arXiv subjects

Rene Marczinzik

Publications and source records attributed to Rene Marczinzik.

46 records · Page 3Linked to original sources

arXiv2017

On the finiteness of the Gorenstein dimension for Artin algebras

In \cite{SSZ}, the authors proved that an Artin algebra $A$ with infinite global dimension has an indecomposable module with infinite projective and infinite injective dimension, giving a new characterisation of algebras with finite global dimension. We prove in this article that an Artin algebra $A$ that is not Gorenstein has an indecomposable $A$-module with infinite Gorenstein projective dimension and infinite Gorenstein injective dimension, which gives a new characterisation of algebras with finite Gorenstein dimension. We show that this gives a proper generalisation of the result in \cite{SSZ} for Artin algebras.

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arXiv2017

On bounds of homological dimensions in Nakayama algebras

Let $A$ be a Nakayama algebra with $n$ simple modules and a simple module $S$ of even projective dimension $m$. Choose $m$ minimal such that a simple $A$-module with projective dimension $2m$ exists, then we show that the global dimension of $A$ is bounded by $n+m-1$. This gives a combined generalisation of results of Gustafson \cite{Gus} and Madsen \cite{Mad}. In \cite{Bro}, Brown proved that the global dimension of quasi-hereditary Nakayama algebras with $n$ simple modules is bounded by $n$. Using our result on the bounds of global dimensions of Nakayama algebras, we give a short new proof of this result and generalise Brown's result from quasi-hereditary to standardly stratified Nakayama algebras, where the global dimension is replaced with the finitistic dimension.

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arXiv2017

On a new formula for the Gorenstein dimension

Let $A$ be a finite dimensional algebra over a field $K$ with enveloping algebra $A^e=A^{op} \otimes_K A$. We call algebras $A$ that have the property that the subcategory of Gorenstein projective modules in $mod-A$ coincide with the subcategory $\{ X \in mod-A | Ext_A^i(X,A)=0 $ for all $i \geq 1 \}$ left nearly Gorenstein. The class of left nearly Gorenstein algebras is a large class that includes for example all Gorenstein algebras and all representation-finite algebras. We prove that the Gorenstein dimension of $A$ coincides with the Gorenstein projective dimension of the regular module as $A^e$-module for left nearly Gorenstein algebras $A$. We give three application of this result. The first generalises a formula by Happel for the global dimension of algebras. The second application generalises a criterion of Shen for an algebra to be selfinjective. As a final application we prove a stronger version of the first Tachikawa conjecture for left nearly Gorenstein algebras.

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arXiv2017

Auslander-Gorenstein algebras from Serre-formal algebras via replication

We introduce a new family of algebras, called Serre-formal algebras. They are Iwanaga-Gorenstein algebras for which applying any power of the Serre functor on any indecomposable projective module, the result remains a stalk complex. Typical examples are given by (higher) hereditary algebras and self-injective algebras; it turns out that other interesting algebras such as (higher) canonical algebras are also Serre-formal. Starting from a Serre-formal algebra, we consider a series of algebras - called the replicated algebras - given by certain subquotients of its repetitive algebra. We calculate the self-injective dimension and dominant dimension of all such replicated algebras and determine which of them are minimal Auslander-Gorenstein, i.e. when the two dimensions are finite and equal to each other. In particular, we show that there exist infinitely many minimal Auslander-Gorenstien algebras in such a series if, and only if, the Serre-formal algebra is twisted fractionally Calabi-Yau. We apply these results to a construction of algebras from Yamagata, called SGC extensions, given by iteratively taking the endomorphism ring of the smallest generator-cogenerator. We give a sufficient condition so that the SGC extensions and replicated algebras coincide. Consequently, in such a case, we obtain explicit formulae for the self-injective dimension and dominant dimension of the SGC extension algebras.

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arXiv2017

Upper bounds for the dominant dimension of Nakayama and related algebras

Optimal upper bounds are provided for the dominant dimensions of Nakayama algebras and more generally algebras $A$ with an idempotent $e$ such that there is a minimal faithful injective-projective module $eA$ and such that $eAe$ is a Nakayama algebra. This answers a question of Abrar and proves a conjecture of Yamagata for monomial algebras.

