Finite projective complex for proper flat coherent cohomology

Setup and the cover. Since $f$ is proper it is separated and of finite type by [F1], hence quasi-compact, and $\mathrm{Spec}A$ is quasi-compact, so $X=f^{-1}(\mathrm{Spec}A)$ is quasi-compact; as affine opens form a basis there are an integer $r\ge 0$ and a finite affine open cover $X=U_0\cup \cdots \cup U_r$, with the empty cover and $r=0$ allowed when $X=\mathrm{\varnothing }$.

The intersections. For every nonempty finite subset $I\subseteq \{0,\dots ,r\}$ the intersection $U_I:={\bigcap }_{i\in I}{U}_i$ is affine, say $U_I=\mathrm{Spec}{B}_I$ with $B_I={\mathcal{O}}_X(U_I)$; an intersection that is empty is the affine scheme $\mathrm{Spec}0$. Every $U_I$ is an affine open subscheme of $X$ mapping into the affine base $\mathrm{Spec}A$.

Flatness of the Čech terms. The coherent module $\mathcal{F}$ is quasi-coherent and its stalks are flat over the local rings of $\mathrm{Spec}A$, so [F4] applied to the affine open $U_I$ over the affine base gives that $\mathcal{F}(U_I)$ is a flat $A$-module. The degree-$p$ term $K^p:={\prod }_{i_0<\cdots <i_p}\mathcal{F}(U_{i_0}\cap \cdots \cap U_{i_p})$ of the ordered Čech complex is a finite product of flat $A$-modules, hence a finite direct sum of flat modules, hence flat by [F4].

The Čech complex is bounded and flat. Put $K^{\bullet }:=C^{\bullet }(\mathcal{U},\mathcal{F})$ with $\mathcal{U}=(U_0,\dots ,U_r)$; by [F3] its differential squares to zero, $K^p=0$ for $p<0$ and for $p>r$ because there are no increasing $(p+1)$-tuples in $\{0,\dots ,r\}$, and each $K^p$ is a flat $A$-module by 1.3. Thus $K^{\bullet }$ is a bounded complex of flat $A$-modules concentrated in degrees $0,\dots ,r$.

The complex computes the cohomology of $X$. The scheme $X$ is quasi-compact and separated by [F1], the cover $\mathcal{U}$ is finite affine with all finite intersections affine by 1.2, and $\mathcal{F}$ is quasi-coherent, so [F3] gives a canonical isomorphism ${\check{H}}^q(\mathcal{U},\mathcal{F})\to H^q(X,\mathcal{F})$, that is, $H^q(K^{\bullet })\cong H^q(X,\mathcal{F})$ for every $q\ge 0$.

Finiteness of the cohomology modules. Since $f$ is proper, $A$ is Noetherian and $\mathcal{F}$ is coherent, [F5] gives $R^q{f}_{\ast }\mathcal{F}\cong \widetilde{H^q(X,\mathcal{F})}$ for the affine open $\mathrm{Spec}A$ and shows that $R^q{f}_{\ast }\mathcal{F}$ is a coherent sheaf on $\mathrm{Spec}A$; by [F6] it is of finite type, so the module $H^q(X,\mathcal{F})$ is finitely generated, and by 1.5 each $H^q(K^{\bullet })$ is a finitely generated $A$-module, while $H^q(K^{\bullet })=0$ for $q\mathrm{\notin }[0,r]$.

The base-changed geometry. Fix a ring map $A\to A^{\mathrm{\prime }}$ and put $S^{\mathrm{\prime }}=\mathrm{Spec}{A}^{\mathrm{\prime }}$, $X_{A^{\mathrm{\prime }}}:=X{\times }_{\mathrm{Spec}A}{S}^{\mathrm{\prime }}$, with projection $p:X_{A^{\mathrm{\prime }}}\to X$, and ${\mathcal{F}}_{A^{\mathrm{\prime }}}:=p^{\ast }\mathcal{F}$. For $I\subseteq \{0,\dots ,r\}$ put $U_I^{\mathrm{\prime }}:=p^{-1}(U_I)$; by [F9] $U_I^{\mathrm{\prime }}$ represents $U_I{\times }_{\mathrm{Spec}A}{S}^{\mathrm{\prime }}\cong \mathrm{Spec}(B_I{\otimes }_A{A}^{\mathrm{\prime }})$, so it is affine with ring $B_I{\otimes }_A{A}^{\mathrm{\prime }}$ and $U_{i_0}^{\mathrm{\prime }}\cap \cdots \cap U_{i_p}^{\mathrm{\prime }}=U_{i_0\cdots i_p}^{\mathrm{\prime }}$ for increasing tuples. The $U_0^{\mathrm{\prime }},\dots ,U_r^{\mathrm{\prime }}$ cover $X_{A^{\mathrm{\prime }}}$, since $p^{-1}$ commutes with unions and $U_0\cup \cdots \cup U_r=X$, so they form a finite affine open cover of $X_{A^{\mathrm{\prime }}}$.

