Serre's cohomological criterion for affineness (Tag T-5IOUR)
Let \(X\) be a Noetherian scheme. The following are equivalent:
\(X\) is affine.
\(H^p(X, \mcf) = 0\) for all quasicoherent \(\mcf\) and all \(p > 0\).
\(H^1(X, \mci) = 0\) for every coherent sheaf of ideals \(\mci\).
[@Har10a, Theorem III.3.7]
(Scope of the hypotheses)
Weakening Noetherian. The criterion holds for quasicompact quasi-separated schemes, with \(\mci\) ranging over quasicoherent ideals rather than coherent ones. Noetherian is used only to have coherent ideals available and to extract finite subcovers, and both can be arranged directly.
Without quasicompactness. The conclusion fails. An infinite disjoint union of affine schemes has vanishing higher cohomology for every quasicoherent sheaf, being a disjoint union of affines, but it is not affine: its ring of global sections is an infinite product, and the comparison map is not an isomorphism. The finite subcover in the proof is where quasicompactness enters, and without it the \(f_i\) need not be finite in number.
The criterion holds for quasicompact quasi-separated schemes, with \(\mci\) ranging over quasicoherent ideal sheaves. It fails without quasicompactness: \(X=\coprod_{n\in\NN}\Spec k\) has \(H^p(X,\mcf)=\prod_nH^p(\Spec k,\mcf|_{\Spec k})=0\) for every quasicoherent \(\mcf\) on \(X\) and every \(p>0\), and \(X\) is not quasicompact, hence not affine.
Serre duality and Riemann–Roch for curves
Serre duality in dimension $n$ (Tag T-COHSD)
Let \(X\) be a smooth proper variety of dimension \(n\) over a field \(k\), with canonical sheaf \(\omega_X = \Omega^n_{X/k}\), and let \(\mcl\) be locally free of finite rank. Then the cup product pairing \[ H^i(X, \mcl) \tensor H^{n-i}\qty{X, \omega_X \tensor \dualof{\mcl}} \to H^n(X,\omega_X)\xrightarrow{t_X} k \] is perfect, so \(H^i(X,\mcl) \cong H^{n-i}\dualof{\qty{X, \omega_X \tensor \dualof{\mcl}}}\), by duality for proper schemes over fields. The trace \(t_X\) corresponds under \(H^n(X,\omega_X)\cong H^0(X,\OO_X)^\vee\) to evaluation on the constant section \(1\). It is an isomorphism exactly when \(H^0(X,\OO_X)=k\). For example, for \(X=\Spec K\) with \(K/k\) a finite separable extension of degree greater than one, \(H^0(X,\omega_X)=K\) has that greater dimension over \(k\), so the trace cannot be an isomorphism.
(Dualizing sheaf)
Let \(X\) be a proper scheme of dimension \(n\) over a field \(k\). A dualizing sheaf for \(X\) is a coherent sheaf \(\omega_X^\circ\) with a \(k\)-linear trace map \(t \colon H^n(X, \omega_X^\circ) \to k\) such that for every coherent sheaf \(\mcf\) the pairing \[\Hom(\mcf, \omega_X^\circ) \times H^n(X, \mcf) \to H^n(X, \omega_X^\circ) \xrightarrow{t} k\] induces an isomorphism \(\Hom(\mcf, \omega_X^\circ) \cong \dualof{H^n(X, \mcf)}\). A dualizing sheaf is unique up to unique isomorphism compatible with the trace maps.
(Serre duality for projective schemes)
Let \(X\) be a projective scheme of dimension \(n\) over an algebraically closed field \(k\).
\(X\) has a dualizing sheaf \(\omega_X^\circ\).
For every \(i\) and every coherent \(\mcf\) there are natural maps \(\theta^i \colon \Ext^i(\mcf, \omega_X^\circ) \to \dualof{H^{n-i}(X, \mcf)}\), and \(\theta^0\) is an isomorphism.
The maps \(\theta^i\) are isomorphisms for all \(i\) and all coherent \(\mcf\) if and only if \(X\) is Cohen–Macaulay and equidimensional. In that case, for \(\mcf\) locally free, \(H^i(X, \mcf) \cong \dualof{H^{n-i}(X, \omega_X^\circ \otimes \dualof{\mcf})}\).
