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Let $A$ be a finite dimensional selfinjective algebra and $M$ an indecomposable non-projective $A$-module such that the algebra $B:=End_A(A \oplus M)$ is standardly stratified.

Examples of such $B$ include the representation-finite blocks of Schur algebras or the case when $A$ is commutative and $M$ an ideal with simple top (see also http://www.sciencedirect.com/science/article/pii/S0021869305003200 for examples).

Let $e$ be an idempotent of $B$ such that $eB$ is minimal faithful projective-injective. Now I can prove that all (basic) tilting modules of $B$ are isomorphic to modules of the form $eB \oplus \Omega^{-n}(B)$ for some $n=0,...,d$ when $d$ is the dominant dimension of $B$. (this is part of a more general result which has nothing to do with $B$ being standardly stratified).

I wonder whether one can decide what the characteristic tilting module $T$ of $B$ is, or equivalently what is $n$ such that $T \cong eB \oplus \Omega^{-n}(B)$? I think this $n$ can somehow be related to the homological dimensions of $B$. When $B$ is Gorenstein, $n$ is characterised by the property $F(\Delta)=Proj_n$, the subcategory of module of projective dimension at most $n$. Can one characterise what $n$ is directly just by properties depending on $M$ in $mod-A$?

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  • $\begingroup$ Is it clear that in this setting there is only one partial order making $B$ standardly stratified? I wouldn't expect a homological characterisation in general as different partial orders on the simples yield different characteristic tilting modules. $\endgroup$ Oct 23, 2017 at 17:22
  • $\begingroup$ @JulianKuelshammer would be interesting to see an example if that can happen. In case the algebra is Gorenstein and properly stratified and has a simple preserving duality the characteristic tilting module should be unique. $\endgroup$
    – Mare
    Oct 23, 2017 at 17:37

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