Base change of the sections. For every $I$ there is a canonical isomorphism ${\mathrm{\Phi }}_I:\mathcal{F}(U_I){\otimes }_A{A}^{\mathrm{\prime }}\to {\mathcal{F}}_{A^{\mathrm{\prime }}}(U_I^{\mathrm{\prime }})$: the affine open $U_I=\mathrm{Spec}{B}_I$ satisfies $\mathcal{F}{\mathrm{\mid }}_{U_I}\cong \widetilde{\mathcal{F}(U_I)}$ because $\mathcal{F}$ is quasi-coherent, so [F10] applied to the morphism $p$ and the affine opens $U_I^{\mathrm{\prime }}\subseteq X_{A^{\mathrm{\prime }}}$, $U_I\subseteq X$ gives $p^{\ast }\mathcal{F}{\mathrm{\mid }}_{U_I^{\mathrm{\prime }}}\cong \widetilde{(B_I{\otimes }_A{A}^{\mathrm{\prime }}){\otimes }_{B_I}\mathcal{F}(U_I)}$, and since $\mathrm{\Gamma }(U_I^{\mathrm{\prime }},\tilde{N})\cong N$ and $(B_I{\otimes }_A{A}^{\mathrm{\prime }}){\otimes }_{B_I}\mathcal{F}(U_I)\cong \mathcal{F}(U_I){\otimes }_A{A}^{\mathrm{\prime }}$ by [F10], taking global sections yields ${\mathrm{\Phi }}_I$.

Compatibility with restrictions and the base-changed complex. The isomorphisms ${\mathrm{\Phi }}_I$ of 1.8 are compatible with restriction: for $I\subseteq J$ the square with the restriction maps $\mathcal{F}(U_I)\to \mathcal{F}(U_J)$ and ${\mathcal{F}}_{A^{\mathrm{\prime }}}(U_I^{\mathrm{\prime }})\to {\mathcal{F}}_{A^{\mathrm{\prime }}}(U_J^{\mathrm{\prime }})$ commutes, because both composites are obtained by applying the affine equivalence and the base change $(-){\otimes }_A{A}^{\mathrm{\prime }}$ to the same restriction map, and these constructions are natural in the open subscheme. Consequently the ${\mathrm{\Phi }}_I$, one for each increasing tuple, assemble into an isomorphism of complexes $K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }}\cong C^{\bullet }({\mathcal{U}}^{\mathrm{\prime }},{\mathcal{F}}_{A^{\mathrm{\prime }}})$, where ${\mathcal{U}}^{\mathrm{\prime }}=(U_0^{\mathrm{\prime }},\dots ,U_r^{\mathrm{\prime }})$ and the differentials on both sides are the componentwise alternating sums of restriction maps.

The base-changed complex computes the base-changed cohomology. The scheme $X_{A^{\mathrm{\prime }}}$ is quasi-compact and separated by [F9], the cover ${\mathcal{U}}^{\mathrm{\prime }}$ is finite affine with affine intersections by 1.7, and ${\mathcal{F}}_{A^{\mathrm{\prime }}}=p^{\ast }\mathcal{F}$ is quasi-coherent by [F10], so [F3] gives a canonical isomorphism $H^q(C^{\bullet }({\mathcal{U}}^{\mathrm{\prime }},{\mathcal{F}}_{A^{\mathrm{\prime }}}))\cong H^q(X_{A^{\mathrm{\prime }}},{\mathcal{F}}_{A^{\mathrm{\prime }}})$ for every $q\ge 0$. Combined with 1.9 there are canonical isomorphisms $H^q(K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }})\cong H^q(X_{A^{\mathrm{\prime }}},{\mathcal{F}}_{A^{\mathrm{\prime }}})$ for every ring map $A\to A^{\mathrm{\prime }}$; they are natural in $A^{\mathrm{\prime }}$, because the section isomorphisms of 1.8, their assembly in 1.9 and the comparison map of [F3] are all built from pullbacks and restrictions and commute with composition of ring maps.

