Cousin Complexes and almost flat rings
Let (R, m) be a d-dimensional Noetherian local ring and T be a commutative strict algebra with unit element 1T over R such that mT 6= T. We define almost exact sequences of T-modules and characterize almost flat T-modules. Moreover, we define almost (faithfully) flat homomorphisms between R-algebras T and W, where W has similar properties that T has as an R-algebra. By almost (faithfully) flat homomorphisms and almost flat modules, we investigate Cousin complexes of T and W-modules. Finally, for a finite filtration F = (Fi)i≥0 of length less than d of Spec(T) such that it admits a T-module X, we show that IE 2 p,q := TorT p M, H d−q (CT (F, X)) p⇒ Hp+q(Tot(T )) and IIE 2 p,q := Hd−p TorT q (M, CT (F, X)) p⇒ Hp+q(Tot(T )), where M is an any flat T-module and as a result we show that IE 2 p,q and IIE 2 p,q are almost zero, when M is almost flat
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