Petter Andreas Bergh

On support varieties for modules over complete intersections

This PhD-thesis consists of the five papers- On the Hochschild (co)homology of quantum exterior algebras, to appear in Comm. Algebra,-Complexity and periodicity, Coll. Math. 104 (2006), no. 2, 169-191,-Twisted support varieties,-Modules with reducible complexity, to appear in J. Algebra,- On support varieties for modules over complete intersections, to appear in Proc. Amer. Math. Soc.These papers are roughly divided into two groups; the ¯rst three study modules over Artin algebras using techniques from Hochschild cohomology, whereas the last two papers study modules over commutative Noetherian local rings, in particular modules over complete intersections.

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 Igusas formula relating the Euler characteristic of relative cyclic homology to the graded Cartan determinant.

Hochschild homology and truncated cycles

We study algebras having 2-truncated cycles and show that these algebras have infinitely many nonzero Hochschild homology groups. Consequently, algebras of finite global dimension have no 2-truncated cycles and therefore satisfy a higher version of the “no loops conjecture”.

Higher n-angulations from local rings

We show that the category of finitely generated free modules over certain local rings is n-angulated for every n at least 3. In fact, we construct several classes of n-angles, parametrized by equivalence classes of units in the local rings. Finally, we show that for odd values of n some of these n-angulated categories are not algebraic.

Tate-Hochschild homology and cohomology of Frobenius algebras

We study Tate-Hochschild homology and cohomology for a two-sided Noetherian Gorenstein algebra. These (co)homology groups are defined for all degrees, non-negative as well as negative, and they agree with the usual Hochschild (co)homology groups for all degrees larger than the injective dimension of the algebra. We prove certain duality theorems relating the Tate-Hochschild (co)homology groups in positive degree to those in negative degree, in the case where the algebra is Frobenius. We explicitly compute all Tate-Hochschild (co)homology groups for certain classes of Frobenius algebras, namely, certain quantum complete intersections.

On growth in minimal totally acyclic complexes

Given a commutative Noetherian local ring, we provide a criterion under which a minimal totally acyclic complex of free modules has symmetric growth. As a special case, we show that whenever an image in the complex has finite complete intersection dimension, then the complex has symmetric polynomial growth.

The negative side of cohomology for Calabi-Yau categories

We study integer-graded cohomology rings defined over Calabi-Yau categories. We show that the cohomology in negative degree is a trivial extension of the cohomology ring in non-negative degree, provided the latter admits a regular sequence of central elements of length two. In particular, the product of elements of negative degrees are zero. As corollaries we apply this to Tate-Hochschild cohomology rings of symmetric algebras, and to Tate cohomology rings over group algebras. We also prove similar results for Tate cohomology rings over commutative local Gorenstein rings.

The stable Auslander–Reiten quiver of a quantum complete intersection

We completely describe the tree classes of the components of the stable Auslander-Reiten quiver of a quantum complete intersection. In particular, we show that the tree class is always

Realizability and the Avrunin–Scott theorem for higher-order support varieties

A_{\infty}

A Generalized Dade’s Lemma for Local Rings

whenever the algebra is of wild representation type. Moreover, in the tame case, there is one component of tree class