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Mohan Kumar–Murthy rank-three Chow classification

Statement

Assume AC. Let $k$ be an algebraically closed field of arbitrary characteristic and $Y$ a smooth affine threefold over $k$ . Then $C_{3} : {V e c t}_{3} (Y) \to {C H}^{1} (Y) \times {C H}^{2} (Y) \times {C H}^{3} (Y)$ , $[G]_{i s o} \mapsto (c_{1} (G), c_{2} (G), c_{3} (G))$ , is bijective. The classes are the ordinary integral Chow Chern classes with the quotient-projective-bundle normalization. Thus every triple occurs, and two such bundles are isomorphic exactly when their triples agree.

External input

Only the established classification statement is imported, via the exact authoritative subsequent statement. Standard ordinary Chern classes, rather than abstract unnamed isomorphic invariants, are specified there. The local projective-bundle coefficient interface fixes the same sign convention. This is not a comparison with a singular motivic Chern operation. The surface branch below uses a cylinder reduction; it does not claim to reproduce either the original Murthy–Swan argument or the original Mohan Kumar–Murthy proof.

Facts & Assumptions

Given: The data and hypotheses in the statement.

Goal: Establish the stated conclusion.

[F1]

The exact classical rank-three Chern bijection is the external input stated by Asok–Fasel p. 2 and attributed to Mohan Kumar–Murthy.

[F2]

The ordinary Chow Chern map and its coefficient convention are fixed in The ordinary Chow Chern map on isomorphism classes.

Proof

1.1F1F2given

Apply the external rank-three classification with the given algebraically closed field and smooth affine threefold. The recalled classical theorem imposes no characteristic restriction; the separate nearby rank-two theorem is not being applied. Its map is the ordinary Chern tuple map fixed by [F2].

2.1F1F2step 1.1∎

Surjectivity supplies a rank-three bundle for each triple, and injectivity says that equal triples have the same bundle isomorphism class. Conversely an isomorphism gives equal Chern classes by [F2]. These are the asserted existence and uniqueness conclusions.

Depends on

Used by · 1

Classical smooth critical-rank classifications

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

External dependencies

Aravind Asok and Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds · p. 2, Introduction, paragraph beginning For context, immediately after Theorem 1; compare v1 p. 2 opening paragraph

For a smooth affine threefold X over an algebraically closed field, the map assigning the ordinary Chow Chern triple to a rank-three bundle is a bijection onto CH^1(X) x CH^2(X) x CH^3(X); this is attributed there to N. Mohan Kumar and M. P. Murthy [KM82].

Hypotheses: X a smooth affine threefold over an algebraically closed field, with no characteristic exclusion in the recalled classical result. Bundles have rank three. The nearby Theorem 1 has characteristic unequal to two for rank-two bundles on threefolds, and is not the theorem imported here.

Reading limits: Read v5 p. 2 introduction, including the exact displayed rank-three map and the surrounding distinction from rank-two Theorem 1; also read v1 p. 2 introduction, which starts with arbitrary algebraically closed k and states unique rank-three existence for given Chern classes. The Mohan Kumar–Murthy original paper, Ann. Math. 116 (1982), 579–591, DOI 10.2307/2007024, Theorem 2.1(iii), was not accessed beyond its publisher metadata. No original proof or obstruction-theory reconstruction is claimed.

Sources

Aravind Asok and Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds · p. 2, Introduction, paragraph beginning For context, immediately after Theorem 1; compare v1 p. 2 opening paragraph