Finiteness of cohomology for coherent sheaves on a projective scheme (Tag T-COHFIN)
Let \(A\) be Noetherian, \(X\) projective over \(A\), and \(\mcf \in \Coh(X)\). Then each \(H^i(X, \mcf)\) is a finitely generated \(A\dash\)module. Over a field, every \(h^i(X,\mcf)\) is finite and vanishes for \(i > \dim X\).
(Proof shape)
Reduce to \(X = \PP^n_A\) by pushing forward along the closed immersion, which changes no cohomology. Any coherent \(\mcf\) on \(\PP^n\) sits in \(0 \to \mcr \to \mce \to \mcf \to 0\) with \(\mce\) a finite sum of twists \(\OO(-d_i)\), and \(\mcr\) is again coherent. The twists have finitely generated cohomology by the explicit computation, and descending induction on \(i\) from Grothendieck vanishing runs the long exact sequence down.
(Hypotheses)
Properness is what fails first and it fails already in degree \(0\): on \(X = \AA^1_k\), \(H^0(X,\OO) = k[x]\) is infinite dimensional. So finiteness is a statement about proper schemes, not about nice sheaves, and the correct general form is for \(f: X \to \Spec A\) proper with \(\mcf\) coherent.
Coherence is the second hypothesis, and it fails on quasicoherent sheaves for the same reason: \(\bigoplus_{d \geq 0} \OO(d)\) on \(\PP^n\) has infinite dimensional \(H^0\).
The practical payoff is that \(h^0, h^1, \ldots\) are numbers, so \(\chi\) exists and Riemann–Roch can be a statement about integers.
Serre vanishing for large twists (Tag T-COHSVAN)
Let \(A\) be noetherian, let \(X\) be projective over \(A\) with \(\OO_X(1)\) very ample, and let \(\mcf\) be coherent. Then there is \(n_0\) such that \[ H^i(X, \mcf(n)) = 0 \quad \text{for all } i > 0 \text{ and all } n \geq n_0 . \] [@Har10a, Theorem III.5.2]
(Dependence of the bound on the sheaf)
For a field \(k\) and an integer \(m\ge0\), the sheaf \(\OO_{\PP_k^1}(-m)\) has vanishing positive-degree cohomology after twisting by every \(n\ge m\). At \(n=m-2\), its first cohomology has dimension one [@Har10a, Theorem III.5.1]. Thus this family on \(\PP_k^1\) has no common twist bound as \(m\) varies; T-MODSERRE gives the explicit cohomology and global-generation bounds.
(Euler characteristics over a field)
Suppose \(A=k\) is a field. The dimensions \(h^i(X,\mcf(n))=\dim_k H^i(X,\mcf(n))\) are finite by T-COHFIN. For all sufficiently large \(n\), Serre vanishing gives \[ \chi(X,\mcf(n))=\sum_i(-1)^i h^i(X,\mcf(n))=h^0(X,\mcf(n)). \] If \(S\) is a finitely generated graded \(k\)-algebra with \(S_0=k\), generated in degree one, and \(M\) is a finite graded \(S\)-module with \(X=\Proj S\) and \(\mcf=\widetilde M\), then P-AGH259GAMMASTAR identifies the last term with \(\dim_k M_n\) for all sufficiently large \(n\). Consequently this Euler characteristic agrees in large degrees with the Hilbert polynomial of \(M\) [@Har10a, Chapter I, §7].
(Cohomological criterion for ampleness)
Let \(A\) be noetherian, let \(Y\) be proper over \(A\), and let \(\mcl\) be an invertible sheaf on \(Y\). Then \(\mcl\) is ample if and only if, for every coherent sheaf \(\mcg\) on \(Y\), there exists an integer \(N\) such that \[ H^i(Y,\mcg\otimes_{\OO_Y}\mcl^{\otimes n})=0 \quad\text{for every }i>0\text{ and every }n\ge N. \] [@Har10a, III.5.3]
For \(X\) projective over a field \(k\) and \(\mcf\) coherent, the two theorems give finite numbers \(h^i(X,\mcf(n))=\dim_kH^i(X,\mcf(n))\) and \(\chi(X,\mcf(n))=h^0(X,\mcf(n))\) for \(n\ge n_0\). The threshold \(n_0\) depends on \(\mcf\): \(H^1(\PP^1_k,\OO(-m)(m-2))\cong H^1(\PP^1_k,\OO(-2))\cong k\) for every \(m\ge0\), so no single \(n_0\) serves all the sheaves \(\OO_{\PP^1_k}(-m)\).
Euler characteristic and Hilbert polynomial
The Euler characteristic and the Hilbert polynomial (Tag D-COHEULER)
For a scheme \(X\) projective over a field \(k\) and a coherent sheaf \(\mcf\), its Euler characteristic is \[ \chi(X,\mcf)\coloneqq\sum_{i\ge0}(-1)^i\dim_kH^i(X,\mcf). \] The sum is finite and its terms are finite by cohomological vanishing and finiteness [@Har10a, Theorems III.2.7 and III.5.2]. Write \(\chi(\mcf)\) when the scheme is fixed.
