the behavior of homological dimensions
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abstract
let r be a commutative noetherian ring. we study the behavior of injectiveand at dimension of r-modules under the functors homr(-,-) and -×r-.
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The behavior of homological dimensions
Let R be a commutative noetherian ring. We study the behavior of injectiveand at dimension of R-modules under the functors HomR(-,-) and -×R-.
full textHomological dimensions of complexes of R-modules
Let R be an associative ring with identity, C(R) be the category of com-plexes of R-modules and Flat(C(R)) be the class of all at complexes of R-modules. We show that the at cotorsion theory (Flat(C(R)); Flat(C(R))−)have enough injectives in C(R). As an application, we prove that for each atcomplex F and each complex Y of R-modules, Exti (F,X)= 0, whenever Ris n-perfect and i > n.
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We define homological dimensions for S-algebras, the generalized rings that arise in algebraic topology. We compute the homological dimensions of a number of examples, and establish some basic properties. The most difficult computation is the global dimension of real K-theory KO and its connective version ko at the prime 2. We show that the global dimension of KO is 2 or 3, and the global dimen...
full textHomological Dimensions and Regular Rings
A question of Avramov and Foxby concerning injective dimension of complexes is settled in the affirmative for the class of noetherian rings. A key step in the proof is to recast the problem on hand into one about the homotopy category of complexes of injective modules. Analogous results for flat dimension and projective dimension are also established.
full texthomological dimensions of complexes of r-modules
let r be an associative ring with identity, c(r) be the category of com-plexes of r-modules and flat(c(r)) be the class of all at complexes of r-modules. we show that the at cotorsion theory (flat(c(r)); flat(c(r))−)have enough injectives in c(r). as an application, we prove that for each atcomplex f and each complex y of r-modules, exti (f,x)= 0, whenever ris n-perfect and i > n.
full textThe Frobenius endomorphism and homological dimensions
In 1969 Kunz [Ku] proved a fundamental result, connecting the regularity of a local ring of positive characteristic with the flatness of its Frobenius endomorphism φ. This was a first indication of the important role that φ would play in homological commutative algebra, especially in reflecting basic homological properties of the ring. Some results in Peskine and Szpiro’s groundbreaking thesis,...
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Journal title:
نظریه تقریب و کاربرد های آنجلد ۷، شماره ۱، صفحات ۱-۱۰
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