The descending construction: the invariant. For a partial complex $\tilde{F}$ concentrated in degrees $\ge n$ with finite free terms write $Z^n(\tilde{F})=\ker ({\tilde{F}}^n\to {\tilde{F}}^{n+1})$. We construct, by descending induction on $n\le r+1$, partial complexes $F^{\ge n}$ with $F^i$ a finite free $A$-module for $n\le i\le r$ and $F^i=0$ for $i>r$, together with maps ${\alpha }^i:F^i\to K^i$ commuting with the differentials, such that $H^i(\alpha )$ is an isomorphism for every $i>n$ and surjective for $i=n$. The stage $n=r+1$ is the zero complex, the invariant being vacuous and $H^{r+1}$ being zero on both sides.

The extension step: killing the kernel of $H^n(\alpha )$. Suppose the invariant of 1.11 holds at stage $n$. The module $H^n(F^{\ge n})=Z^n(F)\mathrm{/}\mathrm{im}(d^{n-1})$ (with $F^{n-1}=0$) is a subquotient of the finitely generated free module $F^n$, hence finitely generated over the Noetherian ring $A$ by [F7], and so is the kernel of $H^n(\alpha ):H^n(F)\to H^n(K)$; choose finitely many generators of this kernel, lift them to cycles $x_1,\dots ,x_s\in Z^n(F)$, and let $F_0^{n-1}:=A^s\to Z^n(F)\subseteq F^n$ send $e_j\mapsto x_j$, a surjection onto their span. Each $x_j$ maps to a boundary in $K^n$ because its class lies in the kernel of $H^n(\alpha )$, so choose $y_j\in K^{n-1}$ with $d_K^{n-1}{y}_j={\alpha }^n(x_j)$ and set ${\alpha }_0^{n-1}(e_j):=y_j$; then ${\alpha }^n{d}_0^{n-1}=d_K^{n-1}{\alpha }_0^{n-1}$, so $\alpha$ is a map of complexes on the extended partial complex, and the extension makes $H^n(\alpha )$ an isomorphism while leaving $H^i(\alpha )$ unchanged for $i>n$.

The extension step: making $H^{n-1}(\alpha )$ surjective. In the extended partial complex of 1.12 the cokernel of $H^{n-1}(\alpha ):H^{n-1}(F)\to H^{n-1}(K)$ is a quotient of the finitely generated module $H^{n-1}(K)$ (1.6), hence finitely generated by [F7]; choose finitely many generators and cycles $z_1,\dots ,z_t\in Z^{n-1}(K)$ representing them, and replace $F^{n-1}$ by $F^{n-1}\oplus A^t$, the new basis vectors mapping to $0$ in $F^n$ and to $z_j$ in $K^{n-1}$. The differentials still commute with the $\alpha$'s, since $d_K{z}_j=0$, the homology $H^n(\alpha )$ remains an isomorphism because the image of $d^{n-1}$ in $F^n$ is unchanged, and $H^{n-1}(\alpha )$ becomes surjective because the classes of the $z_j$ generate its cokernel; hence the invariant of 1.11 holds at stage $n-1$.

The finite free replacement. The induction of 1.11-1.13 yields a complex $F^{\bullet }$ of finite free $A$-modules with $F^i=0$ for $i>r$, each degree being fixed after finitely many steps, together with a map of complexes $\alpha :F^{\bullet }\to K^{\bullet }$. For every $i\le r$ the stage $n=i-1$ exhibits $H^i(\alpha )$ as an isomorphism, and for $i>r$ both sides vanish; hence $\alpha$ is a quasi-isomorphism.

