Seminormality and seminormalization

PDF p. 5: Seminormalization in Lemma 2.4

A commutative ring $A$ is seminormal if, whenever $x^3=y^2$ in $A$, there is a unique $a\in A$ with $a^2=x$ and $a^3=y$. A scheme is seminormal if every point has an affine open neighbourhood whose coordinate ring is seminormal. In particular it is reduced: its affine coordinate rings have no nonzero nilpotents.

A morphism $f:Y\to X$ is subintegral if it is integral, is bijective on underlying points, and induces an isomorphism $\kappa (x)\to \kappa (y)$ at every point $y$ over $x$. Integral means affine and given on affine inverse images by integral ring maps; an element is integral when it satisfies a monic polynomial over the base ring. A universal homeomorphism is a morphism whose pullback along every scheme morphism is a homeomorphism on underlying spaces. Integral, bijective morphisms inducing residue-field isomorphisms are universal homeomorphisms, and conversely a universal homeomorphism with residue-field isomorphisms is integral and bijective.

For a scheme $X$, its seminormalization $X^{\mathrm{s}\mathrm{n}}\to X$ is the initial object in the category of subintegral morphisms to $X$: it has a unique $X$-morphism to every object of that category. Its source is seminormal. The construction is functorial: every morphism $h:Y\to X$ induces a unique compatible $h^{\mathrm{s}\mathrm{n}}:Y^{\mathrm{s}\mathrm{n}}\to X^{\mathrm{s}\mathrm{n}}$. The map factors through the reduction $X_{\mathrm{r}\mathrm{e}\mathrm{d}}$, the scheme obtained by quotienting the structure sheaf by its nilradical. When $X$ is locally of finite type over a field, the resulting map $X^{\mathrm{s}\mathrm{n}}\to X_{\mathrm{r}\mathrm{e}\mathrm{d}}$ is finite. In this scope a normal scheme is seminormal, and a reduced scheme is seminormal exactly when every finite subintegral morphism to it from a reduced scheme is an isomorphism.

The initial object and functoriality are the established input Seminormalization and the finite-type interface; Uniqueness of an initial object proves uniqueness up to unique isomorphism. The standard characterization of universal homeomorphisms as integral, surjective and universally injective identifies the two formulations: residue-field isomorphisms make the fibers universally injective. A map from a reduced scheme factors uniquely through the reduction, because its structure sheaf has no nonzero nilpotents.

The square-cube condition implies reducedness by The square-cube condition implies reducedness, while Uniqueness from squares and cubes explains uniqueness in any reduced ring. As a local illustration, every integrally closed domain satisfies the condition by Square-cube roots in a normal domain. The scheme-level normality implication is supplied by the cited input. Its finite-type construction uses an intermediate ring in a finite normalization; Intermediate rings in finite algebras supplies the noetherian finiteness step. If a reduced finite-type scheme passes the finite-morphism test, apply that test to its finite seminormalization; the resulting isomorphism makes it seminormal. Conversely, the input proves the stronger test for all subintegral morphisms with reduced source.