3
$\begingroup$

Given a finite dimensional algebra $A$ with finite global dimension such that there are only finitely many basic tilting modules. Then every selforthogonal indecomposable module $M$ (that is a module with $\mathrm{Ext}_A^i(M,M)=0$ for all $i \geq 1$) is a direct summand of a tilting module and thus there are only finitely many such indecomposable selforthogonal modules. See 3.9 in https://arxiv.org/abs/1803.10707.

Question 1: Is there an effective method to obtain every indecomposable selforthogonal module in a nice way? For example using the GAP-package QPA.

(A nice method would for example be an "Auslander-Reiten" theory for indecomposable selforthogonal modules in such algebras, where one can obtain every such module using certain exact sequences starting from just one such module.)

Question 2: Is there an example of an Auslander algebra with infinitely many selforthogonal indecomposable modules?

$\endgroup$
2
$\begingroup$

Concerning your second question, it is claimed in the proof of Corollary 4.8 in a recent preprint by Iyama-Zhang that Kajita showed in his Master's thesis that the Auslander algebra of the linearly oriented path with at least 6 vertices has infinitely many classical tilting modules. So it has in particular infinitely many indecomposable selforthogonal modules.

Let's now turn to the first question. Considering the set $\mathfrak{T}$ of (isomorphism classes of basic) tilting $A$-modules with the partial order induced by the inclusion order of the corresponding perpendicular subcategories, $\mathfrak{T}$ becomes a finite lattice. According to Happel-Unger there is an arrow $T \to T'$ in the Hasse diagram of $\mathfrak{T}$ if and only if $T = X \oplus M$ and $T' = Y \oplus M$ with $X$, $Y$ indecomposable fitting into an exact sequence $$ 0 \to X \xrightarrow{f} M' \xrightarrow{g} Y \to 0 $$ where $f$ is a minimal left and $g$ a minimal right $\mathrm{add}(M)$-approximation.

Given that QPA can compute minimal approximations, we can start with the set $\{P_1,\ldots,P_n\}$ of indecomposable projective $A$-modules and use BFS (or another graph-traversal algorithm) to find all sets $\{T_1,\ldots,T_n\}$ of indecomposable $A$-modules such that $T = \bigoplus_i T_i$ is a tilting module. The union of all these sets $\{T_1,\ldots,T_n\}$ is then the set of indecomposable selforthogonal $A$-modules.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.