The negative side of cohomology for Calabi-Yau categories

We study integer-graded cohomology rings defined over Calabi-Yau categories. We show that the cohomology in negative degree is a trivial extension of the cohomology ring in non-negative degree, provided the latter admits a regular sequence of central elements of length two. In particular, the product of elements of negative degrees are zero. As corollaries we apply this to Tate-Hochschild cohomology rings of symmetric algebras, and to Tate cohomology rings over group algebras. We also prove similar results for Tate cohomology rings over commutative local Gorenstein rings.Comment: 14 page

Spectral Types of Field and Cluster O-Stars

The recent catalog of spectral types of Galactic O-type stars by Mai'z-Apella'niz et al. has been used to study the differences between the frequencies of various subtypes of O-type stars in the field, in OB associations and among runaway stars. At a high level of statistical significance the data show that O-stars in clusters and associations have earlier types (and hence presumably larger masses and/or younger ages) than those that are situated in the general field. Furthermore it is found that the distribution of spectral subtypes among runaway O-stars is indistinguishable from that among field stars, and differs significantly from that of the O-type stars that are situated in clusters and associations. The difference is in the sense that runaway O-stars, on average, have later subtypes than do those that are still located in clusters and associations.Comment: To be published in the October 2004 issue of the Astronomical Journal Included Figure 1, page

The Gorenstein defect category

We consider the homotopy category of complexes of projective modules over a Noetherian ring. Truncation at degree zero induces a fully faithful triangle functor from the totally acyclic complexes to the stable derived category. We show that if the ring is either Artin or commutative Noetherian local, then the functor is dense if and only if the ring is Gorenstein. Motivated by this, we define the Gorenstein defect category of the ring, a category which in some sense measures how far the ring is from being Gorenstein.Comment: 11 pages, updated versio