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Alexander Zimmermann

Publications and source records attributed to Alexander Zimmermann.

At least 19 recordsLinked to original sources
arXiv2026

Differential graded Brauer groups over dg-rings

Brauer groups of graded rings were defined and studied by Caenepeel and van Oystaeyen. We study the question what happens to this group if a ${\mathbb Z}$-graded ring carries a differential graded structure in addition. We define a Brauer group for differential graded algebras over differential graded graded-commutative or commutative base rings. Based on previous work we give an explicit classification of dg-fields, and compute as an example the so-defined Brauer group in each case explicitly.

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arXiv2025

DG-Semiprimary DG-Algebras, Acyclicity and Hopkins-Levitzki Theorem for DG-Algebras

We study the analogue of the Hopkins-Levitzky Theorem for dg-algebras $(A,d)$. We first consider the Hopkins approach. Here we show that for acyclic dg-algebras with graded-Artinian algebras of cycles $\ker(d)$, we also have that $(A,d)$ is left dg-Noetherian, and we show that acyclic dg-Artinian dg-algebras are dg-Noetherian. Then, studying the Levitzki approach, we consider a definition of a dg-semiprimary algebra. For dg-semiprimary dg-artinian dg-algebras $(A,d)$, we show that all dg-simple dg-modules are acyclic, and so are all dg-modules with finite dg-composition length. We finally show that dg-Artinian dg-semiprimary dg-algebras with nilpotent dg-radical $dgrad_2(A,d)$ are dg-Noetherian and acyclic.

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arXiv2024

Dg-separable dg-extensions

We define and characterise completely dg-separable dg-extensions $\varphi:(A,d_A)\rightarrow (B,d_B)$. We completely characterise the case of graded commutative dg-division algebras in characteristic different from $2$. We prove that for a dg-separable extension a short exact sequence of dg-modules over $(B,d_B)$ splits if and only if the restriction to $(A,d_A)$ splits.

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arXiv2024

Differential graded division algebras, their modules, and dg-simple algebras

We give the definition of a dg-division algebra, that is a concept of a differential graded algebra which may serve as an analogue of a division algebra. We classify them completely, and show that they are either acyclic or have differential $0$. Further, we prove that the graded centre of dg-simple dg-algebras is a dg-division algebra, and also the dg-endomorphism ring of a dg-simple module is a dg-division algebra. We also shall give a Jacobson-Chevalley density theorem for acyclic dg-algebras.

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arXiv2023

Differential graded orders, their class groups and id\`eles

For a Dedekind domain $R$ with field of fractions $K$ a classical $R$-order in a semisimple $K$-algebra $A$ is an $R$-projective $R$-subalgebra $\Lambda$ of $A$ such that $K\Lambda=A$. We study differential graded $K$-algebras which are semisimple as $K$-algebras and define differential graded $R$-orders as a differential graded $R$-subalgebras, which are in addition classical $R$-orders in $A$. We give a series of examples for such differential graded algebras and orders. We show that any differential graded $R$-order is contained in a maximal differential graded order. We develop parts of the classical ring theory in the differential graded setting, in particular the properties of analogues of the Jacobson radical. We further define class groups of differential graded orders as subgroups of the Grothendieck group of locally free differential graded modules. We define id\`eles in this setting showing that these id\`ele groups maps surjectively to the differential graded class group. Finally we give a homomorphism to the class group of the homology of the differential graded order and prove a Mayer-Vietoris like sequence for each central idempotent of $A$, including the analogous one for the kernel groups of these morphisms.

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arXiv2023

Differential graded Brauer groups

We consider central simple $K$-algebras which happen to bedifferential graded $K$-algebras. Two such algebras $A$ and $B$are considered equivalent if there are bounded complexes of finite dimensional$K$-vector spaces $C_A$ and $C_B$ such that the differential graded algebras $A\otimes_K {\rm End}_K^\bullet(C_A)$ and $B\otimes_K {\rm End}_K^\bullet(C_B)$ are isomorphic.Equivalence classes form an abelian group, which we call thedg Brauer group.We prove that this group is isomorphic to the ordinary Brauer group of the field $K$.

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arXiv2022

Clifford's theorem for orbit categories

Clifford theory relates the representation theory of finite groups to those of a fixed normal subgroup by means of induction and restriction, which is an adjoint pair of functors. We generalize this result to the situation of a Krull-Schmidt category on which a finite group acts as automorphisms. This then provides the orbit category introduced by Cibils and Marcos, and studied intensively by Keller in the context of cluster algebras, and by Asashiba in the context of Galois covering functors. We formulate and prove Clifford's theorem for Krull-Schmidt orbit categories with respect to a finite group $\Gamma$ of automorphisms, clarifying this way how the image of an indecomposable object in the original category decomposes in the orbit category. The pair of adjoint functors appears as the Kleisli category of the naturally appearing monad given by $\Gamma$.

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arXiv2020

Remarks on a triangulated version of Auslander-Kleiner's Green correspondence

For a finite group $G$ and an algebraically closed field $k$ of characteristic $p>0$ for any indecomposable finite dimensional $kG$-module $M$ with vertex $D$ and a subgroup $H$ of $G$ containing $N_G(D)$ there is a unique indecomposable $kH$-module $N$ of vertex $D$ being a direct summand of the restriction of $M$ to $H$. This correspondence, called Green correspondence, was generalised by Auslander-Kleiner to the situation of pairs of adjoint functors between additive categories. In the original situation of group rings Carlson-Peng-Wheeler proved that this correspondence is actually restriction of triangle functors between triangulated quotient categories of the corresponding module categories. We review this theory and show how we got a common generalisation of the approaches of Auslander-Kleiner and Carlson-Peng-Wheeler, using Verdier localisations.

