Clopen affine pieces and the zero-dimensional residue-point basis

Given: The field, scheme, clopen subscheme, Choice and compatible size assumptions in the Statement.

Goal: Check the clopen geometric hypotheses, then prove the field decomposition and its actual K-class normalization.

The Axiom of Choice is assumed for the coordinate-ring, idempotent, affine compactness, component, dimension-zero field and smoothness suppliers below and for the perfect K-theory and motivic constructions. Fix their compatible Grothendieck universes with the closure convention of Grothendieck universe closure convention; their existence and required size bounds are separate assumptions. All component fields and complexes stay within those bounds. Only finite grouping and explicitly described coordinate maps are added locally.

Normal dimension strata and the actual residue class model supplies the finite clopen integral components of $X$ and the positive residue-class compatibility for exact clopen zero extension in the current perfect K-model. Here normal means that every local ring is an integrally closed domain, clopen means open and closed, and dimension is the supremum of lengths of strict chains of nonempty irreducible closed subsets. A pure-dimensional scheme has all its irreducible components of the specified dimension. An integral scheme is nonempty, reduced and irreducible, equivalently each nonempty affine open is the spectrum of a domain, as in Integral schemes.

Write $X=\mathrm{Spec}A$ with its structure sheaf. The structure morphism to $\mathrm{Spec}k$ is finite type in the sense of Locally finite type and finite type morphisms: locally of finite type and quasi-compact. Here locally of finite type means that every source point has an affine open neighborhood $\mathrm{Spec}B$ mapping into an affine target neighborhood $\mathrm{Spec}R$ for which $B$ is a finitely generated $R$-algebra. A finitely generated algebra is one whose elements are polynomial expressions in a fixed finite list of generators with coefficients in the base ring. An affine scheme of finite type over a field has a finitely generated coordinate ring makes $A$ a finitely generated $k$-algebra. A clopen decomposition of the spectrum comes from a nontrivial idempotent expresses any clopen subset as $D(1-e)$ for an idempotent $e\in A$, and A principal localization identifies its spectrum with a distinguished open identifies its open subscheme with $\mathrm{Spec}{A}_{1-e}$. This localization is generated over $k$ by generators of $A$ and $(1-e{)}^{-1}$. Its spectrum is therefore locally of finite type over $k$ by the affine-chart definition just stated. The target $\mathrm{Spec}k$ has one point by [F5], so quasi-compactness of the source also gives quasi-compactness of this structure morphism; together these imply finite type. Every affine scheme is quasi-compact and Affine schemes are quasi-separated give the two qcqs conditions: quasi-compactness, and quasi-compact intersections of affine opens. A field has only the zero ideal and itself, hence is Noetherian and Every algebra of finite type over a Noetherian ring is a Noetherian ring make these coordinate algebras Noetherian.

For a prime $\mathfrak{p}\subset B$, put $V(\mathfrak{p})=\{\mathfrak{q}\in \mathrm{Spec}B:\mathfrak{p}\subseteq \mathfrak{q}\}$. A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point identifies these as the nonempty irreducible closed subsets, with their unique prime generic points. Global functions on Spec A recover A identifies $B$ with the global sections on $\mathrm{Spec}B$. A scheme is locally ringed, so each stalk is local by A locally ringed space. By The stalk of a presheaf at a point, a stalk is the colimit of sections over the open neighborhoods of its point. On a one-point space the stalk equals global sections, since the whole space is its only nonempty open neighborhood. A Noetherian local domain has dimension zero exactly when it is a field applies to a Noetherian local domain of ring dimension zero, meaning that there is no strict inclusion between prime ideals, and makes it a field under [A1].

A field finitely generated as a k-algebra is a finite extension of k makes a field finitely generated as a $k$-algebra finite over $k$. Perfectness means that every algebraic extension is separable; the implication from the irreducible-polynomial definition is Every algebraic extension of a perfect field is separable. For a Noetherian local ring with maximal ideal $\mathfrak{m}$ and residue field $L$, regular local means ${\dim }_L(\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2)=\dim R$, as in embedding dimension and regular local ring. Regular equals smooth over a perfect field makes a regular finite-type scheme over the perfect field smooth in its local-standard-smooth convention, under [A1]. Clause 1 of Classical and scheme smoothness over a perfect field identifies this with scheme smoothness for every finite-type field-scheme. The resulting smoothness is that of Smooth morphism of schemes: local finite presentation, flatness and geometric regularity of the fibers. The field components below are clopen smooth neighborhoods of their points.

The spectrum of a field is a one-point affine scheme identifies the unique point, local ring, residue field and global sections of $\mathrm{Spec}L$ for every field $L$. Affine perfect K zero and bundle K zero identifies its spectral $K_0$ with the vector-bundle Grothendieck group and preserves degree-zero bundle classes. On this one-point ringed space, finite locally free sheaves and their short exact sequences are exactly finite-dimensional $L$-vector spaces and their short exact sequences. The Grothendieck ring of finite-dimensional vector spaces identifies their group with $\mathbb{Z}$ by dimension, taking $[L]$ to $1$.

Finite clopen restrictions split the actual motivic image filtration supplies finite clopen restriction decompositions in actual $K_0$ and its image filtration, including the empty partition, with exact zero extension giving the coordinate inclusions. Motivic filtration and image groups identifies stage zero with $K$ and defines $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^0{K}_0$ as the image of that identification, hence as all of $K_0$.