Differentiation and splitting for lattices over orders

Wolfgang Rump¹

¹ Mathematisch-Geographische Fakultät Katholische Universität Eichstätt Ostenstr. 26-28 D-85071 Eichstätt, Germany

Colloquium Mathematicum, Tome 89 (2001) no. 1, pp. 7-42

Résumé

We extend our module-theoretic approach to Zavadskiĭ's differentiation techniques in representation theory. Let $R$ be a complete discrete valuation domain with quotient field $K$, and ${\mit\Lambda}$ an $R$-order in a finite-dimensional $K$-algebra. For a hereditary monomorphism $u: P\hookrightarrow I$ of ${\mit\Lambda}$-lattices we have an equivalence of quotient categories $\widetilde{\partial}_u:{\mit\Lambda}\hbox{-}{\bf lat}/[{\cal H}]\buildrel\sim\over\to \delta_u{\mit\Lambda}\hbox{-}{\bf lat}/[B]$ which generalizes Zavadskiĭ's algorithms for posets and tiled orders, and Simson's reduction algorithm for vector space categories. In this article we replace $u$ by a more general type of monomorphism, and the derived order $\delta_u{\mit\Lambda}$ by some over-order $\partial_u{\mit\Lambda}\supset\delta_u{\mit\Lambda}$. Then $\widetilde{\partial}_u$ remains an equivalence if $\delta_u{\mit\Lambda}\hbox{-}{\bf lat}$ is replaced by a certain subcategory of ${\partial}_u{\mit\Lambda}\hbox{-}{\bf lat}$. The extended differentiation comprises a splitting theorem that implies Simson's splitting theorem for vector space categories.

Détail

DOI : 10.4064/cm89-1-2

Keywords: extend module theoretic approach zavadski differentiation techniques representation theory complete discrete valuation domain quotient field mit lambda r order finite dimensional k algebra hereditary monomorphism hookrightarrow mit lambda lattices have equivalence quotient categories widetilde partial mit lambda hbox lat cal buildrel sim delta mit lambda hbox lat which generalizes zavadski algorithms posets tiled orders simsons reduction algorithm vector space categories article replace general type monomorphism derived order delta mit lambda over order partial mit lambda supset delta mit lambda widetilde partial remains equivalence delta mit lambda hbox lat replaced certain subcategory partial mit lambda hbox lat extended differentiation comprises splitting theorem implies simsons splitting theorem vector space categories