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Transactions of the American Mathematical Society

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Lattices over orders: finitely presented functors and preprojective partitions


Authors: M. Auslander and S. O. Smalø
Journal: Trans. Amer. Math. Soc. 273 (1982), 433-446
MSC: Primary 16A64
MathSciNet review: 667155
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Abstract: Suppose $ R$ is a commutative noetherian equidimensional Gorenstein ring and $ \Lambda $ an $ R$-algebra which is finitely generated as an $ R$-module. A $ \Lambda $-module $ M$ is a lattice if $ {M_{\underline{\underline p} }}$ is $ {\Lambda _{\underline{\underline p} }}$-projective and $ {\text{Ho}}{{\text{m}}_R}{(M,R)_{\underline{\underline p} }}$ is $ \Lambda _{\underline{\underline p} }^{{\text{op}}}$-projective for all nonmaximal prime ideals $ \underline{\underline p} $ in $ R$. We assume that $ \Lambda $ is an $ R$-order in the sense that $ \Lambda $ is a lattice when viewed as a $ \Lambda $-module. The first main result is to show that simple contravariant functors from lattices to abelian groups are finitely presented. This is then applied to showing that if $ R$ is also local and complete, then the category of lattices has a preprojective partition. This generalizes previous results of the authors in the cases $ R$ is artinian or a discrete valuation ring.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0667155-9
Article copyright: © Copyright 1982 American Mathematical Society