Higher direct images (Tag D-COHRIF)
For \(f: X \to Y\), the functor \(f_*\) is left exact; set \(R^i f_* \mcf \da R^i(f_*)(\mcf)\). Equivalently, \(R^i f_* \mcf\) is the sheafification of \[ V \mapsto H^i\qty{\inverseof{f}(V), \restrictionof{\mcf}{\inverseof{f}(V)}} . \]
(Functoriality in the space, and affine morphisms)
Let \(f \colon X \to Y\) be a morphism of schemes.
For a sheaf of abelian groups \(\mcf\) on \(X\) there are natural maps \(H^i(Y, f_* \mcf) \to H^i(X, \mcf)\), and for a sheaf of \(\OO_Y\)-modules \(\mcg\) natural maps \(H^i(Y, \mcg) \to H^i(X, f^* \mcg)\).
If \(X\) and \(Y\) are quasicompact and separated, \(f\) is affine and \(\mcf\) is quasicoherent, then \(H^i(Y, f_* \mcf) \to H^i(X, \mcf)\) is an isomorphism for every \(i\) (P-AGH341AFFINEMORPH).
If \(X\) is Noetherian, \(Y = \Spec A\) is affine, and \(\mcf\) is quasicoherent, then \(R^i f_* \mcf \cong \widetilde{H^i(X,\mcf)}\) [@Har10a, Proposition III.8.5]. If \(f\) is an affine morphism and \(\mcf\) is quasicoherent, then \(R^{i} f_* \mcf = 0\) for \(i>0\) and \(H^i(X,\mcf) \cong H^i(Y, f_*\mcf)\). For a closed subscheme \(i\colon X\hookrightarrow\PP^n\) and a quasicoherent sheaf \(\mcf\) on \(X\), this gives \(H^i(X,\mcf)\cong H^i(\PP^n,i_*\mcf)\).
If \(Y\) is Noetherian, \(f\) is projective, and \(\mcf\) is coherent, then \(R^i f_* \mcf\) is coherent [@Har10a, Theorem III.8.8]; for \(Y=\Spec k\) this is the finite-dimensionality of \(H^i(X,\mcf)\) [@Har10a, Theorem III.5.2].
The Leray spectral sequence (Tag T-COHLERAY)
For \(f: X \to Y\) and \(\mcf\) a sheaf of abelian groups on \(X\), \[ E_2^{p,q} = H^p\qty{Y, R^q f_* \mcf} \Longrightarrow H^{p+q}(X, \mcf) . \]
It is the derived-functor form of \(\globsec{X;\wait} = \globsec{Y;\wait} \circ f_*\), and the only thing usually needed from it is what happens when it degenerates.
If \(R^{q>0} f_* \mcf = 0\), the sequence collapses to the edge and \(H^i(X,\mcf) \cong H^i(Y, f_*\mcf)\) for all \(i\); that is the affine-morphism case above. In low degrees, with no vanishing at all, one always gets the exact sequence \[ 0 \to H^1(Y, f_* \mcf) \to H^1(X, \mcf) \to H^0(Y, R^1 f_* \mcf) \to H^2(Y, f_*\mcf) , \] which is the form most often actually used: a class on \(X\) is either pulled back from the base or detected fibrewise.
If \(f\) is affine and \(\mcf\) is quasicoherent, then \(R^qf_*\mcf=0\) for \(q>0\), the Leray spectral sequence degenerates, and \(H^i(X,\mcf)\cong H^i(Y,f_*\mcf)\). For a closed immersion \(\iota\colon X\hookrightarrow\PP^n\) and a coherent sheaf \(\mcf\) on \(X\), this gives \(H^i(X,\mcf)\cong H^i(\PP^n,\iota_*\mcf)\).
