Upper bounds for dominant dimensions of gendo-symmetric algebras

Abstract

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.