Derived orders and Auslander-Reiten quivers
by
Wolfgang Rump
Mathematisches Institut B/3, Universität Stuttgart, Germany

Let R be a complete discrete valuation domain with quotient field K, and \Lambda an R-order in a finite dimensional K-algebra A. Recall that a \Lambda-lattice E is said to be injective if E ^* : = Hom_R(E, R) is projective. We call a monomorphism u: P --> I of \Lambda-lattices with R-torsion cokernel hereditary if P is projective and I injective, and

Hom\Lambda(P, I/P) = Ext\Lambda(I/P, I) = Ext\Lambda(H, L) = 0

holds for H, L in H_u : = { E in \Lambda-lat | P subset E subset I}.

For any \Lambda-lattice E we define a pair \partial_uE = \binomE⁺E_- of \Lambda-lattices with E_- subset E subset E⁺ and KE_- = KE⁺. Dually, for a right \Lambda-lattice F, the hereditary monomorphism u ^* : I ^* \hra P ^* gives a pair \binomF^-F₊ of right \Lambda-lattices with F₊ subset F subset F^-. Then the R-order

\deltau\Lambda : = æ ç ç ç ç ç è \Lambda+ Lambda+\Lambda- \Lambda- \Lambda- ö ÷ ÷ ÷ ÷ ÷ ø

in M₂(A) is called the derived order of \Lambda with respect to u. The functor

\partialu: \Lambda-lat --> \deltau\Lambda-lat

induces an equivalence of quotient categories

~ \partial   u  : \Lambda-lat / [Hu] --> ~ \deltau\Lambda-lat / [\tbinomIP].

This equivalence generalizes matrix algorithms of A. G. Zavadskij for representations of posets and tiled orders, and D. Simson's differentiation algorithm for vector space categories.

We also discuss the relationship between the Auslander-Reiten quivers of \Lambda and \delta_u\Lambda. The (finitely many) indecomposables in H_u form a rejectible set, i. e. there is an overorder \Lambda' of \Lambda such that ind\Lambda is a disjoint union of ind\Lambda' and indH_u. Recently, O. Iyama has obtained partial results on the structure of finite rejectible sets. In particular, he proved that they can be constructed by an inductive procedure.

Date received: July 12, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafe-28.