If \(X\) is smooth, \(\omega_X^\circ \cong \omega_X = \Omega^n_{X/k}\).
For \(X = \PP^n\) this is the symmetry already visible in the twists, with \(\omega = \OO(-n-1)\); that computation is what the general theorem is modelled on, and it is also the base case of its proof.
Properness cannot be dropped at all: on \(\AA^n\) the top cohomology that would receive the pairing is zero.
Two consequences worth saying immediately: \(h^i(\mcl) = h^{n-i}(\omega \tensor \dualof{\mcl})\) and \(\chi(\mcl) = (-1)^n \chi(\omega \tensor \dualof{\mcl})\), and taking \(\mcl = \OO\) gives \(h^n(\OO_X) = h^0(\omega_X) = p_g\).
Properness is needed for the pairing: \(\AA^1_k\) is smooth of dimension \(1\), \(H^1(\AA^1_k,\omega_{\AA^1_k})=0\) by affine vanishing, and \(H^0(\AA^1_k,\OO)=k[x]\) is infinite-dimensional.
Riemann--Roch and Serre duality on a curve (Tag T-MWDVL)
For \(D\) a divisor on a smooth projective curve \(X\) of genus \(g\) over \(k = \bar{k}\), \[ \ell(D) - \ell(K - D) = \deg D + 1 - g , \] where \(\ell(D) = h^0(X, \OO(D))\) and \(K\) is a canonical divisor [@Har10a, Theorem IV.1.3].
(Serre duality)
\(H^1(X, \OO(D)) \cong \dualof{H^0(X, \OO(K-D))}\), so \(\ell(K-D) = h^1(D)\) and Riemann–Roch reads \[ \chi(\OO(D)) = \deg D + 1 - g . \]
(Hyperplane sections)
If \(X \subseteq \PP^n\) is a curve of degree \(d\) and \(D = X \cap H\) is a hyperplane section, then \(\mcl(D) \cong \OO_X(1)\), \(\deg D = d\), and Riemann–Roch gives \[ \chi(\mcl(D)) = d + 1 - p_a . \]
Over a field \(k\) that is not algebraically closed, adding a closed point \(P\) to \(D\) raises \(\chi(\OO(D))\) by \(\deg P = [\kappa(P) : k]\), and the formula reads \(\chi(\OO(D)) = \deg D + \chi(\OO_X)\) with \(\deg D = \sum n_P \deg P\) (D-CRVDEGREES).
The second form is the one to state first when asked, because it says what the theorem is: the Euler characteristic is linear in the divisor, and the genus is the constant of integration. Duality is what converts the unknown \(h^1\) into a countable \(h^0\).
Two immediate consequences, both asked:
\(D = 0\) gives \(\ell(K) = g\): the space of global regular differentials on a curve of genus \(g\) has dimension exactly \(g\). This is the answer to the question about holomorphic differentials on a Riemann surface, and it is also how \(g\) can be defined so that Riemann–Roch becomes a statement rather than a tautology.
\(D = K\) gives \(\deg K = 2g-2\), which is the input to Riemann–Hurwitz and to every genus computation.
For \(\deg D > 2g - 2\) the correction term vanishes and \(\ell(D) = \deg D + 1 - g\) exactly; this is the range in which everything is computable and where “very ample” questions are settled.
By Serre duality \(h^1(\OO(D))=\ell(K-D)\), so \(\chi(\OO(D))=\deg D+1-g\) is the identity \(\ell(D)-\ell(K-D)=\deg D+1-g\). For \(\deg D>2g-2\), \(\ell(K-D)=0\) and \(\ell(D)=\deg D+1-g\). At \(D=0\), Riemann–Roch gives \(\ell(K)=g\): the global regular differentials on a curve of genus \(g\) form a \(g\)-dimensional space. At \(D=K\), it gives \(\deg K=2g-2\), which enters Riemann–Hurwitz in curves and surfaces.
Riemann–Roch for surfaces
Riemann--Roch for surfaces (Tag T-COHRRS)
Let \(C\) be a smooth projective curve of genus \(g\) and \(D\) a divisor of degree \(d\) on \(C\). Then \[ h^0\qty{\OO_C(D)} - h^0\qty{\OO_C(K_C - D)} = d - g + 1 , \] equivalently \(\chi\qty{\OO_C(D)} = d - g + 1\). In particular \(h^0(D) - h^0(K_C - D) = \deg D - g + 1\).
