On Local Rings

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.

Orthogonal Groups over Local Rings

In an earlier paper [S] we have determined the structure of the linear groups over a local ring. In this note we continue the study of the classical groups over a local ring with the investigation of the orthogonal groups. Our main result (cf. Theorem 6 below) is a complete description of the invariant subgroups of an orthogonal group of noncompact type (i.e., of index ^ 1) over a local ring L ...