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# 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

#### 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.