ugg boots greece Three local ring of homological c

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Three homological characterizations of local rings

. Derived conclusions. Theorem 2 Let (R,[link widoczny dla zalogowanych],,) is a local ring, dimR = n, E-E (), MHomR (, E), the following conclusions established; (1) R is a c-M ring of any finite length Velvet norm Ext ~ (M,[link widoczny dla zalogowanych], R) 一 0 (Kn) and dried bauxite (R) 10 (open eyes); (2) R is a G. renstein ring of any finite length module Gan have E (, R) Rangan H (R): {. ≠:; I ¨ a nJ (3) R is a regular ring for any finite length module Gan have E (,) 10. Proof (1) note Ext, H and the depth of the relationship and depthR ~ dimR,[link widoczny dla zalogowanych], we have: R is a c-M ring ~ = ~ depthR = n ~ ~ min {iIExt '~ (R / m, R) ≠ 0) A rain {iIH (R) ≠ 0} = nC =* Ext '~ (R / m,[link widoczny dla zalogowanych], R) a H (R) 10 ( dimM), to permit (1) as long as another card when the E by the (R /. R) = 0 (<), then for any finite length of the norm Ext ~ (M, R) = o (<n). Constant (K), of f () for induction. Obviously f () l, the establishment of a set. A】 1110,, (1) a f () a l <+ oo, by the long exact sequence with E. rt ~ (1, R) a E = o. (2) first probands equivalence relation. E (R / P)) 10, because Hom (Rim, E (R / P)) = {oT ∈ E (R / P) ITtL, T ~ 0),: 0 ÷ 0 () P, so 0 with same as (1) of the induction to prove that when () <+ oo when: Horn (, E (R / P)) = 0, where p4: So E cockroach (, R) Blue Island Ⅳ (,) A Horn (M, E) = M. . ,), (R) = Ⅳ: (R). 0R0R0 『<R ≈ ... 0 Piece Goods. destroyed (R) 0 and thus H (R)-E has finite flat dimension. H (R = F = given by E Maths knowledge is limited on the length of dual-mode, so E has finite projective dimension. Foxby's conclusions are from the R is a Gorenstein ring. (3) More articles related to topics：

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