Cohomology and base change for proper flat coherent families

Given: The Axiom of Choice and the Axiom of Dependent Choice, a proper morphism of finite presentation $f:X\to S$, a coherent ${\mathcal{O}}_X$-module $\mathcal{F}$ flat over $S$, a point $s\in S$ and an integer $q\ge 0$.

Coherence unpacked and finite presentation: a coherent ${\mathcal{O}}_X$-module is quasi-coherent of finite type, and for every open $U\subseteq X$ and every morphism ${\mathcal{O}}_U^n\to \mathcal{F}{\mathrm{\mid }}_U$ with finite $n\ge 0$ its kernel is of finite type; consequently, on an affine open $U=\mathrm{Spec}A$ with $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$ and $M$ finitely generated, a surjection $A^n\to M$ has finitely generated kernel, so $M$ is finitely presented and $\mathcal{F}$ is finitely presented; restrictions of coherent modules to open subschemes are coherent. (Coherent module sheaves, Finite type and finitely presented module sheaves, Quasi-coherent module on a scheme, Kernel sheaves are objectwise, while cokernels and images are sheafified, Affine quasi-coherent sheaves are modules, Finitely presented modules and finitely presented algebras)

Flatness and localisation: $\mathcal{F}$ is flat over $S$ meaning that each stalk ${\mathcal{F}}_x$ is flat over the local ring ${\mathcal{O}}_{S,f(x)}$; flatness is preserved by restriction to open subschemes and by base change, a localisation $A\to S^{-1}A$ is a flat ring map, and flatness of a module is local on the base ring. (Flat and faithfully flat modules and ring homomorphisms, Every localization is flat, and localizing a flat module preserves flatness, Flatness is transitive under a flat change of rings, A module is flat if and only if all prime localizations are flat, equivalently all maximal localizations are flat, Extension of scalars carries flat modules to flat modules)

Properness, affine bases and fibres: a proper morphism is separated, of finite type and universally closed, and a morphism of finite type is quasi-compact; affine opens form a basis of every scheme, so there is an affine open $U=\mathrm{Spec}A\subseteq S$ containing $s$, with corresponding prime $\mathfrak{m}\subseteq A$; the restriction $X_U:=f^{-1}U\to U=\mathrm{Spec}A$ is proper and of finite presentation, and $X_U$ is quasi-compact and separated; the canonical morphism $\mathrm{Spec}\kappa (s)\to S$ factors through $U$, so the fibre $X_s=X{\times }_S\mathrm{Spec}\kappa (s)$ is canonically identified with $X_U{\times }_{\mathrm{Spec}A}\mathrm{Spec}\kappa (s)$, and $\kappa (s)$ is the residue field of the local ring $A_{\mathfrak{m}}$. (Proper morphisms, Locally finite presentation morphisms, Locally finite type and finite type morphisms, Quasi-compact and quasi-separated morphisms, Schemes, Affine open subschemes, The underlying space of an affine spectrum, The residue field at a point of an affine scheme, Base change of objects, morphisms and properties, Iterated base change, Properness survives arbitrary base change, Localisation at a prime ideal: $R_{\mathfrak{p}}=(R\setminus \mathfrak{p}{)}^{-1}R$)

The perfect complex over an affine base: for the ring $A$, the proper morphism of finite presentation $X_U\to \mathrm{Spec}A$ and the coherent, hence finitely presented [F1], module ${\mathcal{F}}_U:=\mathcal{F}{\mathrm{\mid }}_{X_U}$ flat over $\mathrm{Spec}A$ by [F2], the cited lemma provides an integer $r\ge 0$ and a bounded complex $K^{\bullet }$ of finite projective $A$-modules, concentrated in degrees $0,\dots ,r$ and finite free in all positive degrees, with canonical isomorphisms ${\theta }_{A^{\mathrm{\prime }}}:H^q(K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }})\cong H^q(X_{A^{\mathrm{\prime }}},{\mathcal{F}}_{A^{\mathrm{\prime }}})$ for every $A$-algebra $A^{\mathrm{\prime }}$, natural in $A^{\mathrm{\prime }}$ and compatible with composition of $A$-algebra maps, where $X_{A^{\mathrm{\prime }}}:=X_U{\times }_{\mathrm{Spec}A}\mathrm{Spec}{A}^{\mathrm{\prime }}$ and ${\mathcal{F}}_{A^{\mathrm{\prime }}}$ is the pullback of ${\mathcal{F}}_U$; moreover $K^{\bullet }$ becomes a bounded complex of finite free modules after restricting to the members of an open cover of $\mathrm{Spec}A$ by affine opens. (Universal finite projective cohomology complex over any base, The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

