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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 Λ an R-order in a finite-dimensional K-algebra. For a hereditary monomorphism u: P ↪ I of Λ-lattices we have an equivalence of quotient categories 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 by some over-order . Then remains an equivalence if is replaced by a certain subcategory of . The extended differentiation comprises a splitting theorem that implies Simson’s splitting theorem for vector space categories.
Wolfgang Rump. "Differentiation and splitting for lattices over orders." Colloquium Mathematicae 89.1 (2001): 7-42. <http://eudml.org/doc/283716>.
@article{WolfgangRump2001, 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 Λ an R-order in a finite-dimensional K-algebra. For a hereditary monomorphism u: P ↪ I of Λ-lattices we have an equivalence of quotient categories $∂̃_\{u\}:Λ-lat/[ℋ ] ⭇ δ_\{u\}Λ-lat/[B]$ 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 $δ_\{u\}Λ$ by some over-order $∂_\{u\}Λ ⊃ δ_\{u\}Λ$. Then $∂̃_\{u\}$ remains an equivalence if $δ_\{u\}Λ-lat$ is replaced by a certain subcategory of $∂_\{u\}Λ-lat$. The extended differentiation comprises a splitting theorem that implies Simson’s splitting theorem for vector space categories.}, author = {Wolfgang Rump}, journal = {Colloquium Mathematicae}, keywords = {orders; lattices; differentiation algorithms; splitting theorems; vector space categories}, language = {eng}, number = {1}, pages = {7-42}, title = {Differentiation and splitting for lattices over orders}, url = {http://eudml.org/doc/283716}, volume = {89}, year = {2001}, }
TY - JOUR AU - Wolfgang Rump TI - Differentiation and splitting for lattices over orders JO - Colloquium Mathematicae PY - 2001 VL - 89 IS - 1 SP - 7 EP - 42 AB - 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 Λ an R-order in a finite-dimensional K-algebra. For a hereditary monomorphism u: P ↪ I of Λ-lattices we have an equivalence of quotient categories $∂̃_{u}:Λ-lat/[ℋ ] ⭇ δ_{u}Λ-lat/[B]$ 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 $δ_{u}Λ$ by some over-order $∂_{u}Λ ⊃ δ_{u}Λ$. Then $∂̃_{u}$ remains an equivalence if $δ_{u}Λ-lat$ is replaced by a certain subcategory of $∂_{u}Λ-lat$. The extended differentiation comprises a splitting theorem that implies Simson’s splitting theorem for vector space categories. LA - eng KW - orders; lattices; differentiation algorithms; splitting theorems; vector space categories UR - http://eudml.org/doc/283716 ER -