Tensoring the replacement with an arbitrary module. For every $A$-module $M$ the map $\alpha {\otimes }_A{\mathrm{id}}_M:F^{\bullet }{\otimes }_{A}M\to K^{\bullet }{\otimes }_{A}M$ is a quasi-isomorphism: choose a projective resolution $Q_{\bullet }\to M$; the complexes $F^{\bullet }$ and $K^{\bullet }$ are bounded above with flat terms and $Q_{\bullet }$ is bounded above with projective, hence flat, terms, so the three maps $F^{\bullet }\otimes Q_{\bullet }\to K^{\bullet }\otimes Q_{\bullet }$, $F^{\bullet }\otimes Q_{\bullet }\to F^{\bullet }\otimes M$ and $K^{\bullet }\otimes Q_{\bullet }\to K^{\bullet }\otimes M$ are quasi-isomorphisms by [F11]; these maps form a commutative square, so $H^q(\alpha \otimes M)$ is conjugate to the isomorphism $H^q(F\otimes Q)\to H^q(K\otimes Q)$ and is itself an isomorphism for every $q\in \mathbb{Z}$.

The replacement is compatible with coefficient change. Taking $M=A^{\mathrm{\prime }}$ in 1.15, for every ring map $A\to A^{\mathrm{\prime }}$ the induced maps $H^q(F^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }})\to H^q(K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }})$ are isomorphisms, canonical in the sense that they are induced by the single map of complexes $\alpha$ and hence natural in $A^{\mathrm{\prime }}$; composing with the isomorphism of 1.10 expresses $H^q(F^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }})$ canonically and naturally in terms of $H^q(X_{A^{\mathrm{\prime }}},{\mathcal{F}}_{A^{\mathrm{\prime }}})$.

Truncation of the replacement. Let $E^{\bullet }:={\tau }^{\ge 0}{F}^{\bullet }$ be the canonical truncation of [F13], so that $E^i=0$ for $i<0$, $E^0=\mathrm{Coker}(F^{-1}\to F^0)$ and $E^i=F^i$ for $i>0$; then $E^{\bullet }$ is concentrated in degrees $0,\dots ,r$, its positive-degree terms are finite free, and $E^0$ is finitely generated as a quotient of the finitely generated module $F^0$. The natural map $\pi :F^{\bullet }\to E^{\bullet }$ is the quotient in degree $0$, the identity in positive degrees and zero in negative degrees. It is a quasi-isomorphism: in degree $0$ the differential induced on $E^0$ has kernel $\ker (d_F^0)\mathrm{/}\mathrm{im}(d_F^{-1})$, so $H^0(E)=H^0(F)$, and cohomology in positive degrees is unchanged. In negative degrees $E$ vanishes and $H^i(F)=H^i(K)=0$ by 1.14 and 1.4.

Flatness of the truncation term. For every $A$-module $M$ the complex $\cdots \to F^{-2}\to F^{-1}\to F^0\to E^0\to 0$ is a projective resolution of $E^0$, because $E^0=\mathrm{Coker}(d^{-1})$ and $H^i(F)=0$ for all $i<0$; hence by [F12] the module ${\mathrm{Tor}}_1^A(E^0,M)$ is the homology at $F^{-1}{\otimes }_{A}M$ of the complex $F^{\bullet }{\otimes }_{A}M$, namely $H^{-1}(F^{\bullet }{\otimes }_{A}M)$, and $\alpha \otimes M$ is a quasi-isomorphism by 1.15 while $K^{\bullet }{\otimes }_{A}M$ is concentrated in degrees $0,\dots ,r$, so $H^{-1}(F^{\bullet }{\otimes }_{A}M)\cong H^{-1}(K^{\bullet }{\otimes }_{A}M)=0$. Therefore ${\mathrm{Tor}}_1^A(E^0,M)=0$ for every $M$, and [F12] makes $E^0$ a flat $A$-module.

The truncation term is finite projective. The module $E^0$ is finitely generated (1.17) over the Noetherian ring $A$, hence finitely presented by [F6], and it is flat by 1.18; therefore it is a finite projective $A$-module by [F7].

