension \(\dim A_j\) is the same in every chart, and by §Dimension, ⁋Proposition 2 this common value is \(\dim Z\). If \(\dim Z=0\) then \(Z\) is a single point and \(\ell\) does not vanish there, so \(Z\cap V_+(\ell)=\emptyset\); conversely, if \(Z\cap V_+(\ell)= \emptyset\) then necessarily \(\dim Z=0\). This is because \(Z\) is affine as a closed subscheme of the affine scheme \(D_+(\ell)\), and its coordinate ring is a finite-dimensional \(\mathbb{K}\)-vector space by Theorem 8 and an integral domain by the integrality of \(Z\), hence a field by [Field Theory] §Algebraic Extensions, ⁋Proposition 3. Therefore if \(\dim Z\geq 1\) then \(Z\cap V_+(\ell)\) is nonempty, and choosing a chart \(D_+(\x_j)\) containing a point of this set, the element \(\ell/\x_j\in A_j\) is neither \(0\) nor a unit since it vanishes at that point; hence by §Dimension, ⁋Proposition 11,

\[\dim\bigl(Z\cap V_+(\ell)\cap D_+(\x_j)\bigr)=\dim A_j-1=\dim Z-1.\]

This value is the same in every chart meeting \(Z\cap V_+(\ell)\), so by §Dimension, ⁋Proposition 2 we have \(\dim(Z\cap V_+(\ell))=\dim Z-1\). In other words, if \(r=0\) then no component meets \(V_+(\ell)\) and \(\mathcal{F}''=0\), while if \(r\geq1\) the components of dimension \(r\) yield dimension \(r-1\), so \(\dim\supp\mathcal{F}''=r-1\).

We now actually run the induction. First consider the case \(r=0\). Here \(\mathcal{F}''=0\), so the short exact sequence obtained by twisting (\(\ast\ast\)) by \(d\) gives an isomorphism \(\mathcal{F}(d-1)\cong\mathcal{F}(d)\); hence setting \(g(d)=\rchi(\mathcal{F}(d))\), the function \(g\) is constant independent of \(d\). Letting this constant be \(P\), then \(P\) is a numerical polynomial satisfying \(\rchi(\mathcal{F}(d))=P(d)\) for all integers \(d\). Moreover, by Theorem 9 (Serre Vanishing) the sheaf \(\mathcal{F}(d)\) is globally generated for sufficiently large \(d\), and since \(\mathcal{F}\neq0\) we have \(\Gamma(X, \mathcal{F}(d))\neq0\); hence by the last claim \(P>0\), so \(\deg P=0=r\).

Now assume \(r\geq1\) and that the theorem holds for support of dimension \(r-1\). As seen above, \(\dim\supp\mathcal{F}''=r-1\) and in particular \(\mathcal{F}''\neq0\); hence by the inductive hypothesis \(\rchi(\mathcal{F}''(d))\) agrees for all \(d\) with some numerical polynomial \(Q\) of degree \(\deg Q=r-1\). By part 1 of Proposition 14 we have \(g(d)-g(d-1)=Q(d)\) for all integers \(d\), so \(g(d+1)-g(d)=Q(d+1)\) is a numerical polynomial; then by the second result of [Commutative Algebra] §Hilbert–Samuel Function, ⁋Lemma 2 there exists a numerical polynomial \(P\) agreeing with \(g\) for sufficiently large \(d\). Since \(\deg Q=r-1\geq0\) we have \(Q\neq0\), so \(\deg P=\deg Q+1=r\). For sufficiently large \(d\) we have \(P(d+1)-P(d)=g(d+1)-g(d)=Q(d+1)\), so this equality holds as polynomials, and therefore stepping \(d\) down one by one we obtain

\[g(d)=g(d+1)-Q(d+1)=P(d+1)-Q(d+1)=P(d)\]

so \(g\) and \(P\) agree at all integers.

This polynomial \(P_\mathcal{F}\) is called the Hilbert polynomial of \(\mathcal{F}\). Corollary 15 is the case \(\mathcal{F}=\mathcal{O}_{\mathbb{P}^n}\), for which \(P_{\mathcal{O}_{\mathbb{P}^n}}(t)=\binom{n+t}{n}\) and its degree is \(\dim\mathbb{P}^n=n\). The last claim of Theorem 16 (Hilbert) says that this polynomial measures, for sufficiently large degree, the dimension of the space of global sections of \(\mathcal{F}(d)\); this is the point of contact with the classical viewpoint of introducing the Hilbert polynomial via the Hilbert function of the homogeneous coordinate ring. ([Commutative Algebra] §Hilbert–Samuel Function, ⁋Definition 4) In particular, when \(\mathcal{F}=\mathcal{O}_X\) this polynomial becomes an invariant of \(X\) itself.

