Gena Puninski

Classifying projective modules over some semilocal rings

We classify all (finitely generated or not) projective modules over a class of semilocal ring constructed using nealy simple uniserial domains. They in turn are connected with noncommutative valuations constructed using embeddings of right ordered groups into skew fields.

The Krull–Gabriel Dimension of a Serial Ring

Abstract We prove that every serial ring R has the isolation property: every isolated point in any theory of modules over R is isolated by a minimal pair. Using this we calculate the Krull–Gabriel dimension of the module category over serial rings. For instance, we show that this dimension cannot be equal to 1.

The Ziegler Spectrum and Ringel’s Quilt of the A-infinity Plane Curve Singularity

We describe the Cohen–Macaulay part of the Ziegler spectrum and calculate Ringel’s quilt of the category of finitely generated Cohen–Macaulay modules over the A-infinity plane curve singularity.

ITERATED POWER INTERSECTIONS OF IDEALS IN RINGS OF ITERATED DIFFERENTIAL POLYNOMIALS

Let R be an n-iterated ring of differential polynomials over a commutative noetherian domain which is a ℚ-algebra. We will prove that for every proper ideal I of R, the (n + 1)-iterated intersection I(n + 1) of powers of I equals zero. A standard application includes the freeness of non-finitely generated projective modules over such rings. If I is a proper ideal of the universal enveloping algebra of a finite-dimensional solvable Lie algebra over a field of characteristic zero, then we will improve the above estimate by showing that I(2) = 0.