Naturality of the comparison with base-change maps: the naturality clause of [F4] says that for $A$-algebras $A^{\mathrm{\prime }}\to A^{\mathrm{\prime }\mathrm{\prime }}$ the square with the maps ${\theta }_{A^{\mathrm{\prime }}}\otimes \mathrm{id}$ and ${\theta }_{A^{\mathrm{\prime }\mathrm{\prime }}}$ commutes, where the algebraic horizontal arrow $H^q(K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }}){\otimes }_{A^{\mathrm{\prime }}}{A}^{\mathrm{\prime }\mathrm{\prime }}\to H^q(K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }\mathrm{\prime }})$ is induced by tensoring representatives of cohomology classes, and the geometric horizontal arrow $A^{\mathrm{\prime }\mathrm{\prime }}{\otimes }_{A^{\mathrm{\prime }}}{H}^q(X_{A^{\mathrm{\prime }}},{\mathcal{F}}_{A^{\mathrm{\prime }}})\to H^q(X_{A^{\mathrm{\prime }\mathrm{\prime }}},{\mathcal{F}}_{A^{\mathrm{\prime }\mathrm{\prime }}})$ is the extension of scalars of the pullback of cohomology classes along $X_{A^{\mathrm{\prime }\mathrm{\prime }}}\to X_{A^{\mathrm{\prime }}}$ composed with the canonical map $g^{-1}{\mathcal{F}}_{A^{\mathrm{\prime }}}\to {\mathcal{F}}_{A^{\mathrm{\prime }\mathrm{\prime }}}$, that is, the map on global sections induced by the base-change map of the cohomology-and-base-change definition for the Cartesian square over $\mathrm{Spec}{A}^{\mathrm{\prime }}\to \mathrm{Spec}{A}^{\mathrm{\prime }\mathrm{\prime }}$; the pullback of classes is the map supplied by the contravariance of sheaf cohomology in the space. (Universal finite projective cohomology complex over any base, Cohomology and base-change map, Variance of sheaf cohomology, Associativity of tensor products for compatible bimodules, The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$)

The finite-free criterion: for a ring $B$ with maximal ideal $\mathfrak{n}\subseteq B$ and residue field $\lambda =B\mathrm{/}\mathfrak{n}$ and a bounded complex $L^{\bullet }$ of finite free $B$-modules with differentials $d^q$, the map ${\phi }_L^q:H^q(L){\otimes }_B\lambda \to H^q(L{\otimes }_B\lambda )$ induced by tensoring representatives satisfies: (1) it is surjective if and only if there is $t\in B\setminus \mathfrak{n}$ such that over $B_t$ there are bases of $L_t^q$ and $L_t^{q+1}$ in which $d^q$ is $\left(\begin{array}{cc}{I}_r& 0\\ 0& 0\end{array}\right)$; (2) if so, then $H^q(L{)}_t$ is a finitely generated $B_t$-module and the natural map $H^q(L{)}_t{\otimes }_{B_t}{B}^{\mathrm{\prime }\mathrm{\prime }}\to H^q(L{\otimes }_B{B}^{\mathrm{\prime }\mathrm{\prime }})$ is an isomorphism for every $B_t$-algebra $B^{\mathrm{\prime }\mathrm{\prime }}$; (3) given (1), $H^q(L{)}_t$ is a finite projective $B_t$-module for some $t\in B\setminus \mathfrak{n}$ if and only if, after shrinking further, the analogous map ${\phi }_L^{q-1}$ is also surjective, and if $L^{q-1}=0$ this surjectivity is automatic. The Axiom of Choice enters only through the Nakayama lemma and its corollary. (Finite-free local criterion for cohomology and base change, Assuming the Axiom of Choice, Nakayama's lemma, Assuming the Axiom of Choice, generators modulo an ideal in the Jacobson radical lift to generators, Cohomology object of a cochain complex, The Axiom of Choice)

Higher direct images and the fibre map: for $f$ quasi-compact and separated and $\mathcal{F}$ quasi-coherent, each $R^q{f}_{\ast }\mathcal{F}$ is quasi-coherent, $R^q{f}_{\ast }\mathcal{F}=0$ for $q<0$, and for an affine open $V=\mathrm{Spec}B\subseteq S$ there is a canonical isomorphism $(R^q{f}_{\ast }\mathcal{F}){\mathrm{\mid }}_V\cong \widetilde{H^q(f^{-1}V,\mathcal{F})}$ with the associated sheaf of the $B$-module $H^q(f^{-1}V,\mathcal{F})$; hence $(R^q{f}_{\ast }\mathcal{F})(V)=H^q(f^{-1}V,\mathcal{F})$, the fibre at $s$ is $(R^q{f}_{\ast }\mathcal{F})(s)=H^q(f^{-1}V,\mathcal{F}){\otimes }_B\kappa (s)$ for every affine open $V\ni s$, and the fibre map ${\phi }_s^q$ of the cohomology-and-base-change definition is the extension of scalars of the pullback-of-classes map $H^q(f^{-1}V,\mathcal{F})\to H^q(X_s,{\mathcal{F}}_s)$, its colimit description being independent of the chosen affine neighbourhood of $s$. (Higher direct image of a sheaf, Higher direct images localize over an affine base, Cohomology and base-change map, Fibre of a module sheaf at a point, Module sheaf on an affine scheme, The stalk of an associated sheaf is the localisation, Pullback of a module along a morphism of ringed spaces)

Finite locally free modules on affine schemes: for a ring $B$ and a finitely presented $B$-module $P$, the associated sheaf $\tilde{P}$ is locally free of finite rank near $\mathfrak{p}\in \mathrm{Spec}B$ if and only if $P_{\mathfrak{p}}$ is free over $B_{\mathfrak{p}}$; the locus of primes at which a finitely presented module is free of a fixed rank is open, and the module is free of that rank on an open neighbourhood of any such prime. (Locally free sheaves of finite rank, Openness of the finite free locus, Affine quasi-coherent sheaves are modules, Module sheaf on an affine scheme, The stalk of an associated sheaf is the localisation)

Finitely generated projective modules: a finitely generated module is a quotient of a finite free module; a surjection onto a projective module splits, exhibiting the projective module as a direct summand of a free module under the Axiom of Choice; over a local ring $(R,\mathfrak{n})$ a finitely generated module $N$ generated by elements whose classes generate $N\mathrm{/}\mathfrak{n}N$ is generated by them; passing to the residue field is right exact; and a direct summand of a finitely generated free module is finitely generated, while a finitely presented module has finitely generated syzygies. (Generated submodule, cyclic and finitely generated modules, module basis and free module, The splitting lemma for short exact sequences of modules, Equivalent characterizations of projective modules, Projective modules and the lifting property, Assuming the Axiom of Choice, Nakayama's lemma, Assuming the Axiom of Choice, generators modulo an ideal in the Jacobson radical lift to generators, The Jacobson radical of a ring, A local ring is a nonzero commutative ring with a unique maximal ideal, Tensoring is right exact, Universal property of a direct sum of modules, Finitely presented modules and finitely presented algebras, The Axiom of Choice)

The Axiom of Choice and the Axiom of Dependent Choice are the choice principles named in the statement. (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)