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Smooth affine components and gluing a trivial line

Statement

Assume AC. A nonempty smooth affine finite-type scheme $X$ over an algebraically closed field has a finite disjoint decomposition $X = ∐_{j} X_{j}$ into integral open and closed smooth affine schemes. Put $d = \dim X$ and $d_{j} = \dim X_{j} \leq d$ . Bundles, bundle isomorphisms and finite direct sums are given componentwise. Define ${C H}^{i} (X) = ⨁_{j} {C H}^{i} (X_{j})$ in the codimension convention; this agrees with the usual Chow ring for pure-dimensional $X$ . Chern classes are given componentwise. If a rank- $d$ bundle restricts to a bundle with a trivial line summand on every $X_{j}$ , these decompositions glue to a trivial line summand on $X$ . For $d_{j} < d$ , its restriction always splits off a line.

Facts & Assumptions

Given: The hypotheses and data in the statement, with AC and the article's fixed universes wherever perfect K-theory is used.

Goal: Establish the asserted conclusion.

[F1]

Smooth finite-type schemes over a perfect field are regular; regular local rings are domains. Regular equals smooth over a perfect field, regular local rings are domains and cohen macaulay

[F2]

A noetherian space has finitely many irreducible components. A Noetherian space is a finite union of irreducible closed subsets

[F3]

Chow groups are cycles modulo divisors of rational functions on integral closed subschemes. Rational equivalence and the Chow group of cycles

[F4]

A bundle of rank greater than the dimension on a noetherian affine scheme splits off a trivial line. Serre splitting above the dimension

Proof

1.1F1F2construct

A finite-type scheme over a field is noetherian. Distinct irreducible components cannot meet: at an intersection the local ring would have distinct minimal primes, contrary to being a domain by [F1]. Finiteness by [F2] makes each component both closed and open. Such a component is affine: glue the functions 1 and 0 on it and its complement to obtain a global idempotent, whose corresponding ring factor has that spectrum. Regularity makes each component reduced, hence integral.

2.1F3step 1.1construct

Sheaves on a finite disjoint open cover are precisely the tuples of their restrictions: there are no nonempty overlaps imposing further gluing conditions. Thus bundle maps and isomorphisms, and decompositions $E_{j} ≅ O_{X_{j}} \oplus F_{j}$ , glue. An integral closed subscheme and each rational-function relation lie in a single component, so the cycle and relation groups split componentwise. This justifies the stated Chow convention and componentwise Chern operations.

3.1F4step 1.1step 2.1∎

If $d_{j} < d$ , the restricted rank- $d$ bundle satisfies the strict rank inequality in [F4] and therefore splits off a line. This also handles all components below maximal dimension without a critical-rank theorem.

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Used by · 2

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