Upper bounds for dominant dimensions of gendo-symmetric algebras

Upper bounds for dominant dimensions of gendo-symmetric algebras The famous Nakayama conjecture states that the dominant dimension of a non-selfinjective finite dimensional algebra is finite. Yamagata (Frobenius algebras handbook of algebra, vol 1. Elsevier/North-Holland, Amsterdam, pp 841–887, 1996) 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 also give new results for showing the non-vanishing of $$Ext^{1}(M,M)$$ E x t 1 ( M , M ) for certain modules in local symmetric algebras, which specializes to show that blocks of category $$\mathcal {O}$$ O and 1-quasi-hereditary algebras with a special duality have dominant dimension exactly 2. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archiv der Mathematik Springer Journals

Upper bounds for dominant dimensions of gendo-symmetric algebras

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer International Publishing AG
Subject
Mathematics; Mathematics, general
ISSN
0003-889X
eISSN
1420-8938
D.O.I.
10.1007/s00013-017-1073-9
Publisher site
See Article on Publisher Site

Abstract

The famous Nakayama conjecture states that the dominant dimension of a non-selfinjective finite dimensional algebra is finite. Yamagata (Frobenius algebras handbook of algebra, vol 1. Elsevier/North-Holland, Amsterdam, pp 841–887, 1996) 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 also give new results for showing the non-vanishing of $$Ext^{1}(M,M)$$ E x t 1 ( M , M ) for certain modules in local symmetric algebras, which specializes to show that blocks of category $$\mathcal {O}$$ O and 1-quasi-hereditary algebras with a special duality have dominant dimension exactly 2.

Journal

Archiv der MathematikSpringer Journals

Published: Aug 4, 2017

References

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