Formal functions and semicontinuity
The theorem on formal functions (Tag T-COHFF)
Let \(f: X \to Y\) be proper with \(Y\) Noetherian, \(\mcf\) coherent on \(X\), and \(y \in Y\) with maximal ideal \(\mfm_y\). Write \(X_n\) for the subscheme cut out by \(\mfm_y^n \OO_X\). Then the completion of the stalk satisfies \[ \hat{\qty{R^i f_* \mcf}_y} \cong \varprojlim_n H^i\qty{X_n, \mcf/\mfm_y^n \mcf} . \]
The theorem identifies the completed stalk \(\hat{(R^if_*\mcf)_y}\) with the inverse limit of the cohomology groups \(H^i(X_n,\mcf/\mfm_y^n\mcf)\) over all infinitesimal thickenings \(X_n\) of the fibre. Properness is load-bearing and is what makes the inverse limit converge to something algebraic; the statement is false for open immersions.
Two corollaries are the reason it is on a syllabus. Zariski’s connectedness theorem: if \(f: X \to Y\) is proper with \(f_* \OO_X = \OO_Y\), then every fibre is connected. And the version of Zariski’s main theorem for a birational proper morphism to a normal variety: the fibres are connected, so no fibre breaks into pieces, which is what “no blowing down to two points” means in practice.
Semicontinuity, constancy of $\chi$, and cohomology and base change (Tag T-COHBC)
Let \(f: X \to T\) be projective with \(T\) Noetherian and \(\mcf\) coherent on \(X\), flat over \(T\). Then \(t \mapsto h^i(X_t, \mcf_t)\) is upper semicontinuous on \(T\), and \(t \mapsto \chi(\mcf_t)\) is locally constant.
(Grauert and base change)
If \(T\) is reduced and \(t \mapsto h^i(X_t,\mcf_t)\) is constant, then \(R^i f_* \mcf\) is locally free and \[ R^i f_* \mcf \tensor k(t) \iso H^i(X_t, \mcf_t) \] for all \(t\). Without reducedness there is still Grothendieck’s criterion: if the base change map in degree \(i\) is surjective at \(t\), it is an isomorphism near \(t\), and \(R^i f_* \mcf\) is locally free at \(t\) exactly when the map in degree \(i-1\) is surjective as well.
The shape to hold: individual \(h^i\) can only jump up on closed subsets, the alternating sum cannot jump at all, and cohomology commutes with restriction to a fibre exactly when nothing jumps.
Flatness is the load-bearing hypothesis throughout, and dropping it destroys even the constancy of \(\chi\). Reducedness of the base is what Grauert needs, and it is where non-reduced bases genuinely fail.
The standard specimen is a family of line bundles of degree \(0\) on a curve: \(h^0\) is \(0\) at a general point and \(1\) at the trivial bundle, a jump on a closed subset, while \(\chi = 1 - g\) throughout. That single example answers “give an example where cohomology does not commute with base change”.
Without flatness, \(\chi\) need not be locally constant. For distinct points \(p,q\in\PP^1\), let \(X=(\AA^1\times\{p\})\cup(\{0\}\times\{q\})\subset\AA^1\times\PP^1\) with the projection \(f\colon X\to\AA^1\). Then \(f\) is projective, \(\chi(\OO_{X_t})=1\) for \(t\neq0\), and \(\chi(\OO_{X_0})=2\); the sheaf \(\OO_X\) is not flat over \(\AA^1\).
Cohomology need not commute with base change when \(h^i\) jumps. For a smooth projective curve \(C\) of genus \(g\ge1\) and the Poincaré line bundle \(\mathcal P\) on \(C\times\Pic^0(C)\), \(h^0(C,\mathcal P_t)\) is \(1\) at the trivial bundle \(t=0\) and \(0\) elsewhere, while \(\chi(\mathcal P_t)=1-g\) for every \(t\). The pushforward \(\pi_*\mathcal P\) to \(\Pic^0(C)\) is torsion-free of generic rank \(0\), so \(\pi_*\mathcal P=0\), and the base change map \(\pi_*\mathcal P\otimes k(0)\to H^0(C,\OO_C)=k\) is not surjective.
Base change theorems
Flat, proper and smooth base change (Tag T-BASECHANGE)
Throughout, consider a Cartesian square \[ \begin{array}{ccc} X' & \xrightarrow{g'} & X \\ \downarrow{\scriptstyle f'} & & \downarrow{\scriptstyle f} \\ Y' & \xrightarrow{g} & Y \end{array} \] and the base change map \(g^* R^i f_* \mathcal{F} \to R^i f'_* g'^* \mathcal{F}\).
