Weight filtrations via commuting automorphisms

We consider a filtration of theK-theory space for a regular Noetherian ring proposed by Goodwillie and Lichtenbaum and show that its successive quotients are geometric realizations of explicit simplicial Abelian groups. The filtration in weightt involvest-tuples of commuting automorphisms of projectiveR-modules. It remains to show that the Adams operations act appropriately on the filtration.Supported by NSF grant DMS 90-02715. I thank Tom Goodwillie, Stephen Lichtenbaum, Friedhelm Waldhausen, Steven Landsburg, and Stephen Ullom for useful discussions and ideas.     

On the boundary value problem for the spectrally loaded heat conduction operator

We consider the boundary value problems in a quarter-plane for a loaded heat conduction operator (one-dimensional in the space variable). A peculiarity of the operator in question is as follows: first, the spectral parameter is the coefficient of the loaded summand; second, the order of the derivative in the loaded summand is equal to that of the differential part of the operator, and third, the load point moves with a variable velocity. We demonstrate that the boundary value problem under study is Noetherian.     

Homological behavior of idempotent subalgebras and Ext algebras

Let A be a(left and right) Noetherian ring that is semiperfect. Let e be an idempotent of A and consider the ring Γ :=(1-e)A(1-e) and the semi-simple right A-module Se := e A/e rad A. In this paper, we investigate the relationship between the global dimensions of A and Γ, by using the homological properties of Se. More precisely, we consider the Yoneda ring Y(e) := Ext_A~*(Se, Se) of e. We prove that if Y(e) is Artinian of finite global dimension, then A has finite global dimension if and only if so does Γ. We also investigate the situation where both A and Γ have finite global dimension. When A is Koszul and finite dimensional, this implies that Y(e) has finite global dimension. We end the paper with a reduction technique to compute the Cartan determinant of Artin algebras. We prove that if Y(e) has finite global dimension, then the Cartan determinants of A and Γ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture.     

Some Comments on the Jacobson Conjecture

Noetherian rings with Krull dimension one are shown to have closed left ideals in the J-adic topology. The radical of these rings also satisfies the AR property.     

Application of Full Quivers of Representations of Algebras,to Polynomial Identities

In [7 Belov , A. , Rowen , L. H. , Vishne , U. Full quivers of representations of algebras. To appear in Trans. Amer. Math. Soc.  [Google Scholar]] we introduced the notion of full quivers of representations of algebras, which are more explicit than quivers of algebras, and better suited for algebras over finite fields. Here, we consider full quivers as a combinatorial tool in order to describe PI-varieties of algebras. We apply the theory to clarify the proofs of diverse topics in the literature: Determining which relatively free algebras are weakly Noetherian, determining when relatively free algebras are finitely presented, presenting a quick proof for the rationality of the Hilbert series of a relatively free PI-algebra, and explaining counterexamples to Specht's conjecture for varieties of Lie algebras.     

On the Finiteness of Noetherian Rings with Finitely Many Regular Elements

Let R be a left Noetherian ring and ZD(R) be the set of all zero-divisors of R. In this paper, it is shown that if R \ ZD(R) is finite, then R is finite.     

The Ideal Completion of a Noetherian Local Domain

The ideal topology on a integral domain R is the linear topology which has as a fundamental system of neighborhoods of 0 the nonzero ideals of R. We investigate the properties of the ideal topology on a Noetherian local domain (R, 𝔪), and we establish connections between the 𝔪-adic completion and the ideal completion. We give conditions under which the completion in the ideal topology is Noetherian, and we show that, unlike the 𝔪-adic completion, the completion in the ideal topology is not always Noetherian.     

Extension closure of k-torsionfree modules

Let Λ be a left and right Noetherian ring. For a positive integer k, we give an equivalent condition that flat dimensions of the first k terms in the minimal injective resolution of Λ are less than or equal to k. In this case we show that the subcategory consisting of k-torsionfree modules is extension closed. Moreover we prove that for a Noetherian algebra every subcategory consisting of i-torsionfree modules is extension closed for any 1 ≤ i ≤ k if and only if every subcategory consisting of i-th syzygy modules is extension closed for any 1 ≤ i ≤ k. Our results generalize the main results in Auslander and Reiten [4].