Intersection sheaves are usually defined for a proper flat surjective morphism of Noetherian schemes of relative dimension d and for d + 1 invertible sheaves on the ambient scheme. In this article, the construction is generalized to the equidimensional proper surjective morphisms over normal separated Noetherian schemes. Applications to the studies on family of effective algebraic cycles and on...

We study row-finite Leavitt path algebras. We characterize the row-finite graphs E for which the Leavitt path algebra is weakly Noetherian. Our main result is that a Leavitt path algebra is weakly Noetherian if and only if there is ascending chain condition on the hereditary and saturated closures of the subsets of the vertices of the graph E.

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.

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...

Throughout this paper all rings are commutative, with identity, and Noetherian, unless otherwise specified. In particular, “local ring” always means Noetherian local ring, unless otherwise specified. Our objective is to prove a generalization of Faltings’ connectedness theorem [Fal1, Fal2], which asserts that in a complete local domain (R,m,K) of dimension n, if I ⊆ m is an ideal generated by a...