(Flat base change)
If \(f\) is quasicompact and separated, \(g\) is flat, and \(\mathcal{F}\) is quasicoherent on \(X\), the base change map is an isomorphism for all \(i\). [@Har10a, Proposition III.9.3]
(Extension of the base field)
Let \(X\) be a quasicompact separated scheme over a field \(k\), \(\mathcal{F}\) a quasicoherent sheaf on \(X\), and \(K/k\) a field extension, with \(X_K = X \times_k \Spec K\) and \(\mathcal{F}_K\) the pullback of \(\mathcal{F}\). Then \(H^i(X_K, \mathcal{F}_K) \cong H^i(X, \mathcal{F}) \otimes_k K\) for every \(i\); in particular \(\dim_k H^i(X, \mathcal{F}) = \dim_K H^i(X_K, \mathcal{F}_K)\) [@Har10a, Proposition III.9.3].
(Proper base change)
If \(f\) is proper, \(g\) is arbitrary, and \(\mathcal{F}\) is a torsion étale sheaf on \(X\), the base change map \(g^* R^i f_* \mathcal{F} \to R^i f'_* g'^* \mathcal{F}\) is an isomorphism for all \(i\). The same holds for sheaves of abelian groups on locally compact Hausdorff spaces when \(f\) is proper. In particular the stalk of \(R^i f_* \mathcal{F}\) at a geometric point \(\bar{y}\) is \(H^i(X_{\bar{y}}, \mathcal{F})\).
(Smooth base change)
If \(g\) is smooth, \(f\) is quasicompact and quasiseparated, and \(\mathcal{F}\) is a torsion étale sheaf with torsion orders invertible on \(Y\), the base change map is an isomorphism for all \(i\).
Without flatness of \(g\), quasicoherent base change fails. Let \(E\) be an elliptic curve over an algebraically closed field, \(p \in E\), \(f \colon E \times E \to E\) the second projection, and \(\mathcal{L} = \OO(\Delta - \{p\} \times E)\). The restriction of \(\mathcal{L}\) to the fibre \(E \times \{t\}\) is \(\OO_E(t - p)\), which has no nonzero sections for \(t \neq p\), so the torsion-free sheaf \(f_* \mathcal{L}\) is \(0\). For \(g\) the inclusion of the point \(p\), \(g^* f_* \mathcal{L} = 0\), while the fibre gives \(H^0(E, \OO_E) = k\). The coherent statement comparing \(R^i f_* \mathcal{F} \otimes \kappa(y)\) with \(H^i(X_y, \mathcal{F}_y)\) at arbitrary points is the theorem on cohomology and base change.
The projection formula
The projection formula (Tag PR-SCHPROJFORM)
Let \(f \colon X \to Y\) be a morphism of ringed spaces, \(\mathcal{F}\) an \(\OO_X\)-module, and \(\mathcal{E}\) a locally free \(\OO_Y\)-module of finite rank. Then there are natural isomorphisms \[ R^i f_* (\mathcal{F} \otimes_{\OO_X} f^* \mathcal{E}) \cong R^i f_* \mathcal{F} \otimes_{\OO_Y} \mathcal{E} \] for all \(i \geq 0\). [@Har10a, Exercise III.8.3]
For \(\pi \colon \PP(\mathcal{V}) \to Y\) a projective bundle and \(\mathcal{L}\) a line bundle on \(Y\), \(\pi_*(\OO(m) \otimes \pi^* \mathcal{L}) \cong \Sym^m \mathcal{V} \otimes \mathcal{L}\) for \(m \geq 0\).
Deformations
The cotangent complex and its dual (Tag D-COTCPLX)
For a morphism of schemes \(f \colon X \to Y\), the cotangent complex \(L_{X/Y}\) is an object of the derived category of \(\OO_X\)-modules, concentrated in nonpositive degrees, with \(H^0(L_{X/Y}) = \Omega_{X/Y}\). It is built from a simplicial resolution of \(\OO_X\) by free \(f^{-1}\OO_Y\)-algebras, by applying \(\Omega\) termwise.
If \(f\) is smooth, \(L_{X/Y} \simeq \Omega_{X/Y}[0]\).