(Riemann--Roch for surfaces)
Let \(X\) be a smooth projective surface with canonical divisor \(K\), and \(D\) any divisor. Then \[ \chi\qty{\OO_X(D)} = \chi\qty{\OO_X} + \frac{1}{2} D \cdot (D - K) . \]
The curve case is the same statement with \(\deg\) in place of the intersection pairing, and the surface case is the first one where the correction term is quadratic rather than linear. \(\chi(\OO_X) = 1 - q + p_g\) is the constant, with \(q = h^1(\OO_X)\) the irregularity.
What it is used for is existence, not counting. Serre duality turns \(h^2(D)\) into \(h^0(K-D)\), so \[ h^0(D) \geq \chi(\OO_X) + \tfrac{1}{2} D\cdot(D-K) - h^0(K - D) , \] and once \(D\) is positive enough that \(K - D\) has no sections, a large right-hand side forces \(h^0(D) > 0\) and hence \(D\) effective. That inequality gives a standard sufficient criterion for nonemptiness of a linear system.
By Serre duality \(h^2(\OO_X(D))=h^0(\OO_X(K-D))\), so \[ h^0(\OO_X(D))\ge\chi(\OO_X)+\tfrac12D\cdot(D-K)-h^0(\OO_X(K-D)). \] If \(h^0(\OO_X(K-D))=0\) and \(\chi(\OO_X)+\tfrac12D\cdot(D-K)>0\), then \(h^0(\OO_X(D))>0\), and \(D\) is linearly equivalent to an effective divisor.
Finiteness and vanishing for ample twists
Finiteness of coherent cohomology (Cartan--Serre) (Tag T-CARTSERRE)
Let \(X\) be a compact complex manifold, or more generally a compact complex space, and let \(\mathcal{F}\) be a coherent analytic sheaf on \(X\). Then \(H^i(X, \mathcal{F})\) is a finite-dimensional complex vector space for every \(i\).
(Serre's algebraic counterpart)
Let \(X\) be a projective scheme over a Noetherian ring \(A\) and \(\mathcal{F}\) a coherent sheaf on \(X\). Then each \(H^i(X, \mathcal{F})\) is a finitely generated \(A\)-module. With \(\OO_X(1)\) very ample, \(\mathcal{F}(m)\) is generated by global sections for \(m \gg 0\), and \(H^i(X, \mathcal{F}(m)) = 0\) for \(i > 0\) and \(m \gg 0\).
On a compact complex manifold the theorem gives finite-dimensionality of \(H^0(X, \Omega^p)\) and of the Dolbeault groups \(H^q(X, \Omega^p)\), and so the Hodge numbers \(h^{p,q}\) are defined. The analytic statements for Stein manifolds are Cartan’s Theorems A and B: every coherent analytic sheaf is generated by global sections at each stalk, and has vanishing cohomology in positive degrees.
Kodaira vanishing (Tag T-KODVAN)
Let \(X\) be a smooth projective variety of dimension \(n\) over a field of characteristic zero, and let \(L\) be an ample line bundle on \(X\). Then \[ H^i(X, \omega_X \otimes L) = 0 \quad \text{for } i > 0 . \] Equivalently, by Serre duality, \(H^i(X, L^{-1}) = 0\) for \(i < n\).
On a smooth projective curve of genus \(g\), an ample line bundle is one of positive degree, and \(H^1(C, \omega_C \otimes L) \cong \dualof{H^0(C, L^{-1})} = 0\) because \(L^{-1}\) has negative degree. On \(\PP^n\), with \(\omega = \OO(-n-1)\) and \(L = \OO(m)\) for \(m > 0\), the theorem says \(H^i(\PP^n, \OO(m-n-1)) = 0\) for \(i > 0\), which agrees with the Čech computation of the cohomology of projective space.
The characteristic-zero hypothesis is necessary: Raynaud constructed smooth projective surfaces in characteristic \(p\) with an ample line bundle \(L\) and \(H^1(X, L^{-1}) \neq 0\).
GAGA (Tag T-GAGA)
Let \(X\) be a projective scheme over \(\CC\) and \(X^{an}\) its associated complex analytic space, with the morphism of ringed spaces \(h \colon X^{an} \to X\).
