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Register a new userA generalization of Wattss Theorem: Right exact functors on module categories
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Wattss Theorem says that a right exact functor F:Mod R-->Mod S that commutes with direct sums is isomorphic to -otimes_R B where B is the R-S-bimodule FR. The main result in this paper is the following: if A is a cocomplete abelian category and F:Mod R --> A is a right exact functor commuting with direct sums, then F is isomorphic to - otimes_R B where B is a suitable R-module in A, i.e., a pair (B,f) consisting of an object B in A and a ring homomorphism f:R --> Hom_A(B,B). Part of the point is to give meaning to the notation -otimes_R B. That is done in the paper by Artin and Zhang on Abstract Hilbert Schemes. The present paper is a natural extension of some of the ideas in the first part of their paper.
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In this paper, we first introduce $mathcal {W}_F$-Gorenstein modules to establish the following Foxby equivalence: $xymatrix@C=80pt{mathcal {G}(mathcal {F})cap mathcal {A}_C(R) ar@<0.5ex>[r]^{Cotimes_R-} & mathcal {G}(mathcal {W}_F) ar@<0.5ex>[l]^{textrm{Hom}_R(C,-)}} $ where $mathcal {G}(mathcal {F})$, $mathcal {A}_C(R) $ and $mathcal {G}(mathcal {W}_F)$ denote the class of Gorenstein flat modules, the Auslander class and the class of $mathcal {W}_F$-Gorenstein modules respectively. Then, we investigate two-degree $mathcal {W}_F$-Gorenstein modules. An $R$-module $M$ is said to be two-degree $mathcal {W}_F$-Gorenstein if there exists an exact sequence $mathbb{G}_bullet=indent ...longrightarrow G_1longrightarrow G_0longrightarrow G^0longrightarrow G^1longrightarrow...$ in $mathcal {G}(mathcal {W}_F)$ such that $M cong$ $im(G_0rightarrow G^0) $ and that $mathbb{G}_bullet$ is Hom$_R(mathcal {G}(mathcal {W}_F),-)$ and $mathcal {G}(mathcal {W}_F)^+otimes_R-$ exact. We show that two notions of the two-degree $mathcal {W}_F$-Gorenstein and the $mathcal {W}_F$-Gorenstein modules coincide when R is a commutative GF-closed ring.
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