Cousin Complexes and almost flat rings

Abstract

Let $(R,\mathfrak{m})$ is a $d$-dimensional Noetherian local ring and $T$ be a commutative strict algebra with unit element $1_T$  over $R$ such that  $\mathfrak{m}T\neq T$. 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 of length less than $d$ of $\mathrm{Spec}(T)$, $\mathcal{F}=(F_i)_{i\geq0}$  such that admits a $T$-module $X$, we show that $^I\mathrm{E}_{p,q}^2:=\mathrm{Tor}_p^T \left(M,\mathrm{H}^{d-q}\left(\mathcal{C}_T\left(\mathcal{F},X\right)\right)\right) \stackrel{p}{\Rightarrow}\mathrm{H}_{p+q}(\mathrm{Tot}(\mathcal{T}))$ and $^{II}\mathrm{E}_{p,q}^2:=\mathrm{H}^{d-p}\left(\mathrm{Tor}_q^T\left(M,\mathcal{C}_T\left(\mathcal{F},X\right)\right)\right)   \stackrel{p}{\Rightarrow}\mathrm{H}_{p+q}(\mathrm{Tot}(\mathcal{T}))$, where $M$ is an any flat $T$-module and as result we show that $^I\mathrm{E}_{p,q}^2$ and $^{II}\mathrm{E}_{p,q}^2$ are almost zero, when $M$ is almost flat.