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Suslin cancellation over affine algebras

Statement

Let $k$ be an algebraically closed field of arbitrary characteristic, let $A$ be a commutative unital finite-type $k$ -algebra of Krull dimension $d$ , and let $P$ be a finite projective $A$ -module of constant rank $r \geq d$ . Then $P$ is cancellative: for every finite projective $Q$ and every integer $N \geq 0$ , an isomorphism $P \oplus A^{N} ≅ Q \oplus A^{N}$ implies $P ≅ Q$ .

External input

This is an explicit established external input, attributed to Suslin 1977 through Keshari. The numbered argument only translates the exact statement and cancellation convention. The five local application items establish the separate affine/K-zero application interface; they do not reconstruct Suslin's deep cancellation argument. The factorial restriction in Keshari's abstract concerns a different rank-(d-1) problem and is absent from Theorem 1.1(i).

Facts & Assumptions

Given: The data and hypotheses in the statement.

Goal: Establish the stated conclusion.

[F1]

The external input is exactly Keshari Theorem 1.1(i), with cancellative defined immediately above it.

Proof

1.1F1given

The affine algebra, base field and rank bound are the hypotheses of the imported theorem. Its conclusion says precisely that $P$ is cancellative in the source definition.

2.1F1step 1.1∎

Expand the source definition in [F1], which explicitly uses a common finite free summand. It yields the displayed implication for every common finite free stabilization, including $N = 0$ .

Depends on

Manoj Kumar Keshari, Cancellation problem for projective modules over affine algebras · p. 1, Introduction: definition of cancellative and Theorem 1.1(i); p. 16, reference [20]

Used by · 1

Classical smooth critical-rank classifications

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

Manoj Kumar Keshari, Cancellation problem for projective modules over affine algebras · p. 1, Introduction: definition of cancellative and Theorem 1.1(i); p. 16, reference [20]

Over a finite-type affine algebra A of dimension d over an algebraically closed field, every finite projective A-module of constant rank at least d is cancellative: a common finite free stabilization can be removed.

Hypotheses: A commutative unital affine algebra over an algebraically closed field of arbitrary characteristic, finite Krull dimension d; finitely generated projective modules of constant rank r >= d. No smoothness, normality, reducedness or invertibility of d! is assumed in Theorem 1.1(i).

Reading limits: In this adjudication, read arXiv:0708.2961v1 p. 1 definition and Theorem 1.1(i). This is the exact attributed external theorem; Suslin’s original 1977 proof and Keshari’s later proofs were not reconstructed. Earlier authoring reported additional bibliography/hosted-introduction inspection; that is historical, not fresh reading here.

Sources

Manoj Kumar Keshari, Cancellation problem for projective modules over affine algebras · p. 1, Introduction: definition of cancellative and Theorem 1.1(i); p. 16, reference [20]