… fact that every complete local ring is the quotient of a regular local ring, we prove theorems on
… regular ring, naturally extending the definition from local rings to arbitrary noetherian rings. …

… rings (with special emphasis on local rings) which we have obtained using the concepts
and techniques of homological algebra, recently introduced by H. Cartan and S. Eilenberg. …

… homological dimension in local rings. In the first section, I show that no non-zero ideal of finite
homological dimension in a local ring … ideal of homological dimension one in a local ring Q …

… coincide with their homology groups. The result for Ext is proved similarly. THEOREM 11. …
Examples and applications As in ?2 we assume that A is a graded ring with radical N = No + A' …

… neat over a local ring has finite homological dimension if and only if the maximal ideal do
es… For the rest of this chapter R denotes a commutative, noetherian, local ring with maximal …

… The characterization of regular local rings as those of finite homological dimension, first …
of local complete intersections Here, we will restrict ourselvesto local rings which can be …

… Since for each m we have that Rm is a local ring and hdRm Em < oo, it follows from [2, 1.7]
that hdRm Em?codim Rm for all m. From this we conclude that hdR E? <supm codimRm Rm = …

… In this paper we consider local rings and, unless otherwise … dimension over a ring R
characterizes regular rings in the … In the first part of this paper we introduce homological dimension …

… homological dimensions in ring theory were in the study of commutative noetherian rings …
The outstanding result obtained by these methods was the theorem that a regular local ring …

Let (R,m) be a Noetherian local ring of depth d and C a semidualizing R-… C -dimension of M
is at most t or C is a dualizing complex. Analogous results for other homological dimensions …