The complex and its base-change property. The complex $E^{\bullet }$ is a bounded complex of finite projective $A$-modules, concentrated in degrees $0,\dots ,r$ and finite free in all positive degrees. Since $K^{-1}=0$ and $\alpha :F\to K$ is a chain map, ${\alpha }^0{d}_F^{-1}=d_K^{-1}{\alpha }^{-1}=0$. Thus ${\alpha }^0$ factors through $E^0=\mathrm{Coker}(d_F^{-1})$ and, together with ${\alpha }^i$ for $i>0$, gives a direct chain map $\gamma :E^{\bullet }\to K^{\bullet }$ with $\alpha =\gamma \pi$. Steps 1.14 and 1.17 make $\gamma$ a quasi-isomorphism. For every $A$-module $M$, both $F$ and $E$ are bounded above complexes of flat modules, the latter by 1.19. Apply the projective-resolution comparison from step 1.15 to the quasi-isomorphism $\pi$, using [F11] on both complexes, to see that $\pi {\otimes }_{A}M$ is a quasi-isomorphism; step 1.15 already proves the same for $\alpha {\otimes }_{A}M$. Since $\alpha {\otimes }_{A}M=(\gamma {\otimes }_{A}M)(\pi {\otimes }_{A}M)$, two-out-of-three makes $\gamma {\otimes }_{A}M$ a quasi-isomorphism. Taking $M=A^{\mathrm{\prime }}$ and composing with 1.10 gives canonical isomorphisms $H^q(E^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }})\cong H^q(X_{A^{\mathrm{\prime }}},{\mathcal{F}}_{A^{\mathrm{\prime }}})$ for all $q\in \mathbb{Z}$, natural in $A^{\mathrm{\prime }}$ and compatible with composition of ring maps. For the local-freeness clause, fix a prime $\mathfrak{p}\subseteq A$: the ring $A_{\mathfrak{p}}$ is Noetherian and $E_{\mathfrak{p}}^0$ is a finite flat module over this Noetherian local ring, hence free by [F14]; moreover $\widetilde{E^0}$ is finitely presented, so by [F14] every point of $\mathrm{Spec}A$ has an affine open neighbourhood $V$ on which $\widetilde{E^0}{\mathrm{\mid }}_V$ is free of finite rank, and then $E^{\bullet }{\otimes }_A\mathcal{O}(V)$ is a bounded complex of finite free $\mathcal{O}(V)$-modules.

Boundaries and choice accounting. If $X=\mathrm{\varnothing }$ the cover is empty, $K^{\bullet }=0$ by [F3], the construction of 1.11-1.14 returns $E^{\bullet }=0$, and for every $A^{\mathrm{\prime }}$ the scheme $X_{A^{\mathrm{\prime }}}$ is empty with zero sheaf, so both sides of the isomorphism vanish and the claims hold with $r=0$. If $\mathcal{F}=0$ then $K^{\bullet }=0$ and $E^{\bullet }=0$ for the same reason. A one-member cover, $r=0$, is included: then $K^{\bullet }=K^0=\mathcal{F}(U_0)$ is a single flat module in degree $0$ with finitely generated cohomology and the induction starts at $n=1$, producing $E^{\bullet }$ concentrated in degree $0$. The zero ring $A=0$ is Noetherian and $\mathrm{Spec}0=\mathrm{\varnothing }$, so $X=\mathrm{\varnothing }$ and the empty case applies, and for $A^{\mathrm{\prime }}=0$ the scheme $X_{A^{\mathrm{\prime }}}$ is empty and $K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }}=0$. For $q<0$ both sides vanish by 1.9 and 1.10, and for $q>r$ the complex $K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }}$ is concentrated in degrees $0,\dots ,r$, so its cohomology vanishes and by 1.10 so does $H^q(X_{A^{\mathrm{\prime }}},{\mathcal{F}}_{A^{\mathrm{\prime }}})$. The Axiom of Choice [F15] is used for the finite affine cover, the affine equivalence and associated sheaves of [F6] and [F10], the finite free covers of finitely generated modules in [F8], and the projective resolutions of [F11]; the Axiom of Dependent Choice [F15] is used for the Tor and projective-resolution criteria [F11, F12] and for the descending recursion of 1.11-1.13, which makes finitely many choices at each of countably many stages.