For every coherent sheaf \(\mathcal{F}\) on \(X\), the natural maps \(H^i(X, \mathcal{F}) \to H^i(X^{an}, \mathcal{F}^{an})\) are isomorphisms, where \(\mathcal{F}^{an} = h^* \mathcal{F}\).
The functor \(\mathcal{F} \mapsto \mathcal{F}^{an}\) is an equivalence from coherent \(\OO_X\)-modules to coherent analytic sheaves on \(X^{an}\).
(Chow's theorem)
A closed analytic subspace of \(\PP^n(\CC)\) is algebraic, and a holomorphic map between projective varieties is a morphism.
Holomorphic line bundles on a projective variety are algebraic, so \(\Pic(X) \cong H^1(X^{an}, \OO^*_{X^{an}})\). Global meromorphic functions on a smooth projective variety are rational functions. Projectivity cannot be dropped: on \(X = \AA^1_\CC\), \(H^0(X, \OO_X) = \CC[z]\) while \(H^0(X^{an}, \OO_{X^{an}})\) is the ring of entire functions, and \(\exp \colon \CC \to \CC^*\) is holomorphic but not a morphism \(\AA^1 \to \AA^1 \setminus \{0\}\).
Topology of hyperplane sections
The Lefschetz hyperplane theorem (Tag T-LEFHYP)
Let \(X \subseteq \PP^N\) be a projective variety of dimension \(n\) over \(\CC\), and let \(Y = X \cap H\) be a hyperplane section such that \(X \setminus Y\) is smooth. Then the restriction maps \[ H^i(X, \ZZ) \to H^i(Y, \ZZ) \] are isomorphisms for \(i < n - 1\) and injective for \(i = n - 1\), and the maps \(\pi_i(Y) \to \pi_i(X)\) are isomorphisms for \(i < n - 1\) and surjective for \(i = n - 1\).
A smooth hypersurface \(Y \subseteq \PP^{n}\) with \(n \geq 3\) has \(\pi_1(Y) \cong \pi_1(\PP^{n}) = 1\) and \(H^2(Y, \ZZ) \cong H^2(\PP^{n}, \ZZ) \cong \ZZ\) when \(n \geq 4\). Iterating, a smooth complete intersection of dimension at least \(2\) in projective space is simply connected. A smooth plane curve of degree \(d \geq 3\) is a hyperplane section of dimension \(1\), and the theorem makes no claim about its fundamental group, which is nontrivial.
Every hypersurface section is a hyperplane section after a Veronese embedding, so the theorem applies to \(Y = X \cap V(F)\) for any homogeneous \(F\) with \(X \setminus Y\) smooth. The proof uses that \(X \setminus Y\) is an affine variety of dimension \(n\), which has the homotopy type of a CW complex of dimension at most \(n\).
The hard Lefschetz theorem (Tag T-HARDLEF)
Let \(X\) be a smooth projective complex variety of dimension \(n\), and let \(\omega \in H^2(X, \QQ)\) be the class of a hyperplane section. For every \(0 \leq k \leq n\), cup product with \(\omega^k\) is an isomorphism \[ L^k \colon H^{n-k}(X, \QQ) \xrightarrow{\ \sim\ } H^{n+k}(X, \QQ) . \]
The Betti numbers satisfy \(b_{i-2} \leq b_i\) for \(i \leq n\), and \(H^\bullet(X, \QQ)\) has the Lefschetz decomposition into primitive classes \(P^{n-k} = \ker\big(L^{k+1} \colon H^{n-k} \to H^{n+k+2}\big)\): \[ H^m(X, \QQ) = \bigoplus_{j \geq 0} L^j P^{m-2j} . \]
The weak Lefschetz theorem is the Lefschetz hyperplane theorem T-LEFHYP: restriction to a smooth hyperplane section \(Y\) is an isomorphism \(H^i(X) \to H^i(Y)\) for \(i < n-1\) and injective for \(i = n-1\). The hard Lefschetz theorem fails for compact complex manifolds that are not Kähler, and for singular projective varieties unless ordinary cohomology is replaced by intersection cohomology. On a smooth projective curve it says \(H^0 \cong H^2\) via the class of a point; on a surface it says \(H^1 \cong H^3\) via a hyperplane class, so \(b_1 = b_3\), which Poincaré duality also gives.