Maximal selforthogonal modules and a new generalization of tilting modules
arXiv preprint, February 2023. Link: arXiv:2301.13498
Citation: H. Enomoto, Maximal selforthogonal modules and a new generalization of tilting modules, arXiv:2301.13498.
Comment
I study selforthogonal modules using socalled projective Wakamatsu tilting modules and their relation to Wakamatsu tilting modules and maximal selforthogonal modules.
$T$ is projectively Wakamatsu tilting if $T$ is an Extprogenerator of the category $T^\perp$ consisting of $X$ with $\operatorname{Ext}^{>0}(T, X) = 0$. Then we have
{tilting} $\subseteq$ {projectively Wakamatsu tilting} $\subseteq$ {Wakamatsu tilting}.
My interest is mainly in the representationfinite case, and I show that the following are equivalent for $T \in \mathsf{mod} \, \Lambda$ if $\Lambda$ is representationfinite.
 $T$ is projectively Wakamatsu tilting.
 $T$ is Wakamatsu tilting.
 $T$ is maximal selforthogonal module, that is, selforthogonal, and if $T \oplus M$ is selforthogonal, then $M \in \mathsf{add} \, T$ holds.
 $T$ is selforthogonal with $M = \Lambda$.
3rd and 4th conditions are convenient for actual computation. I give many examples in this paper using a computer program (which I developed but in preparation). This was the initial motivation of this paper (I was wondering how to compute all Wakamatsu tilting modules over repfin algebras).
Also, this has applications in Gorenstein homological algebra (e.g. this immediately deduces the fact that the category of Gorensteinprojective modules coincide with $^\perp \Lambda$ if $^\perp\Lambda$ has only finitely many indecomposables).
I also study a binary relation on the set of projectively Wakamatsu tilting modules, defined by the vanishing of Ext, so generalizing the poset of tilting modules. Unfortunately, this is not a poset even if the algebra is repfin, and I gave such an example. Using this relation, I prove that any selforthogonal module over an IwanagaGorenstein algebra $\Lambda$ has finite projective dimension if $\Lambda$ is representationfinite (I think there may be a direct proof, but I cannot find it).
Some conjectures related on selforthogonal modules are discussed in detail.
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