The paper begins in §1 with a foundational discussion of a new notion, that of a semistandard filtration in a highest weight category. The main result is Theorem 1, which says that “multiplicities” of standard modules in such filtration are well-defined. In §2, we specialize to the case of semistandard filtrations of maximal submodules of standard modules. The main result is Theorem 2, which un...

In this paper, we investigate the maximal Cohen-Macaulay property of tensor products modules, and then give criteria for projectivity modules in terms vanishing Ext modules. One applications shows that Auslander-Reiten conjecture holds normal rings.

let $r$ be a ring‎, ‎and let $n‎, ‎d$ be non-negative integers‎. ‎a right $r$-module $m$ is called $(n‎, ‎d)$-projective if $ext^{d+1}_r(m‎, ‎a)=0$ for every $n$-copresented right $r$-module $a$‎. ‎$r$ is called right $n$-cocoherent if every $n$-copresented right $r$-module is $(n+1)$-coprese-nted‎, ‎it is called a right co-$(n,d)$-ring if every right $r$-module is $(n‎, ‎d)$-projective‎. ‎$r$ ...

Given a finite dimensional Lie algebra $L$ let $I$ be the augmentation ideal in universal enveloping $U(L)$. We study conditions on under which $Ext$-groups $Ext (k,k)$ for trivial $L$-module $k$ are same when computed category of all $U(L)$-modules or $I$-torsion $U(L)$-modules. also prove that Rees $\oplus _{n\geq 0}I^n$ is Noetherian if and only nilpotent. An application to cohomology equiva...

 Let $R$ be a ring‎, ‎and let $n‎, ‎d$ be non-negative integers‎. ‎A right $R$-module $M$ is called $(n‎, ‎d)$-projective if $Ext^{d+1}_R(M‎, ‎A)=0$ for every $n$-copresented right $R$-module $A$‎. ‎$R$ is called right $n$-cocoherent if every $n$-copresented right $R$-module is $(n+1)$-coprese-nted‎, ‎it is called a right co-$(n,d)$-ring if every right $R$-module is $(n‎, ‎d)$-projective‎. ‎$R$...

Auslander’s depth formula for pairs of Tor-independent modules over a regular local ring, depth(M ⊗R N) = depth M + depth N − depthR, has been generalized in several directions; most significantly it has been shown to hold for pairs of Tor-independent modules over complete intersection rings. In this paper we establish a depth formula that holds for every pair of Tate Tor-independent modules ov...