Euler characteristic in a proper flat family is locally constant

Given: The Axiom of Choice and the Axiom of Dependent Choice, a proper morphism of finite presentation $f:X\to S$ with $S$ arbitrary, an ${\mathcal{O}}_X$-module $\mathcal{F}$ of finite presentation that is flat over $S$, 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 $\kappa (\mathfrak{m})\cong A_{\mathfrak{m}}\mathrm{/}\mathfrak{m}{A}_{\mathfrak{m}}$; 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)

The fibres are proper over their residue fields and the pulled-back sheaves are coherent: for $s\in S$ the base change $X_s\to \mathrm{Spec}\kappa (s)$ of the proper morphism $f$ is proper, hence of finite type and quasi-compact, so $X_s$ is covered by affine charts $\mathrm{Spec}B$ with $B$ a finitely generated $\kappa (s)$-algebra; a field is Noetherian and a finitely generated algebra over a Noetherian ring is Noetherian, so each such $B$ is Noetherian and $X_s$ is locally Noetherian. The pullback ${\mathcal{F}}_s$ is quasi-coherent, and it is of finite type: over an affine chart $\mathrm{Spec}B$ of $X_s$ mapping into an affine chart $\mathrm{Spec}{B}_0$ of $X$ on which $\mathcal{F}\cong \tilde{M}$ with $M$ finitely generated, the pullback is the associated sheaf of $B{\otimes }_{B_0}M$, which is finitely generated because extension of scalars is right exact; on the locally Noetherian scheme $X_s$ a quasi-coherent module of finite type is coherent. (Properness survives arbitrary base change, Proper morphisms, Locally finite type and finite type morphisms, A field has only the zero ideal and itself, hence is Noetherian, Every algebra of finite type over a Noetherian ring is a Noetherian ring, Locally Noetherian and Noetherian schemes, Scheme pullback preserves quasi-coherence, Affine quasi-coherent sheaves are modules, Module sheaf on an affine scheme, Tensoring is right exact, Coherent sheaves on a locally Noetherian scheme, Finite type and finitely presented module sheaves)

Rank-nullity: for a linear map $T:V\to W$ of vector spaces over a field whose domain $V$ is finite-dimensional, $\dim V=\dim \ker T+\dim \mathrm{im}T$, both summands being natural numbers; in particular the image of a surjection from a finite-dimensional vector space is finite-dimensional. (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)

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, for a finitely generated projective $N$, choose lifts of a basis of the finite-dimensional residue vector space $N\mathrm{/}\mathfrak{n}N$. Nakayama makes the resulting map $R^m\to N$ surjective, and projectivity splits it. Its kernel $N^{\mathrm{\prime }}$ is a direct summand of $R^m$, hence finitely generated. After reducing this split sequence modulo $\mathfrak{n}$, the map on the chosen basis is an isomorphism, so $N^{\mathrm{\prime }}\mathrm{/}\mathfrak{n}{N}^{\mathrm{\prime }}=0$. Nakayama now gives $N^{\mathrm{\prime }}=0$, hence $N\cong R^m$ 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 $r\ge 0$ the locus $Z_r=\{x\in \mathrm{Spec}A:{\tilde{P}}_x\cong {\mathcal{O}}_{\mathrm{Spec}A,x}^r\}$ is open and $\tilde{P}$ is locally free of rank $r$ on it. For the prime $\mathfrak{p}\subseteq A$ corresponding to $x$, the stalk is ${\tilde{P}}_x\cong P_{\mathfrak{p}}$ and the fibre is $\tilde{P}(x)=P_{\mathfrak{p}}{\otimes }_{A_{\mathfrak{p}}}\kappa (\mathfrak{p})\cong P{\otimes }_A\kappa (\mathfrak{p})$; on the free locus of rank $r$ this fibre is $\kappa (\mathfrak{p}{)}^r$, so the function $\mathfrak{p}\mapsto {\dim }_{\kappa (\mathfrak{p})}(P{\otimes }_A\kappa (\mathfrak{p}))$ is constant with value $r$ on $Z_r$. (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)

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)