(Hilbert polynomial)
Choose a very ample invertible sheaf \(L\) on the projective \(k\)-scheme \(X\) and put \(\mcf(n)=\mcf\otimes L^{\otimes n}\), using the dual of \(L\) for negative powers. The Hilbert polynomial of \(\mcf\) with respect to \(L\) is the unique \(P_{\mcf,L}\in\QQ[z]\) satisfying \[ P_{\mcf,L}(n)=\chi(X,\mcf(n))\qquad(n\in\ZZ) \] [@Har10a, Exercise III.5.2]. It depends on the chosen twisting sheaf; its value at zero is the intrinsic Euler characteristic \(\chi(X,\mcf)\).
(Arithmetic and geometric genus)
For a nonempty projective scheme \(X\) over a field of dimension \(n\), the arithmetic genus is \(p_a(X)\coloneqq(-1)^n(\chi(\OO_X)-1)\), so \(p_a(X)=1-\chi(\OO_X)\) for a curve. For a nonsingular projective integral variety \(X\) of dimension \(n\) over an algebraically closed field, the geometric genus is \(p_g(X)\coloneqq h^0(X,\omega_X)\), and Serre duality gives \(p_g(X)=h^n(X,\OO_X)\). [@Har10a, §II.8, Exercise III.5.3]
For a nonsingular projective integral curve over an algebraically closed field, \(p_a=p_g\). For a nonsingular projective integral surface over that field, \(p_a=h^2(\OO_X)-h^1(\OO_X)=p_g-q\) with \(q=h^1(X,\OO_X)\). For example, an abelian surface has \(h^0(\OO_X)=h^2(\OO_X)=1\) and \(h^1(\OO_X)=2\), hence \(p_g=1\) and \(p_a=-1\). The product of two nonsingular plane cubics gives an explicit example, computed in P-AGH283PRODDIFF.
\(\chi\) is additive: \(\chi(\mcf) = \chi(\mcf') + \chi(\mcf'')\) for every short exact sequence \(0 \to \mcf' \to \mcf \to \mcf'' \to 0\). This follows by taking the alternating sum of the dimensions in the finite long exact cohomology sequence, as proved in P-AGH351EULERCHAR.
For sufficiently large \(n\), Serre vanishing gives \(H^i(X,\mcf(n))=0\) for all \(i>0\), so \(P_{\mcf,L}(n)=h^0(X,\mcf(n))\) [@Har10a, Theorem III.5.2]. For small \(n\) this can fail: on \(X=\PP^1\) with \(L=\OO(1)\), \(P_{\OO_X,L}(n)=n+1\), so \(P_{\OO_X,L}(-2)=-1\), while \(h^0(\PP^1,\OO(-2))=0\).
Flatness is constancy of the Hilbert polynomial (Tag T-COHFLATCHI)
Let \(T\) be an integral Noetherian scheme and \(\mcf\) a coherent sheaf on \(\PP^n_T\). Then \(\mcf\) is flat over \(T\) exactly when the Hilbert polynomial of \(\restrictionof{\mcf}{\PP^n_t}\) is the same for every \(t \in T\).
This is the theorem that makes “flat family” mean “family in which the numerical invariants do not jump”, and it is the reason the Hilbert scheme is stratified by Hilbert polynomials.
The invariants that do stay constant are exactly the ones read off the Hilbert polynomial: degree, dimension, arithmetic genus, \(\chi(\OO)\). Everything else can degenerate. Irreducibility, reducedness, the geometric genus, and the Picard number are all lost in flat limits, and a flat family with smooth general fibre can have a singular or non-reduced special fibre. A conic degenerating to a double line keeps \(\chi\) and loses reducedness.
A blowup is not flat because the fibre dimension jumps over the centre.
For \(f\colon X\to T\) projective with \(T\) noetherian and \(\mcf\) coherent on \(X\) and flat over \(T\), each \(h^i(X_t,\mcf_t)\) is upper semicontinuous in \(t\) and \(\chi(X_t,\mcf_t)\) is locally constant (Semicontinuity, constancy of $\chi$, and cohomology and base change). For a finitely generated graded \(k\)-algebra \(S\) generated in degree one and a finite graded \(S\)-module \(M\), the sheaf \(\mcf=\widetilde M\) on \(\Proj S\) has \(\chi(\mcf(n))\) equal to the Hilbert polynomial of \(M\) (dimension and degree) at every integer \(n\). Over an integral noetherian base, Flatness is constancy of the Hilbert polynomial gives the converse to local constancy: a coherent sheaf on \(\PP^n_T\) whose fibres all have the same Hilbert polynomial is flat over \(T\).