Differentiation and splitting for lattices over orders
Volume 89 / 2001
Abstract
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.