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arXiv2016

Auslander-Gorenstein algebras, standardly stratified algebras and dominant dimensions

We give new properties of algebras with finite Gorenstein dimension coinciding with the dominant dimension $\geq 2$, which are called Auslander-Gorenstein algebras in the recent work of Iyama and Solberg, see \cite{IyaSol}. In particular, when those algebras are standardly stratified, we give criteria when the category of (properly) (co)standardly filtered modules has a nice homological description using tools from the theory of dominant dimensions and Gorenstein homological algebra. We give some examples of standardly stratified algebras having dominant dimension equal to the Gorenstein dimension, including examples having an arbitrary natural number as Gorenstein dimension and blocks of finite representation-type of Schur algebras.

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arXiv2016

Upper bounds for dominant dimensions of gendo-symmetric algebras

The famous Nakayama conjecture states that the dominant dimension of a non-selfinjective finite dimensional algebra is finite. In \cite{Yam}, Yamagata stated the stronger conjecture that the dominant dimension of a non-selfinjective finite dimensional algebra is bounded by a function depending on the number of simple modules of that algebra. With a view towards those conjectures, new bounds on dominant dimensions seem desirable. We give a new approach to bounds on the dominant dimension of gendo-symmetric algebras via counting non-isomorphic indecomposable summands of rigid modules in the module category of those algebras. On the other hand, by Mueller's theorem, the calculation of dominant dimensions is directly related to the calculation of certain Ext-groups. Motivated by this connection, we generalize a theorem of Tachikawa about non-vanishing of $Ext^{1}(M,M)$ for a non-projective module $M$ in group algebras of $p$-groups to local Hopf algebras and we also give new results for showing the non-vanishing of $Ext^{1}(M,M)$ for certain modules in other local selfinjective algebras, which specializes to show that blocks of category $\mathcal{O}$ and 1-quasi-hereditary algebras with a special duality have dominant dimension exactly 2. In the final section we raise different questions with the hope of a new developement on those conjectures in the future.

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arXiv2016

On representation-finite gendo-symmetric biserial algebras

Gendo-symmetric algebras were introduced by Fang and Koenig as a generalisation of symmetric algebras. Namely, they are endomorphism rings of generators over a symmetric algebra. This article studies various algebraic and homological properties of representation-finite gendo-symmetric biserial algebras. We show that the associated symmetric algebras for these gendo-symmetric algebras are Brauer tree algebras, and classify the generators involved using Brauer tree combinatorics. We also study almost $ν$-stable derived equivalences, introduced by Hu and Xi, between representation-finite gendo-symmetric biserial algebras. We classify these algebras up to almost $ν$-stable derived equivalence by showing that the representative of each equivalence class can be chosen as a Brauer star with some additional combinatorics. We also calculate the dominant, global, and Gorenstein dimensions of these algebras. In particular, we found that representation-finite gendo-symmetric biserial algebras are always Iwanaga-Gorenstein algebras.

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arXiv2016

A bocs theoretic characterization of gendo-symmetric algebras

Gendo-symmetric algebras were recently introduced by Fang and König. An algebra is called gendo-symmetric in case it is isomorphic to the endomorphism ring of a generator over a finite dimensional symmetric algebra. We show that a finite dimensional algebra $A$ over a field $K$ is gendo-symmetric if and only if there is a bocs-structure on $(A,D(A))$, where $D=Hom_K(-,K)$ is the natural duality. Assuming that $A$ is gendo-symmetric, we show that the module category of the bocs $(A,D(A))$ is isomorphic to the module category of the algebra $eAe$, when $e$ is an idempotent such that $eA$ is the unique minimal faithful projective-injective right $A$-module. We also prove some new results about gendo-symmetric algebras using the theory of bocses.

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arXiv2016

On a conjecture about dominant dimensions of algebras

For every $n \geq 1$, we present examples of algebras $A$ having dominant dimension $n$, such that the algebra $B=End_A(I_0 \oplus Ω^{-n}(A))$ has dominant dimension different from $n$, where $I_0$ is the injective hull of $A$. This gives a counterexample to conjecture 2 of Chen and Xi. While the conjecture is false in general, we show that a large class of algebras containing higher Auslander algebras satisfies the property in the conjecture.

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