Global functions on geometrically connected and geometrically reduced proper schemes

Given: A field $k$ with algebraic closure $\bar{k}$, a nonempty scheme $X$ proper over $k$ whose geometric fibre $X_{\bar{k}}=X{\times }_{\mathrm{Spec}k}\mathrm{Spec}\bar{k}$ is connected and reduced, and the rings $B=H^0(X,{\mathcal{O}}_X)$ and $R=H^0(X_{\bar{k}},{\mathcal{O}}_{X_{\bar{k}}})$; the Axiom of Choice is declared in the statement.

Properness unpacked: a proper morphism is separated, of finite type and universally closed, and a morphism of finite type is locally of finite type, so every point of $X$ has an affine open chart $\mathrm{Spec}A$ with $A$ a finitely generated $k$-algebra; the field $k$ is Noetherian and a finitely generated algebra over a Noetherian ring is Noetherian, so these coordinate rings are Noetherian and $X$ is locally Noetherian. (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)

Over a locally Noetherian scheme a quasi-coherent module of finite type is coherent; the structure sheaf is quasi-coherent, being $\tilde{A}$ on an affine chart $\mathrm{Spec}A$, and of finite type, being generated by the unit section in the local form of finite generation by finitely many sections. Hence ${\mathcal{O}}_X$ is coherent. (Quasi-coherent module on a scheme, Finite type and finitely presented module sheaves, Coherent module sheaves, Coherent sheaves on a locally Noetherian scheme)

For a field $F$, a scheme $Y$ proper over $F$ and a coherent $\mathcal{G}$ on $Y$, every $H^q(Y,\mathcal{G})$ is a finite-dimensional $F$-vector space and the groups vanish past the length of any finite affine open cover of $Y$; in particular $H^0(Y,\mathcal{G})$ is a finite-dimensional $F$-vector space. (Finite-dimensional coherent cohomology over a field, Sheaf cohomology as right derived global sections, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis)

A nonempty scheme has a point $x$, and every point of a scheme has an affine open neighbourhood, so there is an affine open $U\ni x$ with $(U,{\mathcal{O}}_X{\mathrm{\mid }}_U)\cong \mathrm{Spec}A$ and ${\mathcal{O}}_X(U)\cong A$; since $\mathrm{Spec}0=\mathrm{\varnothing }$, this forces $A\mathrm{\ne }0$, so $1_A\mathrm{\ne }0$. The restriction map $\rho :{\mathcal{O}}_X(X)\to {\mathcal{O}}_X(U)$ is a unital ring homomorphism, being a restriction map of the sheaf of rings ${\mathcal{O}}_X$, so $\rho (1_X)=1_A\mathrm{\ne }0$ and $1_X\mathrm{\ne }0$: the ring $H^0(X,{\mathcal{O}}_X)$ is nonzero. (Schemes, The underlying space of an affine spectrum, Global functions on Spec A recover A, Presheaves and sheaves of groups, rings, and modules)

Base change along the field extension $\bar{k}\mathrm{/}k$: the base-change map ${\kappa }^q:H^q(X,{\mathcal{O}}_X){\otimes }_k\bar{k}\to H^q(X_{\bar{k}},{\mathcal{O}}_{X_{\bar{k}}})$ is an isomorphism for every $q\ge 0$, and $X_{\bar{k}}$ is proper over $\bar{k}$. It is the $\bar{k}$-linear extension of the pullback map on cohomology classes, and the pullback map on global sections is a unital ring homomorphism (pullback of structure-sheaf sections is composition of functions); since $B{\otimes }_k\bar{k}$ carries a commutative $\bar{k}$-algebra structure, the degree-zero map ${\kappa }^0$ is an isomorphism of unital commutative $\bar{k}$-algebras. (Flat field extension commutes with coherent cohomology, Cohomology and base-change map, Variance of sheaf cohomology, The tensor product of $R$-algebras has multiplication $(a\otimes b)(a^{\mathrm{\prime }}\otimes b^{\mathrm{\prime }})=a{a}^{\mathrm{\prime }}\otimes b{b}^{\mathrm{\prime }}$, Algebras over a commutative ring, central structure maps, and algebra homomorphisms)

Extension of scalars of a finite-dimensional space: for a finite-dimensional $k$-vector space $V$ with a basis $v_1,\dots ,v_n$, the $k$-bilinear map $V{\otimes }_k\bar{k}\to {\bar{k}}^n$ with $v_i\otimes c\mapsto c{e}_i$ is an isomorphism, so ${\dim }_{\bar{k}}(V{\otimes }_k\bar{k})={\dim }_{k}V$; this uses only the unit and direct-sum compatibilities of the tensor product together with right exactness. (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, Every spanning subset of a vector space contains a basis, Linear combination of a finite list, and the span $\mathrm{span}(S)$ as the smallest linear subspace containing $S$, 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, Tensoring is right exact)

