On Cohen–Macaulay Local Rings with Embedding Dimensione+d−2

On Sums of Squares in Local Rings

Let A be a semilocal ring. We compare the set of positive semidefinite (psd) elements of A and the set of sums of squares in A. For psd f ∈ A, whether f is a sum of squares or not depends only on the behavior of f in an infinitesimal neighborhood of the real zeros of f in Spec A. We apply this observation, first to 1-dimensional local rings, then to 2-dimensional regular semilocal rings. For th...

Convergent Higher Derivations on Local Rings

On Some Local Cohomology Invariants of Local Rings

Let A be a commutative Noetherian local ring containing a field of characteristic p > 0. The integer invariants λi,j(A) have been introduced in an old paper of ours. In this paper we completely describe λd,d(A) where d = dimA in terms of the topology of SpecA.

On K2 of 1-dimensional local rings

We study K2 of one-dimensional local domains over a field of characterstic 0, introduce a conjecture, and show that this conjecture implies Geller’s conjecture. We also show that Berger’s conjecture implies Geller’s conjecture, and hence verify it in many new cases.

Embedding Hypercubes onto Rings and Toruses

Many parallel algorithms use hypercubes as the communication topology among processes. When such algorithms are executed on hypercube multicomputers the communication cost is kept minimum since processes can be allocated to processors in such a way that only communication between neighbor processors is required. However, the scalability of hypercube multicomputers is constrained by the fact tha...