Projective dimension is a lattice invariant

We show that, for a free abelian group $G$ and prime power $p^\nu$, every direct sum decomposition of the group $G/p^\nu G$ lifts to a direct sum decomposition of $G$. This is the key result we use to show that, if $R$ is a commutative von Neumann regular ring, and $\mathcal{E}$ a set of idempotents in $R$, then the projective dimension of the ideal $\mathcal{E} R$ as an $R$-module is the same as the projective dimension of the ideal $\mathcal{EB}$, where $\mathcal{B}$ is the boolean algebra generated by $\mathcal{E} \cup \{1\}$. This answers a thirty year old open question of R. Wiegand. The proof is based on gaussian elimination on an $\omega \times \omega$ matrix, with adaptations enabling one to pass from the integers modulo $p^\nu$ to the integers.Comment: LaTex. 16 page

A hierarchy of parametrizing varieties for representations

The primary purpose is to introduce and explore projective varieties, $\text{GRASS}_{\bf d}(\Lambda)$, parametrizing the full collection of those modules over a finite dimensional algebra $\Lambda$ which have dimension vector $\bf d$. These varieties extend the smaller varieties previously studied by the author; namely, the projective varieties encoding those modules with dimension vector $\bf d$ which, in addition, have a preassigned top or radical layering. Each of the $\text{GRASS}_{\bf d}(\Lambda)$ is again partitioned by the action of a linear algebraic group, and covered by certain representation-theoretically defined affine subvarieties which are stable under the unipotent radical of the acting group. A special case of the pertinent theorem served as a cornerstone in the work on generic representations by Babson, Thomas, and the author. Moreover, applications are given to the study of degenerations

Atomic Resonance and Scattering

Contains reports on six research projects.National Science Foundation (PHY83-06273)Joint Services Electronics Program (DAAL03-86-K-0002)National Science Foundation (PHY84-11483)U.S. Navy-Office of Naval Research (Grant N00014-79-C-0183)Joint Services Electronics Program (Contract DAAG29-83-K-0003)National Science Foundation (Grant PHY83-07172-A01)U.S. Navy - Office of Naval Research (Grant N00014-83-K-0695)National Science Foundation (Grant CHE84-21392

Cross-cutting principles for planetary health education

Since the 2015 launch of the Rockefeller Foundation Lancet Commission on planetary health,1 an enormous groundswell of interest in planetary health education has emerged across many disciplines, institutions, and geographical regions. Advancing these global efforts in planetary health education will equip the next generation of scholars to address crucial questions in this emerging field and support the development of a community of practice. To provide a foundation for the growing interest and efforts in this field, the Planetary Health Alliance has facilitated the first attempt to create a set of principles for planetary health education that intersect education at all levels, across all scales, and in all regions of the world—ie, a set of cross-cutting principles

Projective dimension is a lattice invariant

Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7 Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal