Upper semicontinuity of fibre cohomology dimensions

Given: The Axiom of Choice and the Axiom of Dependent Choice, a proper morphism of finite presentation $f:X\to S$ with $S$ arbitrary, a coherent ${\mathcal{O}}_X$-module $\mathcal{F}$ flat over $S$, an integer $q\ge 0$, and a point $s_0\in S$.

Properness, finite presentation and affine charts: a proper morphism is separated, of finite type and universally closed; a morphism of finite type is quasi-compact; every point of a scheme has an affine open neighbourhood, so there is an affine open $U=\mathrm{Spec}A\subseteq S$ containing $s_0$; the points of $U$ are the primes $\mathfrak{m}\subseteq A$, with residue field $\kappa (\mathfrak{m})\cong A_{\mathfrak{m}}\mathrm{/}\mathfrak{m}{A}_{\mathfrak{m}}$; the open subscheme $X_U:=f^{-1}U$ represents the fibre product $X{\times }_{S}U$, its structure morphism $X_U\to U$ is proper, and base change of a locally finitely presented morphism is again locally of finite presentation, so $X_U\to U$ is proper of finite presentation. (Proper morphisms, Locally finite type and finite type morphisms, Quasi-compact and quasi-separated morphisms, Locally finite presentation morphisms, Schemes, The underlying space of an affine spectrum, The residue field at a point of an affine scheme, Affine open subschemes, Restricting fibre products to open subschemes, Properness survives arbitrary base change, Local finiteness conditions under base change)

Coherence implies finite presentation: a coherent ${\mathcal{O}}_X$-module is quasi-coherent of finite type, and the kernel of every morphism ${\mathcal{O}}_U^n\to \mathcal{F}{\mathrm{\mid }}_U$ is of finite type; hence on an affine open $\mathrm{Spec}B$ with $\mathcal{F}{\mathrm{\mid }}_{\mathrm{Spec}B}\cong \tilde{M}$ and $M$ finitely generated, the kernel of a surjection $B^n\to M$ is finitely generated and $M$ is finitely presented, so $\mathcal{F}$ 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, Finitely presented modules and finitely presented algebras, Affine quasi-coherent sheaves are modules)

The universal perfect complex: for the commutative ring $A$, the proper morphism of finite presentation $X_U\to \mathrm{Spec}A$ and the finitely presented module ${\mathcal{F}}_U:=\mathcal{F}{\mathrm{\mid }}_{X_U}$ that is flat over $A$ (each stalk flat over the corresponding base local ring), there are an integer $r\ge 0$ and a bounded complex $K^{\bullet }$ of finitely generated projective $A$-modules, concentrated in degrees $0,\dots ,r$ and finite free in positive degrees, with canonical isomorphisms ${\theta }_{A^{\mathrm{\prime }}}:H^q(K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }})\cong H^q((X_U{)}_{A^{\mathrm{\prime }}},({\mathcal{F}}_U{)}_{A^{\mathrm{\prime }}})$ for every $A$-algebra $A^{\mathrm{\prime }}$ and every $q\in \mathbb{Z}$, natural in $A^{\mathrm{\prime }}$; here $K^{\bullet }{\otimes }_A{A}^{\mathrm{\prime }}$ is the termwise tensor complex and $H^q$ is the cohomology object of a cochain complex. The Axiom of Choice and the Axiom of Dependent Choice are inherited from this supplier. (Universal finite projective cohomology complex over any base, The tensor product of a right and a left chain complex is totalized by direct sums with the Koszul differential, Cohomology object of a cochain complex, Base change of objects, morphisms and properties, Pullback of a module along a morphism of ringed spaces, 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)

Open restriction commutes with base change: for a morphism $f:X\to S$, an open $U\subseteq S$ and the open subscheme $X_U=f^{-1}U$ representing $X{\times }_{S}U$, and for every $S$-scheme $T$ whose structure morphism to $S$ factors through $U$, there is a canonical identification $X{\times }_{S}T\cong X_U{\times }_{U}T$ compatible with the projections; consequently, for $s\in U$ the fibre of $f$ at $s$ is canonically identified with the fibre of $f_U:X_U\to U$ at $s$, and under this identification the pullback of $\mathcal{F}$ to the fibre of $f$ corresponds to the pullback of ${\mathcal{F}}_U$ to the fibre of $f_U$, by the composition rule $f^{\ast }{g}^{\ast }\cong (g\circ f{)}^{\ast }$ for pullbacks of modules. (Restricting fibre products to open subschemes, Iterated base change, Base change of objects, morphisms and properties, Pullback of a module along a morphism of ringed spaces)

Cohomology is invariant under isomorphism: if a morphism of schemes is an isomorphism and the coefficient sheaves on source and target correspond under the pullback along it, then the pullback maps of sheaf cohomology along the isomorphism and along an inverse are mutually inverse, by the compatibility with composition and the identity clause of the variance of sheaf cohomology. (Variance of sheaf cohomology, Sheaf cohomology as right derived global sections)

