Universal finite projective cohomology complex over any base

Hypothesis transfer. Let $x\in X$ and put $\mathfrak{p}=f(x)$. By [F6] the structure map $A\to {\mathcal{O}}_{X,x}$ factors through the localisation $A\to A_{\mathfrak{p}}$, so ${\mathcal{F}}_x$ is an $A_{\mathfrak{p}}$-module whose restriction to $A$ is its given $A$-module structure; since $A\to A_{\mathfrak{p}}$ is flat and ${\mathcal{F}}_x$ is flat over $A_{\mathfrak{p}}$ by hypothesis, [F4] makes ${\mathcal{F}}_x$ flat over $A$ as well.

The Noetherian stage. By 1.1 the hypotheses of [F1] hold for $f$ and $\mathcal{F}$, so there are a finitely generated $\mathbb{Z}$-subalgebra $A_i\subseteq A$, Noetherian by [F2], a proper morphism of finite presentation $f_i:X_i\to \mathrm{Spec}{A}_i$, a finitely presented ${\mathcal{O}}_{X_i}$-module ${\mathcal{F}}_i$ flat over $A_i$ in the sense of [F1], and identifications $X\cong X_i{\times }_{\mathrm{Spec}{A}_i}\mathrm{Spec}A$ and $\mathcal{F}\cong q^{\ast }{\mathcal{F}}_i$ over $A$, where $q:X\to X_i$ is the projection.

The stage is flat over its base local rings. Let $x\in X_i$ and put ${\mathfrak{p}}_i=f_i(x)$; the ${\mathcal{O}}_{X_i,x}$-module $({\mathcal{F}}_i{)}_x$ has its $A_i$-action factoring through $A_i\to (A_i{)}_{{\mathfrak{p}}_i}$ and is flat over $A_i$ by 1.2. For every prime $\mathfrak{q}\subseteq (A_i{)}_{{\mathfrak{p}}_i}$, corresponding to a prime ${\mathfrak{p}}_i^{\mathrm{\prime }}\subseteq {\mathfrak{p}}_i$ of $A_i$, the change-of-rings isomorphism [F10] together with the identifications of [F5] gives $({\mathcal{F}}_i{)}_x{\otimes }_{(A_i{)}_{{\mathfrak{p}}_i}}(A_i{)}_{{\mathfrak{p}}_i^{\mathrm{\prime }}}\cong ({\mathcal{F}}_i{)}_x{\otimes }_{A_i}(A_i{)}_{{\mathfrak{p}}_i^{\mathrm{\prime }}}$, the localisation of the $A_i$-module $({\mathcal{F}}_i{)}_x$ at ${\mathfrak{p}}_i^{\mathrm{\prime }}$, and this is flat over $(A_i{)}_{{\mathfrak{p}}_i^{\mathrm{\prime }}}=((A_i{)}_{{\mathfrak{p}}_i}{)}_{\mathfrak{q}}$ by [F4] and [F5]; hence $({\mathcal{F}}_i{)}_x$ is flat over $(A_i{)}_{{\mathfrak{p}}_i}$ by the local flatness criterion [F5].

The Noetherian-stage complex. By 1.3 and [F3] applied to $f_i$ and ${\mathcal{F}}_i$ there are an integer $r\ge 0$ and a bounded complex $K_i^{\bullet }$ of finite projective $A_i$-modules, concentrated in degrees $0,\dots ,r$ and finite free in all positive degrees, with canonical isomorphisms $H^q(K_i^{\bullet }{\otimes }_{A_i}{A}_i^{\mathrm{\prime }})\cong H^q((X_i{)}_{A_i^{\mathrm{\prime }}},({\mathcal{F}}_i{)}_{A_i^{\mathrm{\prime }}})$ for every $A_i$-algebra $A_i^{\mathrm{\prime }}$ and every $q$, natural in $A_i^{\mathrm{\prime }}$ and compatible with composition; moreover $K_i^{\bullet }$ becomes a bounded complex of finite free modules on some open cover of $\mathrm{Spec}{A}_i$.