Definition 17 For a nonempty \(r\)-dimensional closed subscheme \(X\) of the projective space \(\mathbb{P}^n_\mathbb{K}\) over a field \(\mathbb{K}\), let \(a_r\) be the leading coefficient of the Hilbert polynomial \(P_{\mathcal{O}_X}\). Then we define the degree of \(X\) by

\[\deg X=r!\cdot a_r\]

and the arithmetic genus of \(X\) by \(p_a(X)=(-1)^r\bigl(P_{\mathcal{O}_X}(0)-1\bigr)\).

Since \(\supp\mathcal{O}_X=X\), Theorem 16 (Hilbert) implies that \(P_{\mathcal{O}_X}\) is a polynomial of degree \(r\), and for sufficiently large \(d\) we have \(P_{\mathcal{O}_X}(d)=\dim_\mathbb{K}\Gamma(X, \mathcal{O}_X(d))>0\), so \(a_r\) is positive. Moreover, writing a numerical polynomial as an integer linear combination of binomial coefficients (the first result of [Commutative Algebra] §Hilbert–Samuel Function, ⁋Lemma 2), we see that \(r!\) times the leading coefficient is an integer. Thus \(\deg X\) is a positive integer. On the other hand, since \(P_{\mathcal{O}_X}(0)=\rchi(\mathcal{O}_X)\), the arithmetic genus is simply a rewriting of the Euler characteristic of the structure sheaf, and the sign \((-1)^r\) is chosen so that for curves we recover the familiar definition \(p_a=1-\rchi(\mathcal{O}_X)\).

The simplest example is projective space itself. From Corollary 15 we have \(P_{\mathcal{O}_{\mathbb{P}^n}}(t)=\binom{n+t}{n}\), so the leading coefficient is \(1/n!\), hence \(\deg\mathbb{P}^n_\mathbb{K}=1\), and from \(P_{\mathcal{O}_{\mathbb{P}^n}}(0)=1\) we get \(p_a(\mathbb{P}^n_\mathbb{K})=0\). A less trivial example is the hypersurface \(X=V_+(f)\subseteq\mathbb{P}^n_\mathbb{K}\) defined by a nonzero homogeneous polynomial \(f\) of positive degree \(e\). The polynomial \(f\) can be a constant multiple of \(\x_j^e\) for at most one \(j\), so if \(n\geq1\) there must exist a chart \(D_+(\x_j)\) where \(f\) is not a constant multiple of \(\x_j^e\). On this chart, \(X\) is the zero set of the dehomogenization \(f/\x_j^e\), which is neither \(0\) nor a unit, so its zero set is nonempty, and thus \(X\) is a nonempty closed subscheme to which Definition 17 applies. Following this through, since \(\mathbb{K}[\x_0,\ldots, \x_n]\) is an integral domain, multiplication by \(f\) is injective on each chart, and therefore we have a short exact sequence

\[0 \rightarrow \mathcal{O}_{\mathbb{P}^n}(-e)\overset{\times f}{\longrightarrow}\mathcal{O}_{\mathbb{P}^n} \rightarrow \mathcal{O}_X \rightarrow 0\]

Twisting this by \(\mathcal{O}(d)\) and applying part 1 of Proposition 14 and Corollary 15 yields

\[P_{\mathcal{O}_X}(t)=\binom{n+t}{n}-\binom{n+t-e}{n}\]

On the right-hand side, the \(t^n\) terms cancel and the coefficient of \(t^{n-1}\) remaining is \(e/(n-1)!\), so together with \(\dim X=n-1\) we obtain \(\deg X=e\). Thus the classical fact that the degree of a hypersurface in \(\mathbb{P}^n\) equals the degree of the defining polynomial is recovered in the language of Hilbert polynomials.

---

References

[Har] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, Springer, 1977.
[Vak] R. Vakil, The rising sea: Foundations of algebraic geometry. Available online.