Fibres of associated sheaves and quotients of finitely generated modules: let $A$ be a commutative ring, $N$ a finitely generated $A$-module and $\mathfrak{p}\subseteq A$ a prime with ${\mathcal{O}}_{\mathrm{Spec}A,\mathfrak{p}}=A_{\mathfrak{p}}$; then the canonical maps $(\tilde{N}{)}_{\mathfrak{p}}\to N_{\mathfrak{p}}$ and $N_{\mathfrak{p}}\to N{\otimes }_A{A}_{\mathfrak{p}}$ are isomorphisms, and the fibre $\tilde{N}(\mathfrak{p})=(\tilde{N}{)}_{\mathfrak{p}}{\otimes }_{A_{\mathfrak{p}}}\kappa (\mathfrak{p})$ is canonically isomorphic to $N{\otimes }_A\kappa (\mathfrak{p})$ through the associativity and unit isomorphisms for tensor products. A quotient module of a finitely generated module is finitely generated: the images of any generating family generate the quotient. (The stalk of an associated sheaf is the localisation, Fibre of a module sheaf at a point, Localisation of modules is extension of scalars, Associativity of tensor products for compatible bimodules, The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$, Module sheaf on an affine scheme, The underlying space of an affine spectrum, Generated submodule, cyclic and finitely generated modules, module basis and free module, Quotient module $M\mathrm{/}N$ with scalar multiplication on additive cosets)

Rank-nullity over a field and cokernels of linear maps: for a linear map $T:V\to W$ of vector spaces over a field with $V$ finite-dimensional, $\dim V=\dim \ker T+\mathrm{rank}T$ where $\mathrm{rank}T=\dim \mathrm{im}T$, and if $T$ is surjective then $W$ is finite-dimensional with $\dim W=\dim V-\dim \ker T$; dimensions of quotient spaces enter the proof only through this identity applied to surjections. For an $R$-module homomorphism $f:M\to N$ the cokernel is the quotient module $\mathrm{coker}f=N\mathrm{/}\mathrm{im}f$, and the canonical projection $N\to \mathrm{coker}f$ is a surjective homomorphism with kernel $\mathrm{im}f$; over a field this is the quotient map of the vector space $N$ by its subspace $\mathrm{im}f$, so the surjective form above applies with $V=N$ and $W=\mathrm{coker}f$. (Rank-nullity: ${\dim }_{F}V=\mathrm{nullity}T+\mathrm{rank}T$, Rank and nullity of a linear map with finite-dimensional domain, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, Module homomorphism and isomorphism, kernel, image and cokernel, Quotient module $M\mathrm{/}N$ with scalar multiplication on additive cosets, The quotient vector space $V\mathrm{/}W$ and its canonical projection, Coset equality, well-defined quotient operations, and the canonical projection with kernel $W$)

Finitely generated projective modules: a finitely generated module is a quotient of a finite free module; a surjection onto a projective module splits, so a finitely generated projective module is a direct summand of a finite free module, hence finitely generated and finitely presented; over a local ring $(R,\mathfrak{n})$, a finitely generated module $N$ whose residue classes generate $N\mathrm{/}\mathfrak{n}N$ is generated by lifts of those classes, and if $R^m\to N$ is a split surjection with kernel $N^{\mathrm{\prime }}$ then the induced surjection $R^m\mathrm{/}\mathfrak{n}{R}^m\to N\mathrm{/}\mathfrak{n}N$ is an isomorphism only if $N^{\mathrm{\prime }}=\mathfrak{n}{N}^{\mathrm{\prime }}$, so $N^{\mathrm{\prime }}=0$ and $N\cong R^m$ is free. The implication from projectivity to the splitting uses the Axiom of Choice. (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, Finitely presented modules and finitely presented algebras, The Axiom of Choice)

The free locus and the fibre dimension: let $P$ be a finitely generated projective $A$-module with associated sheaf $\tilde{P}$ on $\mathrm{Spec}A$; for $\rho \ge 0$ the locus $Z_{\rho }=\{\mathfrak{p}\in \mathrm{Spec}A:{\tilde{P}}_{\mathfrak{p}}\cong {\mathcal{O}}_{\mathrm{Spec}A,\mathfrak{p}}^{\rho }\}$ is open and $\tilde{P}$ is locally free of rank $\rho$ on it. For the prime $\mathfrak{p}$, the stalk is ${\tilde{P}}_{\mathfrak{p}}\cong P_{\mathfrak{p}}$ and the fibre is $\tilde{P}(\mathfrak{p})\cong P{\otimes }_A\kappa (\mathfrak{p})$ by [F6]; on $Z_{\rho }$ this fibre is $\kappa (\mathfrak{p}{)}^{\rho }$, so the function $\mathfrak{p}\mapsto {\dim }_{\kappa (\mathfrak{p})}(P{\otimes }_A\kappa (\mathfrak{p}))$ is constant with value $\rho$ on $Z_{\rho }$. (Openness of the finite free locus, Locally free sheaves of finite rank, Fibre of a module sheaf at a point, The stalk of an associated sheaf is the localisation, Affine quasi-coherent sheaves are modules, Module sheaf on an affine scheme, Localisation of modules is extension of scalars, Associativity of tensor products for compatible bimodules, The regular module is a tensor unit: $R{\otimes }_{R}N\cong N$ and $M{\otimes }_{R}R\cong M$, Tensor products commute with arbitrary direct sums)

