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- W2299813612 abstract "We know that the model theory of modules leads to a way of obtaining definable categories of modules over a ring $R$ as the kernels of certain functors $(Rtextbf{-Mod})^{text{op}}totextbf{Ab}$ rather than of functors $Rtextbf{-Mod}totextbf{Ab}$ which are given by a pp pair. This paper will give various algebraic characterisations of these functors in the case that $R$ is an artin algebra. Suppose that $R$ is an artin algebra. An additive functor $G:(Rtextbf{-Mod})^{text{op}}totextbf{Ab}$ preserves inverse limits and $G|_{(Rtextbf{-mod})^{text{op}}}:(Rtextbf{-mod})^{text{op}}totextbf{Ab}$ is finitely presented if and only if there is a sequence of natural transformations $(-,A)to(-,B)to Gto 0$ for some $A,Bin Rtextbf{-mod}$ which is exact when evaluated at any left $R$-module. Any additive functor $(Rtextbf{-Mod})^{text{op}}totextbf{Ab}$ with one of these equivalent properties has a definable kernel, and every definable subcategory of $Rtextbf{-Mod}$ can be obtained as the kernel of a family of such functors. In the final section a generalised setting is introduced, so that our results apply to more categories than those of the form $Rtextbf{-Mod}$ for an artin algebra $R$. That is, our results are extended to those locally finitely presented $K$-linear categories whose finitely presented objects form a dualising $K$-variety, where $K$ is a commutative artinian ring." @default.
- W2299813612 created "2016-06-24" @default.
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- W2299813612 date "2016-03-18" @default.
- W2299813612 modified "2023-09-27" @default.
- W2299813612 title "Duality and contravariant functors in the representation theory of artin algebras" @default.
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