Base change of finite projective modules. If $P$ is a finitely generated projective $A_i$-module, then $P{\otimes }_{A_i}A$ is a finitely generated projective $A$-module: choosing finitely many generators gives a surjection $A_i^n\to P$ [F7], projectivity of $P$ splits it so that $P$ is a direct summand of $A_i^n$ [F7], tensoring with $A$ is right exact and carries the splitting section to a splitting section [F8], the identification $A_i^n{\otimes }_{A_i}A\cong A^n$ follows from the unit isomorphism $A_i{\otimes }_{A_i}A\cong A$ and compatibility with finite direct sums [F5, F8], so $P{\otimes }_{A_i}A$ is both a quotient and a direct summand of the free $A$-module $A^n$; hence it is finitely generated and projective over $A$ by [F9].

The complex and its coefficient change. Put $K^{\bullet }:=K_i^{\bullet }{\otimes }_{A_i}A$, a bounded complex of finite projective $A$-modules concentrated in degrees $0,\dots ,r$ and finite free in positive degrees by 1.5 and [F13]; for every $A$-algebra $A^{\mathrm{\prime }}$ the associativity and unit isomorphisms of [F5] applied to the $A_i$-module $K_i^j$ and the ring $A$ give canonical isomorphisms $K_i^j{\otimes }_{A_i}{A}^{\mathrm{\prime }}\cong (K_i^j{\otimes }_{A_i}A){\otimes }_A(A{\otimes }_A{A}^{\mathrm{\prime }})\cong K^j{\otimes }_A{A}^{\mathrm{\prime }}$ because $A{\otimes }_A{A}^{\mathrm{\prime }}\cong A^{\mathrm{\prime }}$, natural in $A^{\mathrm{\prime }}$ and compatible with composition, hence an isomorphism of complexes $K_i^{\bullet }{\otimes }_{A_i}{A}^{\mathrm{\prime }}\cong K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }}$.

The base-changed geometry. Under the identification $X\cong X_i{\times }_{\mathrm{Spec}{A}_i}\mathrm{Spec}A$ of 1.2, iterated base change [F11] with $\mathrm{Spec}{A}^{\mathrm{\prime }}\to \mathrm{Spec}A\to \mathrm{Spec}{A}_i$ gives, for every $A$-algebra $A^{\mathrm{\prime }}$, a canonical isomorphism $\psi :X_{A^{\mathrm{\prime }}}=(X_i{\times }_{\mathrm{Spec}{A}_i}\mathrm{Spec}A){\times }_{\mathrm{Spec}A}\mathrm{Spec}{A}^{\mathrm{\prime }}\cong X_i{\times }_{\mathrm{Spec}{A}_i}\mathrm{Spec}{A}^{\mathrm{\prime }}=(X_i{)}_{A^{\mathrm{\prime }}}$ that is compatible with the projections to $X_i$.

The base-changed sheaf. The identification $\mathcal{F}\cong q^{\ast }{\mathcal{F}}_i$ of 1.2 and the composition rule for pullbacks [F12] give canonical identifications ${\mathcal{F}}_{A^{\mathrm{\prime }}}=p^{\ast }\mathcal{F}\cong p^{\ast }{q}^{\ast }{\mathcal{F}}_i\cong (q\circ p{)}^{\ast }{\mathcal{F}}_i$, while $({\mathcal{F}}_i{)}_{A^{\mathrm{\prime }}}=(q^{\mathrm{\prime }}{)}^{\ast }{\mathcal{F}}_i$ for the projection $q^{\mathrm{\prime }}:(X_i{)}_{A^{\mathrm{\prime }}}\to X_i$; since $q^{\mathrm{\prime }}\circ \psi =q\circ p$ by 1.7, the isomorphism $\psi$ identifies these two pullbacks canonically, so ${\mathcal{F}}_{A^{\mathrm{\prime }}}\cong ({\mathcal{F}}_i{)}_{A^{\mathrm{\prime }}}$ naturally in $A^{\mathrm{\prime }}$ and compatibly with composition of $A$-algebra maps.

