On growth in totally acyclic minimal complexes
Given a commutative Noetherian local ring, we provide a criterion under which a totally acyclic minimal complex of free modules has symmetric growth.
math.AC↗
arXiv subjects
Petter Andreas Bergh
Publications and source records attributed to Petter Andreas Bergh.
Dimensions of triangulated categories via Koszul objects
Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras.
math.CT↗
On complexes of finite complete intersection dimension
We study complexes of finite complete intersection dimension in the derived category of a local ring. Given such a complex, we prove that the thick subcategory it generates contains complexes of all possible complexities. In particular, we show that such a complex is virtually small, answering a question raised by Dwyer, Greenlees and Iyengar.
math.AC↗
Ext-symmetry over quantum complete intersections
We show that symmetry in the vanishing of cohomology holds for graded modules over quantum complete intersections. Moreover, symmetry holds for all modules if the algebra is symmetric.
math.KT↗
On the vanishing of cohomology in triangulated categories
We study the vanishing of cohomology in triangulated categories admitting a central ring action. In particular, we study vanishing gaps and symmetry.
math.CT↗
The Avrunin-Scott theorem for quantum complete intersections
We prove the Avrunin-Scott theorem for quantum complete intersections; the rank variety of a module is isomorphic to its support variety.
math.KT↗
Relative support varieties
We define relative support varieties with respect to some fixed module over a finite dimensional algebra. These varieties share many of the standard properties of classical support varieties. Moreover, when introducing finite generation conditions on cohomology, we show that relative support varieties contain homological information on the modules involved. As an application, we provide a new criterion for a selfinjective algebra to be of wild representation type.
math.RT↗
Cohomology of twisted tensor products
It is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies. We look at twisted tensor products, and investigate to which extend this is still true. We give an explicit description of the $\Ext$-algebra of the tensor product of two modules, and under certain additional conditions, describe an essential part of the Hochschild cohomology ring of a twisted tensor product. As an application, we characterize precisely when the cohomology groups over a quantum complete intersection are finitely generated over the Hochschild cohomology ring. Moreover, both for quantum complete intersections and in related cases we obtain a lower bound for the representation dimension of the algebra.
math.KT↗
Hochschild homology and global dimension
We prove that for certain classes of graded algebras (Koszul, local, cellular), infinite global dimension implies that Hochschild homology does not vanish in high degrees, provided the characteristic of the ground field is zero. Our proof uses Igusa's formula relating the Euler characteristic of relative cyclic homology to the graded Cartan determinant.
math.KT↗
Representation dimension and finitely generated cohomology
We consider selfinjective Artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators. For such an algebra, we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules. This provides a unified approach to computing lower bounds for the representation dimension of group algebras, exterior algebras and Artin complete intersections. We also obtain new examples of classes of algebras with arbitrarily large representation dimension.
math.RT↗
The representation dimension of quantum complete intersections
We study the representation dimension of the class of algebras known as quantum complete intersections. For such an algebra, we show that the representation dimension is at most twice its codimension. Moreover, we show that the representation dimension of a "homogeneous" quantum complete intersection is strictly larger than its codimension.
math.QA↗
(Co)homology of quantum complete intersections
We construct a minimal projective bimodule resolution for every finite dimensional quantum complete intersection of codimension two. Then we use this resolution to compute both the Hochschild cohomology and homology for such an algebra. In particular, we show that the cohomology vanishes in high degrees, while the homology is always nonzero.
math.KT↗
Modules with reducible complexity
For a commutative Noetherian local ring we define and study the class of modules having reducible complexity, a class containing all modules of finite complete intersection dimension. Various properties of this class of modules are given, together with results on the vanishing of homology and cohomology.
math.AC↗
On the Hochschild (co)homology of quantum exterior algebras
We compute the Hochschild cohomology and homology of a class of quantum exterior algebras, with coefficients in twisted bimodules. As a result we obtain several interesting examples of the homological behavior of these algebras.
math.KT↗
On support varieties for modules over complete intersections
We show that, over a local complete intersection, every possible variety is realized as the cohomological support variety of some module. Moreover, we show that the projective variety of a complete indecomposable maximal Cohen-Macaulay module is connected.
math.AC↗
Twisted support varieties
We define and study twisted support varieties for modules over an Artin algebra, where the twist is induced by an automorphism of the algebra. Under a certain finite generation hypothesis, we show that the twisted variety of a module satisfies Dade's Lemma and is one dimensional precisely when the module is periodic with respect to the twisting automorphism. As a special case we obtain results on DTr-periodic modules over Frobenius algebras.
math.RA↗
Complexity test modules
A method is provided for computing an upper bound of the complexity of a module over a local ring, in terms of vanishing of certain cohomology modules. We then specialize to complete intersections, which are precisely the rings over which all modules have finite complexity.
math.AC↗