Fitting ideals control fibre dimensions: let $\mathcal{G}$ be a quasi-coherent ${\mathcal{O}}_X$-module of finite type on a scheme $X$, with Fitting ideal sheaves ${\mathrm{Fitt}}_r(\mathcal{G})\subseteq {\mathcal{O}}_X$ and fibre $\mathcal{G}(x)={\mathcal{G}}_x\otimes \kappa (x)$; then for every $r\ge 0$ the locus $D({\mathrm{Fitt}}_r(\mathcal{G}))=\{x\in X:{\dim }_{\kappa (x)}\mathcal{G}(x)\le r\}$ is open in $X$. In particular, for a finitely generated module $N$ over a commutative ring $A$ and a prime $\mathfrak{p}\subseteq A$, the set of primes ${\mathfrak{p}}^{\mathrm{\prime }}$ with ${\dim }_{\kappa ({\mathfrak{p}}^{\mathrm{\prime }})}(N{\otimes }_A\kappa ({\mathfrak{p}}^{\mathrm{\prime }}))\le {\dim }_{\kappa (\mathfrak{p})}(N{\otimes }_A\kappa (\mathfrak{p}))$ is an open neighbourhood of $\mathfrak{p}$ in $\mathrm{Spec}A$, the fibre of $\tilde{N}$ being $N{\otimes }_A\kappa ({\mathfrak{p}}^{\mathrm{\prime }})$ by [F6]. (Fitting ideals control fibre generator loci, Fitting ideal sheaves, Fibre of a module sheaf at a point, Finite type and finitely presented module sheaves)

Upper semicontinuity and openness: a map $g:T\to \mathbb{R}$ on a topological space is upper semicontinuous exactly when every strict sublevel set $\{t:g(t)<a\}$ is open, equivalently when every superlevel set $\{g(t)\ge a\}$ is closed; the underlying topological space of a scheme is a topological space, and an arbitrary union of open sets is open, so a subset of a topological space that contains an open neighbourhood of each of its points is open. (Upper semicontinuous real map on a topological space, Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison, Schemes)

Cohomology and base change (used only in the comparison of 1.10): for the family of the statement, a point $s\in S$ and $q\ge 0$, the cohomology and base-change map is the $\kappa (s)$-linear map ${\phi }_s^q:(R^q{f}_{\ast }\mathcal{F})(s)\to H^q(X_s,{\mathcal{F}}_s)$ out of the fibre of the higher direct image sheaf (Cohomology and base-change map, Higher direct image of a sheaf, Fibre of a module sheaf at a point, Pullback of a module along a morphism of ringed spaces, The residue field at a point of an affine scheme). If ${\phi }_s^q$ and ${\phi }_s^{q-1}$ are surjective (the second condition being automatic for $q=0$), then $R^q{f}_{\ast }\mathcal{F}$ is locally free of finite rank in a neighbourhood of $s$, and there is an affine open neighbourhood $U\subseteq S$ of $s$ such that for every morphism $h:T\to U$, with $g_T:X_T\to X$ the projection, the base-change map $h^{\ast }(R^q{f}_{\ast }\mathcal{F}{\mathrm{\mid }}_U)\to R^q{f}_{T\ast }(g_T^{\ast }\mathcal{F})$ is an isomorphism of ${\mathcal{O}}_T$-modules; for $T=\mathrm{Spec}\kappa (u)$ with $u\in U$ the left side has global sections the fibre $(R^q{f}_{\ast }\mathcal{F})(u)$, by the affine localisation of higher direct images and the pullback formula for associated sheaves, and the right side has global sections $H^q(X_u,{\mathcal{F}}_u)$, so the fibre of the finite locally free sheaf $R^q{f}_{\ast }\mathcal{F}$ at $u$ is identified with $H^q(X_u,{\mathcal{F}}_u)$ and its dimension is the rank. (Cohomology and base change for proper flat coherent families, Higher direct images localize over an affine base, Scheme pullback preserves quasi-coherence, Affine quasi-coherent sheaves are modules, Module sheaf on an affine scheme, Locally free sheaves of finite rank, Base change of objects, morphisms and properties, Restricting fibre products to open subschemes)

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)