Differentiation and splitting for lattices over orders

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 -