Geometric fibres: for $X\to \mathrm{Spec}k$ the fibre at the unique point $s$ is $X_s=X{\times }_{\mathrm{Spec}k}\mathrm{Spec}\kappa (s)\cong X$ because $\kappa (s)=k$, so the geometric fibre is $X_{\bar{s}}=X_s{\times }_{\mathrm{Spec}k}\mathrm{Spec}\bar{k}\cong X{\times }_{\mathrm{Spec}k}\mathrm{Spec}\bar{k}=:X_{\bar{k}}$, and the tests for the chosen algebraic closure agree up to isomorphism with those for any other choice. By the convention of that definition, "geometrically connected" and "geometrically reduced" for this fibre mean that $X_{\bar{k}}$ is connected, respectively reduced, where reducedness of a scheme means that all its local rings have no nonzero nilpotent element. An affine scheme $\mathrm{Spec}A$ is reduced exactly when $A$ is reduced. (Geometric fibres and geometric points, Geometric properties of fibres, Scheme-theoretic fibre, Independence of the chosen algebraic closure, Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets, The reduction of a scheme, Reduced affine schemes)

A reduced scheme has a reduced ring of global sections: if $f\in H^0(Y,{\mathcal{O}}_Y)$ satisfies $f^n=0$, then each germ $f_y$ is nilpotent; in a reduced scheme the local rings have no nonzero nilpotent element, so $f_y=0$ for every $y$; a section of a sheaf of groups whose germs all vanish is zero. Hence $H^0(Y,{\mathcal{O}}_Y)$ has no nilpotent element other than $0$. (The nilradical and reduced rings, The reduction of a scheme, Germs of sections, A section of a sheaf of groups is zero exactly when all of its germs are zero)

Idempotents and connectedness: in a local ring the only idempotents are $0$ and $1$; if $e\in H^0(Y,{\mathcal{O}}_Y)$ is idempotent on a scheme $Y$, then every germ $e_y$ in the local ring ${\mathcal{O}}_{Y,y}$ is $0$ or $1$, the sets $Y_1=\{y:e_y=1\}$ and $Y_0=\{y:e_y=0\}$ are disjoint with union $Y$, and both are open, since on an affine open $U=\mathrm{Spec}A$ the identification ${\mathcal{O}}_{U,\mathfrak{p}}\cong A_{\mathfrak{p}}$ shows $Y_1\cap U=D(e{\mathrm{\mid }}_U)$ and $Y_0\cap U=D(1-e{\mathrm{\mid }}_U)$, which are clopen in $U$; if $e\mathrm{\ne }0,1$ then both are nonempty, because $Y_1=\mathrm{\varnothing }$ forces all germs of $e$ to vanish and hence $e=0$, while $Y_0=\mathrm{\varnothing }$ gives $1-e=0$. Consequently a connected scheme has no idempotent global section other than $0$ and $1$, and for a commutative ring $A$ the spectrum $\mathrm{Spec}A$ is connected if and only if $A$ has no idempotents other than $0$ and $1$. (Assuming the Axiom of Choice, a local ring has no idempotents other than $0$ and $1$, An idempotent partitions the spectrum into complementary clopen subsets, The stalk of the affine structure sheaf at a prime is the localization at that prime, A section of a sheaf of groups is zero exactly when all of its germs are zero, The prime spectrum is connected exactly when the ring has no idempotents other than zero and one, Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets)

Artinian rings and local rings: a finite-dimensional commutative algebra over a field is Artinian, since a strictly descending chain of ideals is a strictly descending chain of subspaces and a strict inclusion of finite-dimensional spaces strictly lowers the dimension; a nonzero commutative ring has a maximal ideal, an Artinian ring has only finitely many maximal ideals, its nilradical is nilpotent, and for a commutative Artinian ring $R\mathrm{\ne }0$ with maximal ideals ${\mathfrak{m}}_1,\dots ,{\mathfrak{m}}_r$ the canonical maps $R\to {\prod }_{j=1}^r{R}_{{\mathfrak{m}}_j}$ and $R\to {\prod }_{j=1}^{r}R\mathrm{/}{\mathfrak{m}}_j^n$ (for $n$ with $\mathrm{Nil}(R{)}^n=0$) are isomorphisms, with every factor $R\mathrm{/}{\mathfrak{m}}_j^n$ nonzero because ${\mathfrak{m}}_j^n\subseteq {\mathfrak{m}}_j\mathrm{\ne }R$; a commutative Artinian local ring has nilpotent maximal ideal; a local ring is a nonzero commutative ring with exactly one maximal ideal, which is the set of its nonunits; and a reduced ring has no nilpotent element other than $0$. (Left and right Artinian rings, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, In a nonzero commutative ring, every proper ideal is contained in a maximal ideal, An Artinian ring has only finitely many maximal ideals, The nilradical of an Artinian ring is a nilpotent ideal, An Artinian ring is canonically the finite product of its localizations at its maximal ideals, An Artinian local ring has nilpotent maximal ideal, and its finite modules have finite length, A local ring is a nonzero commutative ring with a unique maximal ideal, The nilradical and reduced rings)

