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On the genus of the graph of tilting modules. (English) Zbl 1116.16015

From the introduction: Let \(\Lambda\) be a finite dimensional, connected, associative algebra with unit over a field \(k\). Let \(n\) be the number of isomorphism classes of simple \(\Lambda\)-modules. By \(\text{mod\,}\Lambda\) we denote the category of finite dimensional left \(\Lambda\)-modules.
A module \(_\Lambda T\in\text{mod\,}\Lambda\) is called a ‘tilting module’ if (i) the projective dimension \(\text{pd}_\Lambda T\) of \(_\Lambda T\) is finite, and (ii) \(\text{Ext}^i_\Lambda(T,T)=0\) for all \(i>0\), and (iii) there is an exact sequence \(0\to{_\Lambda\Lambda}\to{_\Lambda T^1}\to\cdots\to{_\Lambda T^d}\to 0\) with \(_\Lambda T^i\in\text{add}{_\Lambda T}\) for all \(0\leq i\leq d\). Here \(\text{add}{_\Lambda T}\) denotes the category of direct sums of direct summands of \(_\Lambda T\).
Tilting modules play an important role in many branches of mathematics such as representation theory of Artin algebras or the theory of algebraic groups.
Let \(\bigoplus^m_{i=1}T_i\) be the decomposition of \(_\Lambda T\) into indecomposable direct summands. We call \(_\Lambda T\) ‘basic’ if \(_\Lambda T_i\not\simeq{_\Lambda T_j}\) for all \(i\neq j\). A basic tilting module has \(n\) indecomposable direct summands. A direct summand \(_\Lambda M\) of a basic tilting module \(_\Lambda T\) is called an ‘almost complete tilting module’ if \(_\Lambda M\) has \(n-1\) indecomposable direct summands. Let \(\mathcal T(\Lambda)\) be the set of all non isomorphic basic tilting modules over \(\Lambda\). We associate with \(\mathcal T(\Lambda)\) a quiver \(\overrightarrow{\mathcal K(\Lambda)}\) as follows: The vertices of \(\overrightarrow{\mathcal K(\Lambda)}\) are the tilting modules in \(\mathcal T(\Lambda)\), and there is an arrow \(_\Lambda T'\to{_\Lambda T}\) if \(_\Lambda T\) and \(_\Lambda T'\) have a common direct summand which is an almost complete tilting module and if \(\text{Ext}^1_\Lambda(T,T')\neq 0\). We call \(\overrightarrow{\mathcal K(\Lambda)}\) the ‘quiver of tilting modules’ over \(\Lambda\). With \({\mathcal K}(\Lambda)\) we denote the underlying graph of \(\overrightarrow{\mathcal K(\Lambda)}\). It has been recently shown [D. Happel, L. Unger, Algebr. Represent. Theory 8, No. 2, 147-156 (2005; Zbl 1110.16011)] that \({\mathcal K}(\Lambda)\) is the Hasse diagram of a partial order of tilting modules which was basically introduced by C. Riedtmann and A. Schofield [in Comment. Math. Helv. 66, No. 1, 70-78 (1991; Zbl 0790.16013)]. From this it follows, that \(\overrightarrow{\mathcal K(\Lambda)}\) has no oriented cycles.
If \(\overrightarrow{\mathcal K(\Lambda)}\) is finite, then it is connected. Examples show that \(\overrightarrow{\mathcal K(\Lambda)}\) may be rather complicated. One measure for the complicatedness of a graph \(G\) is its genus \(\gamma(G)\). This is the minimal genus of an orientable surface on which \(G\) can be embedded.
The aim of these notes is to show that there are finite quivers of tilting modules of arbitrary genus. To be precise, we prove:
Theorem 1. For all integers \(r\geq 0\) there is a representation finite, connected algebra \(\Lambda_r\) such that \(\gamma({\mathcal K}(\Lambda_r))=r\).
The proof of the theorem is constructive. For each \(r\in\mathbb{N}\) we give an explicit example of an algebra \(\Lambda_r\) and embed \({\mathcal K}(\Lambda_r)\) in an orientable surface of genus \(r\). This gives an upper bound for \(\gamma({\mathcal K}(\Lambda_r))\). Then we use general results from graph theory to show that the bound is sharp. This is done in Section 3. In Section 1 we recall some basic facts about tilting modules and embeddings of graphs. In Section 2 we introduce the algebras \(\Lambda_r\) and derive some properties of \(\overrightarrow{\mathcal K(\Lambda_r)}\).

MSC:

16G20 Representations of quivers and partially ordered sets
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16E10 Homological dimension in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16D90 Module categories in associative algebras
05C10 Planar graphs; geometric and topological aspects of graph theory
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