Differentiation and splitting for lattices over orders

Volume 89 / 2001

Wolfgang Rump Colloquium Mathematicum 89 (2001), 7-42 MSC: Primary 16G30. DOI: 10.4064/cm89-1-2


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.


  • Wolfgang RumpMathematisch-Geographische Fakultät
    Katholische Universität Eichstätt
    Ostenstr. 26-28
    D-85071 Eichstätt, Germany

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