有限平坦维数的Tor-自正交模(英文)

Tor-self-orthogonal modules of a finite flat dimension

设R为一环,ω_R为Tor-自正交模.引入模的右Tor-正交维数(相对于ω_R)这一概念,并且给出了一种计算模的这种相对右Tor-正交维数的准则.对一个交换、凝聚的半局部环R和一有限表示的Tor-自正交模R-模ω,将证明ω的平坦维数与R/J的右Tor-正交维数(相对于ω_R)是相等的,其中J为环R的Jacobson根.作为上述结果的推论,得到了ω有有限的平坦维数当且仅当每一个(有限表示的)R-模有有限的右Tor-正交维数(相对于ω_R)的结论.最后,若ωR是一Tor-倾斜模,得到了一个左R-模有有限右Tor-正交维数(相对于ω_R)的一些等价条件.

Let R be a ring and ωR a Tor-self-orthogonal module. The notion of the right Tor-orthogonal dimension （relative to ωR） of modules was introduced and a criterion for computing this relative right Tor- orthogonal dimension of modules was given. For a commutative coherent and semilocal ring R and a finitely presented Tor-self-orthogonal R-module w, it was shown that the flat dimension of w and the right Tor-orthogonal dimension （relative to w） of R/J are identical, where J is the Jacobson radical of R. As a consequence, it was obtained that w has a finite flat dimension if and only if every （finitely presented） R-module has a finite right Tor-orthogonal dimension （relative to ω）. If ωR is a Tor-tilting module then some equivalent conditions of a left R-module has a finite right Tor-orthogonal dimension （relative to ωR）.