Mathematical Research Letters

Volume 21 (2014)

Number 4

Jump loci in the equivariant spectral sequence

Pages: 863 – 883

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n4.a13

Authors

Stefan Papadima (Simion Stoilow Institute of Mathematics, Bucharest, Romania)

Alexander I. Suciu (Department of Mathematics, Northeastern University, Boston, Massachusetts, U.S.A.)

Abstract

We study the homology jump loci of a chain complex over an affine $\mathbb{k}$-algebra. When the chain complex is the first page of the equivariant spectral sequence associated to a regular abelian cover of a finite-type CW-complex, we relate those jump loci to the resonance varieties associated to the cohomology ring of the space. As an application, we show that vanishing resonance implies a certain finiteness property for the completed Alexander invariants of the space. We also show that vanishing resonance is a Zariski open condition, on a natural parameter space for connected, finite-dimensional commutative graded algebras.

Keywords

affine algebra, maximal spectrum, homology jump loci, support varieties, equivariant spectral sequence, resonance variety, characteristic variety, Alexander invariants, completion

2010 Mathematics Subject Classification

Primary 55N25. Secondary 14M12, 20J05, 55Txx.

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