Algebraically closed fields: a field $F$ is algebraically closed exactly when it has no nontrivial finite extension; every nonconstant polynomial over an algebraically closed field has a root in that field; and the Frobenius map of a field of characteristic $p>0$ is an injective field endomorphism, so $(a+b{)}^p=a^p+b^p$ and its $n$-fold iterate is $x\mapsto x^{p^n}$. (A field is algebraically closed exactly when every nonconstant polynomial splits, equivalently when it has no nontrivial finite extension, An algebraically closed field: every nonconstant polynomial has a root in the field, Frobenius $x\mapsto x^p$ is an injective endomorphism in characteristic $p$, and an automorphism for finite fields)

Algebras, quotients, affine fibre products and finite morphisms: for commutative $R$-algebras $A,B$ the tensor product $A{\otimes }_{R}B$ is a commutative $R$-algebra; tensoring the presentation $k[T]\mathrm{/}(f)$ of a quotient with a $k$-algebra $C$ gives $(k[T]\mathrm{/}(f)){\otimes }_{k}C\cong C[T]\mathrm{/}(f)$, by right exactness of the tensor product and $k[T]{\otimes }_{k}C\cong C[T]$, and more generally $M{\otimes }_R(R\mathrm{/}I)\cong M\mathrm{/}IM$; for affine schemes $\mathrm{Spec}B{\times }_{\mathrm{Spec}k}\mathrm{Spec}C\cong \mathrm{Spec}(B{\otimes }_{k}C)$ and $\mathrm{\Gamma }(\mathrm{Spec}A,\mathcal{O})\cong A$; and every finite morphism of schemes, meaning over affine opens it is the spectrum of a module-finite algebra, is proper. (The tensor product of $R$-algebras has multiplication $(a\otimes b)(a^{\mathrm{\prime }}\otimes b^{\mathrm{\prime }})=a{a}^{\mathrm{\prime }}\otimes b{b}^{\mathrm{\prime }}$, Universal mapping property of the tensor product of commutative algebras, $M{\otimes }_{R}R\mathrm{/}I\cong M\mathrm{/}IM$ naturally, Tensoring is right exact, Affine fibre products are spectra of tensor products, Global functions on Spec A recover A, Finite morphisms of schemes, Finite morphisms are proper)

The inseparable example: for a prime $p$ the extension ${\mathbb{F}}_p(t)\mathrm{/}{\mathbb{F}}_p(t^p)$ is purely inseparable of degree $p$; writing $u=t^p$ and $k={\mathbb{F}}_p(u)={\mathbb{F}}_p(t^p)$, the element $u$ is not a $p$-th power in $k$ and $L=k[T]\mathrm{/}(T^p-u)\cong {\mathbb{F}}_p(t)$ is a field with $t^p=u$; and the base-change computation for such a field, $L{\otimes }_{k}L\cong L[u]\mathrm{/}(u^p)$ with $u\mathrm{\ne }0$ and $u^p=0$, exhibits the base change of a nontrivial purely inseparable extension as a nonreduced local ring. (${\mathbb{F}}_p(t)\mathrm{/}{\mathbb{F}}_p(t^p)$ is purely inseparable of degree $p$ and separable degree one, Base change of an inseparable field extension is a thickening, Purely inseparable algebraic extensions)

Polynomials: over a field the polynomial ring $\bar{k}[Y]$ is an integral domain, and in a polynomial ring over a domain the degree of a product of nonzero polynomials is the sum of the degrees; hence $Y\mathrm{\notin }(Y^p)$ for $p\ge 2$, because a nonzero multiple $Y^{p}g$ has degree at least $p>1=\mathrm{deg}Y$. (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, A polynomial ring over an integral domain is an integral domain, Over an integral domain, degrees add under multiplication of nonzero polynomials)

Nilpotent ideals with field quotient: if $R$ is a commutative ring and $I\subseteq R$ is an ideal with $I^n=0$ for some $n\ge 1$ while $R\mathrm{/}I$ is a field, then $I$ is a maximal ideal of $R$, and every element of $I$ is nilpotent, hence lies in the nilradical, which is the intersection of all prime ideals, while every maximal ideal is prime; so every maximal ideal contains $I$, making $I$ the unique maximal ideal and $R$ a local ring. Moreover ideals of a quotient ring correspond to ideals containing the kernel, and a unital ring homomorphism out of a polynomial ring is determined by the image of the variable, which identifies $\bar{k}[Y]\mathrm{/}(Y)\cong \bar{k}$ through evaluation at $0$. (Correspondence theorem: ideals of $R\mathrm{/}I$ correspond to ideals of $R$ containing $I$, The nilradical is the intersection of all prime ideals, Every maximal ideal of a commutative ring is prime, A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring, Evaluation and roots of a polynomial in a commutative target ring, A local ring is a nonzero commutative ring with a unique maximal ideal)