If \(X \hookrightarrow P\) is a closed immersion with ideal \(\mathcal{I}\) into a smooth \(Y\)-scheme \(P\) and \(X \to Y\) is a local complete intersection, then \(L_{X/Y} \simeq [\mathcal{I}/\mathcal{I}^2 \to \Omega_{P/Y}|_X]\) in degrees \(-1\) and \(0\).
A composition \(X \to Y \to Z\) gives a distinguished triangle \(f^* L_{Y/Z} \to L_{X/Z} \to L_{X/Y} \to\), extending the two exact sequences of differentials.
(Tangent complex)
The tangent complex is the derived dual \(T_{X/Y} = R\mathcal{H}om_{\OO_X}(L_{X/Y}, \OO_X)\).
The derived dual is required because \(L_{X/Y}\) is an object of the derived category, and a termwise dual of a chosen representative is not invariant under quasi-isomorphism unless the representative consists of locally free sheaves. For the lci representative in part 2 the terms are locally free, and the termwise dual \([T_{P/Y}|_X \to \mathcal{N}_{X/P}]\) does compute \(T_{X/Y}\). For example, let \(k\) have characteristic not \(2\) and \(A = k[x]/(x^2)\), a local complete intersection over \(k\). Its cotangent complex is \(L_{A/k} \simeq [A \xrightarrow{\,2x\,} A]\) in degrees \(-1, 0\), the lci complex \((x^2)/(x^4) \to A\, dx\) with \(x^2 \mapsto 2x\, dx\). So \(\operatorname{Ext}^i_A(L_{A/k}, A)\) is \(\ker(2x) = (x) \cong k\) for \(i = 0\), \(\operatorname{coker}(2x) = A/(x) \cong k\) for \(i = 1\), and \(0\) for \(i \geq 2\): one infinitesimal automorphism, the first-order deformation \(k[x, \varepsilon]/(x^2 - \varepsilon)\), and no obstructions. The naive groups \(\operatorname{Ext}^i_A(\Omega_{A/k}, A)\), with \(\Omega_{A/k} = A\,dx/(2x\,dx) \cong k\), are \(k\) for \(i = 0\) and \(0\) for \(i \geq 1\), because \(A\) is injective as a module over itself; the naive dual misses the first-order deformation.
Infinitesimal automorphisms and first-order deformations (Tag T-DEFEXT)
Let \(k\) be a field, \(D = \Spec k[\varepsilon]/(\varepsilon^2)\), and \(X\) a smooth separated scheme of finite type over \(k\) with tangent sheaf \(T_X = \mathcal{H}om(\Omega_{X/k}, \OO_X)\).
\(\operatorname{Ext}^0(\Omega_{X/k}, \OO_X) = H^0(X, T_X)\), and it is in bijection with the automorphisms of \(X \times D\) over \(D\) that restrict to the identity on \(X\).
\(\operatorname{Ext}^1(\Omega_{X/k}, \OO_X) = H^1(X, T_X)\), and it is in bijection with the isomorphism classes of first-order deformations: flat \(D\)-schemes \(\mathcal{X}\) with \(\mathcal{X} \times_D \Spec k \cong X\).
Let \(A_n = k[t]/(t^{n+1})\) for \(n \geq 1\), so \(A_1 \cong k[\varepsilon]/(\varepsilon^2)\). A flat deformation \(\mathcal{X}_n\) of \(X\) over \(A_n\) determines an obstruction class \(o(\mathcal{X}_n) \in H^2(X, T_X)\), and \(o(\mathcal{X}_n) = 0\) if and only if there is a flat deformation \(\mathcal{X}_{n+1}\) over \(A_{n+1}\) with \(\mathcal{X}_{n+1} \times_{A_{n+1}} \Spec A_n \cong \mathcal{X}_n\).
For a smooth projective curve \(C\) of genus \(g \geq 2\), \(H^0(T_C) = 0\), so \(C\) has no infinitesimal automorphisms, \(H^2(T_C) = 0\), so deformations are unobstructed, and \(h^1(T_C) = h^0(\omega_C^{\otimes 2}) = 3g - 3\) by Serre duality and Riemann–Roch.