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Alexander Zimmermann

Publications and source records attributed to Alexander Zimmermann.

34 records · Page 2Linked to original sources
arXiv2014

The Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra

Analogous to a recent result of N. Kowalzig and U. Krähmer for twisted Calabi-Yau algebras, we show that the Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra, thus generalizing a result of T.Tradler for finite dimensional symmetric algebras. We give a criterion to determine when a Frobenius algebra given by quiver with relations has semisimple Nakayama automorphism and apply it to some known classes of tame Frobenius algebras. We also provide ample examples including quantum complete intersections, finite dimensional Hopf algebras defined over an algebraically closed field of characteristic zero and Koszul duals of Koszul Artin-Schelter regular algebras of dimension three.

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arXiv2013

The Ring of Polynomial Functors of Prime Degree

Let $\hat\Z_p$ be the ring of $p$-adic integers. We prove in the present paper that the category of polynomial functors from finitely generated free abelian groups to $\hat \Z_p$-modules of degree at most $p$ is equivalent to the category of finitely generated modules over a particularly well understood ring, called Green order. That this is the case was conjectured by Yuri Drozd.

math.RT↗
arXiv2013

On singular equivalences of Morita type

Stable equivalences of Morita type preserve many interesting properties and is proved to be the appropriate concept to study for equivalences between stable categories. Recently the singularity category attained much attraction and Xiao-Wu Chen and Long-Gang Sun gave an appropriate definition of singular equivalence of Morita type. We shall show that under some conditions singular equivalences of Morita type have some biadjoint functor properties and preserve positive degree Hochschild homology.

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arXiv2010

On the use of Külshammer type invariants in representation theory

Since 2005 a new powerful invariant of an algebra emerged using earlier work of Horváth, Héthelyi, Külshammer and Murray. The authors studied Morita invariance of a sequence of ideals of the centre of a finite dimensional algebra over a field of finite characteristic. It was shown that the sequence of ideals is actually a derived invariant, and most recently a slightly modified version of it an invariant under stable equivalences of Morita type. The invariant was used in various contexts to distinguish derived and stable equivalence classes of pairs of algebras in very subtle situations. Generalisations to non symmetric algebras and to higher Hochschild (co-)homology was given. This article surveys the results and gives some of the constructions in more detail.

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arXiv2010

Deformed preprojective algebras of type L: Kuelshammer spaces and derived equivalences

Bialkowski, Erdmann and Skowronski classified those indecomposable self-injective algebras for which the Nakayama shift of every (non-projective) simple module is isomorphic to its third syzygy. It turned out that these are precisely the deformations, in a suitable sense, of preprojective algebras associated to the simply laced ADE Dynkin diagrams and of another graph L_n, which also occurs in the Happel-Preiser-Ringel classification of subadditive but not additive functions. In this paper we study these deformed preprojective algebras of type L via their Kuelshammer spaces, for which we give precise formulae for their dimensions. These are known to be invariants of the derived module category, and even invariants under stable equivalences of Morita type. As main application of our study of Kuelshammer spaces we can distinguish many (but not all) deformations of the preprojective algebra of type L up to stable equivalence of Morita type, and hence also up to derived equivalence.

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arXiv2010

Hochschild homology invariants of Külshammer type of derived categories

For a perfect field $k$ of characteristic $p>0$ and for a finite dimensional symmetric $k$-algebra $A$ Külshammer studied a sequence of ideals of the centre of $A$ using the $p$-power map on degree 0 Hochschild homology. In joint work with Bessenrodt and Holm we removed the condition to be symmetric by passing through the trivial extension algebra. If $A$ is symmetric then the dual to the Külshammer ideal structure was generalised to higher Hochschild homology in earlier work. In the present paper we follow this program and propose an analogue of the dual to the Külshammer ideal structure on the degree $m$ Hochschild homology theory also to not necessarily symmetric algebras.

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arXiv2010

Auslander-Reiten conjecture for symmetric algebras of polynomial growth

This paper studies self-injective algebras of polynomial growth. We prove that the derived equivalence classification of weakly symmetric algebras of domestic type coincides with the classification up to stable equivalences (of Morita type). As for weakly symmetric non-domestic algebras of polynomial growth, up to some scalar problems, the derived equivalence classification coincides with the classification up to stable equivalences of Morita type. As a consequence, we get the validity of the Auslander-Reiten conjecture for stable equivalences of Morita type between weakly symmetric algebras of polynomial growth.

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arXiv2008

Generalized Reynolds ideals and derived equivalences for algebras of dihedral and semidihedral type

Generalized Reynolds ideals are ideals of the center of a symmetric algebra over a field of positive characteristic. They have been shown by the second author to be invariant under derived equivalences. In this paper we determine the generalized Reynolds ideals of algebras of dihedral and semidihedral type (as defined by Erdmann), in characteristic 2. In this way we solve some open problems about scalars occurring in the derived equivalence classification of these algebras.

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arXiv2008

On Morita theory for self-dual modules

Let $G$ be a finite group and let $k$ be a field of characteristic $p$. It is known that a $kG$-module $V$ carries a non-degenerate $G$-invariant bilinear form $b$ if and only if $V$ is self-dual. We show that whenever a Morita bimodule $M$ which induces an equivalence between two blocks $B(kG)$ and $B(kH)$ of group algebras $kG$ and $kH$ is self-dual then the correspondence preserves self-duality. Even more, if the bilinear form on $M$ is symmetric then for $p$ odd the correspondence preserves the geometric type of simple modules. In characteristic 2 this holds also true for projective modules.

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arXiv2006

Fine Hochschild invariants of derived categories for symmetric algebras

Let $A$ be a symmetric $k$-algebra over a perfect field $k$. Külshammer defined for any integer $n$ a mapping $ζ\_n$ on the degree 0 Hochschild cohomology and a mapping $κ\_n$ on the degree 0 Hochschild homology of $A$ as adjoint mappings of the respective $p$-power mappings with respect to the symmetrizing bilinear form. In an earlier paper it is shown that $ζ\_n$ is invariant under derived equivalences. In the present paper we generalize the definition of $κ\_n$ to higher Hochschild homology and show the invariance of $κ$ and its generalization under derived equivalences. This provides fine invariants of derived categories.

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arXiv2006

Invariance of generalised Reynolds ideals under derived equivalences

For any algebraically closed field $k$ of positive characteristic $p$ and any non negative integer $n$ Külshammer defined ideals $T\_nA^\perp$ of the centre of a symmetric $k$-algebra $A$. We show that for derived equivalent algebras $A$ and $B$ there is an isomorphism of the centres of $A$ and $B$ mapping $T\_nA^\perp$ to $T\_nB^\perp$ for all $n$. Recently Héthelyi, Horváth, Külshammer and Murray showed that this holds for Morita equivalent algebras.

math.RA↗
arXiv2006

Generalized Reynolds ideals for non-symmetric algebras

We show how to extend the theory of generalized Reynolds ideals, as introduced by B. K"ulshammer, from symmetric algebras to arbitrary finite-dimensional algebras (in positive characteristic). This provides new invariants of the derived module categories of finite-dimensional algebras, extending recent results by the third author for symmetric algebras.

math.RT↗
arXiv2004

Degenerations for derived categories

We propose a theory of degenerations for derived module categories, analogous to degenerations in module varieties for module categories. In particular we define two types of degenerations, one algebraic and the other geometric. We show that these are equivalent, analogously to the Riemann-Zwara theorem for module varieties. Applications to tilting complexes are given, in particular that any two-term tilting complex is determined by its graded module structure.

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