Homologically homogeneous rings

Abstract

In this paper we study the structure of a right Noetherian ring R of finite right global dimesion integral over a central subring C and satisfying the following condition: if V, W are irreducible right K-modules with rc(V) = rc(W) then prdim(K) = pràim(W). 1. Introduction. Let £ be a ring with a central subring C. We shall say that R is homologically homogeneous (horn, horn.) over C if (i) R is right Noetherian; (ii) R is integral over C; (iii) the right global (projective) dimension of R (denoted rtgldim(£)) is finite; and (iv) if V and M are irreducible right £-modules whose annihilators in C, namely rc(V) and rc(W), are equal then prdimR(K) = prdim^H7). (Here, prdimR(F) denotes the projective dimension of the £-module V.) This paper is devoted to the study of such rings. Examples of them include (a) commutative Noetherian rings of finite global dimension; (b) hereditary Noetherian rings integral over their centers with no Artinian ideals; (c) right Noetherian local rings of finite right global dimension which are integral over their centers; and (d) certain rings arising in the representation theory of semisimple Lie groups [2]. In addition, we show in §7 that polynomial rings over horn. horn, rings, and certain group rings, are horn. horn. Rings of type (c) were studied in [5] where we showed that their maximal ideals had rank equal to their grade, defined in terms of maximal lengths of C-sequences. We termed such rings C-Macaulay-a precise definition is given in §2-and showed in [5, §4] that they share many of the properties of commutative Cohen-Macaulay

Reconstruction algebras of type 𝐴

“…We also remark that in general the reconstruction algebra is not homologically homogenous in the sense of Brown-Hajarnavis [BH84]. This should not be surprising, as there are many other examples of non-commutative resolutions of sensible non-Gorenstein Cohen-Macaulay singularities which are not homologically homogeneous ( [QS06] and [SdB06, 5.1(2)]).…”

Group rings which are v-HC orders and Krull orders

“…This result and a reduction from Section 2, in the spirit of that used for Theorem A, together serve to reduce the problem to the case of a group algebra of an abelian-by-finite group over a coefficient field, as is shown in Section 4. Finally this special case is treated in Sections 5 and 6, exploiting the fact that the algebra is a finitely generated Cohen-Macaulay module over its centre [3].…”

“…A Noetherian ring S which is a finite module over a central subring C is centrally Macaulay if S is a Cohen-Macaulay C-module. The key property we require concerning such rings is stated in the next lemma, which, though implicit in [2], [3], [4], is not stated explicitly there. (ii) Let c be a regular element of S. Since S is a finite Z-module, O^cSnZ.…”

Krull and Global Dimensions of Fully Bounded Noetherian Rings

“…There is a finite chain of submodules of X with each factor (Krull) critical, not Artinian, and having prime annihilator [12, 2.5 and 3.2]. It is clearly enough to prove the result when X is itself such a critical module, with prime annihilator P. By [12, 1.1(b) and 2.5] there is an exact sequence of right i?-modules (4) 0 -» X '-* R/P -» W -0.…”