Cyclic Adams operations

Michael K. Brown; Claudia Miller; Peder Thompson; Mark E. Walker

doi:10.1016/j.jpaa.2016.12.018

Cyclic Adams operations

Michael K. Brown, Claudia Miller, Peder Thompson, Mark E. Walker

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

Let Q be a commutative, Noetherian ring and Z⊆Spec(Q) a closed subset. Define K₀^Z(Q) to be the Grothendieck group of those bounded complexes of finitely generated projective Q-modules that have homology supported on Z. We develop “cyclic” Adams operations on K₀^Z(Q) and we prove these operations satisfy the four axioms used by Gillet and Soulé in [9]. From this we recover a shorter proof of Serre's Vanishing Conjecture. We also show our cyclic Adams operations agree with the Adams operations defined by Gillet and Soulé in certain cases. Our definition of the cyclic Adams operators is inspired by a formula due to Atiyah [1]. They have also been introduced and studied before by Haution [10].

Original language	English (US)
Pages (from-to)	1589-1613
Number of pages	25
Journal	Journal of Pure and Applied Algebra
Volume	221
Issue number	7
DOIs	https://doi.org/10.1016/j.jpaa.2016.12.018
State	Published - Jul 1 2017

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jpaa.2016.12.018

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abstract = "Let Q be a commutative, Noetherian ring and Z⊆Spec(Q) a closed subset. Define K0Z(Q) to be the Grothendieck group of those bounded complexes of finitely generated projective Q-modules that have homology supported on Z. We develop “cyclic” Adams operations on K0Z(Q) and we prove these operations satisfy the four axioms used by Gillet and Soul{\'e} in [9]. From this we recover a shorter proof of Serre's Vanishing Conjecture. We also show our cyclic Adams operations agree with the Adams operations defined by Gillet and Soul{\'e} in certain cases. Our definition of the cyclic Adams operators is inspired by a formula due to Atiyah [1]. They have also been introduced and studied before by Haution [10].",

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AU - Brown, Michael K.

AU - Miller, Claudia

AU - Thompson, Peder

AU - Walker, Mark E.

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N2 - Let Q be a commutative, Noetherian ring and Z⊆Spec(Q) a closed subset. Define K0Z(Q) to be the Grothendieck group of those bounded complexes of finitely generated projective Q-modules that have homology supported on Z. We develop “cyclic” Adams operations on K0Z(Q) and we prove these operations satisfy the four axioms used by Gillet and Soulé in [9]. From this we recover a shorter proof of Serre's Vanishing Conjecture. We also show our cyclic Adams operations agree with the Adams operations defined by Gillet and Soulé in certain cases. Our definition of the cyclic Adams operators is inspired by a formula due to Atiyah [1]. They have also been introduced and studied before by Haution [10].

AB - Let Q be a commutative, Noetherian ring and Z⊆Spec(Q) a closed subset. Define K0Z(Q) to be the Grothendieck group of those bounded complexes of finitely generated projective Q-modules that have homology supported on Z. We develop “cyclic” Adams operations on K0Z(Q) and we prove these operations satisfy the four axioms used by Gillet and Soulé in [9]. From this we recover a shorter proof of Serre's Vanishing Conjecture. We also show our cyclic Adams operations agree with the Adams operations defined by Gillet and Soulé in certain cases. Our definition of the cyclic Adams operators is inspired by a formula due to Atiyah [1]. They have also been introduced and studied before by Haution [10].

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Cyclic Adams operations

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