In this paper we classify noetherian hereditary abelian categories satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary categories. As a side result we show that when our hereditary categories have no nonzero projectives or injectives, then the Serre duality property is equivalent to the existence of almost ...

In this note we extend duality theorems for crossed products obtained by M. Koppinen and C. Chen from the case of a base field or a Dedekind domain to the case of an arbitrary noetherian commutative ground ring under fairly weak conditions. In particular we extend an improved version of the celebrated Blattner-Montgomery duality theorem to the case of arbitrary noetherian ground rings.

In this paper, we show the existence of copure injective preenvelopes over noetherian rings and copure flat preenvelopes over commutative artinian rings. We use this to characterize n-Gorenstein rings. As a consequence, if the full subcategory of strongly copure injective (respectively flat) modules over a left and right noetherian ring R has cokernels (respectively kernels), then R is 2-Gorens...

Let A be a semprime, right noetherian ring equipped with an automorphism α, and let B := A[[y;α]] denote the corresponding skew power series ring (which is also semiprime and right noetherian). We prove that the Goldie ranks of A and B are equal. We also record applications to induced ideals.

We classify all noetherian Hopf algebras H over an algebraically closed field k of characteristic zero which are integral domains of GelfandKirillov dimension two and satisfy the condition ExtH(k, k) 6= 0. The latter condition is conjecturally redundant, as no examples are known (among noetherian Hopf algebra domains of GK-dimension two) where it fails.

Let (R,m) be a commutative Noetherian complete local ring, a an ideal of R, and A an Artinian R-module with N-dim A = d. We prove that if d > 0, then Cosupp(H d−1(A)) is finite and if d ≤ 3, then the set Coass(H i (A)) is finite for all i. Moreover, if either d ≤ 2 or the cohomological dimension cd(a) = 1 then H i (A) is a-coartinian for all i; that is, Torj (R/a,H a i (A)) is Artinian for all ...

In this note the Briançon-Skoda theorem is extended to Noetherian filtrations of ideals in a regular ring. The method of proof couples the Lipman-Sathaye approach with results due to Rees. Let A be a regular ring of dimension d and I an ideal of A. Let I denote the integral closure of I. The Briançon-Skoda theorem asserts that if I is generated by l elements then In+l ⊆ I, for all nonnegative i...

Let A be a noetherian AS regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and grA the category of finitely generated graded left A-modules. Following Jørgensen, we define the CastelnuovoMumford regularity reg(M•) of a complex M• ∈ D(grA) in terms of the local cohomologies or the minimal projective resolution of M•. Let A be the quadratic dual ring of A. ...

Given a local Noetherian ring (R, m) of dimension d > 0 and infinite residue field, we study the invariants (dimension and multiplicity) of the Sally module SJ (I) of any m-primary ideal I with respect to a minimal reduction J. As a by-product we obtain an estimate for the Hilbert coefficients of m that generalizes a bound established by J. Elias and G. Valla in a local Cohen-Macaulay setting. ...