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arXiv2020

Green correspondence and relative projectivity for pairs of adjoint functors between triangulated categories

Auslander and Kleiner proved in 1994 an abstract version of Green correspondence for pairs of adjoint functors between three categories. They produce additive quotients of certain subcategories giving the classical Green correspondence in the special setting of modular representation theory. Carlson, Peng and Wheeler showed in 1998 that Green correspondence in the classical setting of modular representation theory is actually an equivalence between triangulated categories with respect to a non standard triangulated structure. In the present note we first define and study a version of relative projectivity, respectively relative injectivity with respect to pairs of adjoint functors. We then modify Auslander Kleiner's construction such that the correspondence holds in the setting of triangulated categories.

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arXiv2019

Degenerating $0$ in Triangulated Categories

In previous work, based on work of Zwara and Yoshino, we defined and studied degenerations of objects in triangulated categories analogous to degeneration of modules. In triangulated categories it is surprising that the zero object may degenerate. We study this systematically. In particular we show that the degeneration of the zero object actually induces all other degenerations by homotopy pullback, that degeneration of $0$ is closely linked, but not equivalent, to having zero image in the Grothendieck group.

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arXiv2018

Verdier quotients of homotopy categories

We study Verdier quotients of diverse homotopy categories of a full additive subcategory $\mathcal E$ of an abelian category. In particular, we consider the categories $K^{x,y}({\mathcal E})$ for $x\in\{\infty, +,-,b\}$, and $y\in\{\emptyset,b,+,-,\infty\}$ the homotopy categories of left, right, unbounded complexes with homology being $0$, bounded, left or right bounded, or unbounded. Inclusion of these categories give a partially ordered set, and we study localisation sequences or recollement diagrams between the Verdier quotients, and prove that many quotients lead to equivalent categories.

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arXiv2017

K{ü}lshammer ideals of algebras of quaternion type

For a symmetric algebra A over a field K of characteristic p > 0 K{ü}lshammer constructed a descending sequence of ideals of the centre of A. If K is perfect this sequence was shown to be an invariant under derived equivalence and for algebraically closed K under stable equivalence of Morita type. Erdmann classified algebras of tame representation type which may be blocks of group algebras, and Holm classified Erdmann's list up to derived equivalence. In both classifications certain parameters occur in the classification, and it was unclear if different parameters lead to different algebras. Erdmann's algebras fall into three classes, namely of dihedral, semidihedral and of quaternion type. In previous joint work with Holm we used K{ü}lshammer ideals to distinguish classes with respect to these parameters in case of algebras of dihedral and semidihedral type. In the present paper we determine the K{ü}lshammer ideals for algebras of quaternion type and distinguish again algebras with respect to certain parameters.

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arXiv2016

Symmetry of the Definition of Degeneration in Triangulated Categories

Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety. Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if it belongs to the Zariski closure of the orbit. Riedtmann and Zwara gave an algebraic characterisation of this concept in terms of the existence of short exact sequences. Jensen, Su and Zimmermann, as well as independently Yoshino, studied the natural generalisation of the Riedtmann-Zwara degeneration to triangulated categories. The definition has an intrinsic non-symmetry. Suppose that we have a triangulated category in which idempotents split and either for which the endomorphism rings of all objects are artinian, or which is the category of compact objects in an algebraic compactly generated triangulated K-category. Then we show that the non-symmetry in the algebraic definition of the degeneration is inessential in the sense that the two possible choices which can be made in the definition lead to the same concept.

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arXiv2015

An axiomatic approach for degenerations in triangulated categories

We generalise Yoshino's definition of a degeneration of two Cohen Macaulay modules to a definition of degeneration between two objects in a triangulated category. We derive some natural properties for the triangulated category and the degeneration under which the Yoshino-style degeneration is equivalent to the degeneration defined by a specific distinguished triangle analogous to Zwara's characterisation of degeneration in module varieties.

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arXiv2015

Remarks on a Categorical Definition of Degeneration in Triangulated Categories

This work reports on joint research with Manuel Saorin. For an algebra A over an algebraically closed field k the set of A-module structures on k d forms an affine algebraic variety. The general linear group Gl d (k) acts on this variety and isomorphism classes correspond to orbits under this action. A module M degenerates to a module N if N belongs to the Zariski closure of the orbit of M. Yoshino gave a scheme-theoretic characterisation, and Saorin and Zimmermann generalise this concept to general triangulated categories. We show that this concept has an interpretation in terms of distinguished triangles, analogous to the Riedtmann-Zwara characterisation for modules. In this manuscript we report on these results and study the behaviour of this degeneration concept under functors between triangulated categories.

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arXiv2015

On a question of Rickard on tensor product of stably equivalent algebras

Let $\overline\F\_p$ be the algebraic closure of the prime field of characteristic $p$. After observing that the principal block $B$ of $\overline\F\_pPSU(3,p^r)$ is stably equivalent of Morita type to its Brauer correspondent $b$, we show however that the centre of $B$ is not isomorphic as an algebra to the centre of $b$ in the cases $p^r\in\{3,4,5,8\}$. As a consequence, the algebra $B\otimes\_{\overline{\F}\_p}\overline \F\_p[X]/X^p$ is not stably equivalent of Morita type to $b\otimes\_{\overline\F\_p}\overline\F\_p[X]/X^p$ in these cases. This yields a negative answer to a question of Rickard.

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