The comparison isomorphism. For every $A$-algebra $A^{\mathrm{\prime }}$ and every $q\in \mathbb{Z}$ compose the isomorphism $H^q(K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }})\cong H^q(K_i^{\bullet }{\otimes }_{A_i}{A}^{\mathrm{\prime }})$ of 1.6 with the isomorphism $H^q(K_i^{\bullet }{\otimes }_{A_i}{A}^{\mathrm{\prime }})\cong H^q((X_i{)}_{A^{\mathrm{\prime }}},({\mathcal{F}}_i{)}_{A^{\mathrm{\prime }}})$ of 1.4 applied to the $A_i$-algebra $A^{\mathrm{\prime }}$, and with the identification $H^q((X_i{)}_{A^{\mathrm{\prime }}},({\mathcal{F}}_i{)}_{A^{\mathrm{\prime }}})\cong H^q(X_{A^{\mathrm{\prime }}},{\mathcal{F}}_{A^{\mathrm{\prime }}})$ induced by 1.7 and 1.8; each constituent is canonical, natural in $A^{\mathrm{\prime }}$ and compatible with composition of $A$-algebra maps, so the composite $H^q(K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }})\cong H^q(X_{A^{\mathrm{\prime }}},{\mathcal{F}}_{A^{\mathrm{\prime }}})$ is as well.

Local finite freeness. By 1.4 there is an open cover of $\mathrm{Spec}{A}_i$ by affine opens $V_j$ such that $K_i^{\bullet }{\otimes }_{A_i}\mathcal{O}(V_j)$ is a bounded complex of finite free $\mathcal{O}(V_j)$-modules; by [F11] the preimages $V_j^{\mathrm{\prime }}\subseteq \mathrm{Spec}A$ are affine opens with $\mathcal{O}(V_j^{\mathrm{\prime }})\cong A{\otimes }_{A_i}\mathcal{O}(V_j)$ and they cover $\mathrm{Spec}A$. For each $j$ the associativity and unit identifications of [F5] give $K^{\bullet }{\otimes }_A\mathcal{O}(V_j^{\mathrm{\prime }})\cong (K_i^{\bullet }{\otimes }_{A_i}\mathcal{O}(V_j)){\otimes }_{\mathcal{O}(V_j)}\mathcal{O}(V_j^{\mathrm{\prime }})$, a bounded complex that is finite free over $\mathcal{O}(V_j^{\mathrm{\prime }})$ because extension of scalars of a free module is free [F8]; hence $K^{\bullet }$ is finite free locally on $\mathrm{Spec}A$.

Boundaries and choice accounting. If $X=\mathrm{\varnothing }$ the stage of 1.2 may be taken with $X_i=\mathrm{\varnothing }$, the stage complex is the zero complex with $r=0$ by 1.4, and for every $A^{\mathrm{\prime }}$ the scheme $X_{A^{\mathrm{\prime }}}$ is empty with vanishing cohomology, so both sides of 1.9 vanish; if $\mathcal{F}=0$ the same argument applies with ${\mathcal{F}}_i=0$; if $A=0$ then $\mathrm{Spec}A=\mathrm{\varnothing }$ and $X=\mathrm{\varnothing }$, and if $A^{\mathrm{\prime }}=0$ then $X_{A^{\mathrm{\prime }}}=\mathrm{\varnothing }$ and $K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }}=0$; the one-member case $r=0$ is covered because 1.4 and 1.5 leave $K^{\bullet }$ concentrated in degree $0$; for $q<0$ both sides of 1.9 vanish, and for $q>r$ the complex $K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }}$ is concentrated in degrees $0,\dots ,r$ by 1.6 while $H^q((X_i{)}_{A^{\mathrm{\prime }}},({\mathcal{F}}_i{)}_{A^{\mathrm{\prime }}})=0$ by 1.4; if $A$ is finitely generated over $\mathbb{Z}$ the stage $A_i=A$ of [F1] is trivial, 1.2 and 1.3 are identities, and 1.9 reduces to [F3]. The Axiom of Choice is used through [F1] in 1.2 and through the projectivity characterisation [F9] in 1.5; the Axiom of Dependent Choice is consumed by [F3] in 1.4; no other selection is made.