Totally acyclic complexes

For a given class of modules \A, we denote by \widetilde{\A} the class of exact complexes $X$ having all cycles in \A, and by dw(\A) the class of complexes $Y$ with all components $Y_j$ in \A. We consider a two sided noetherian ring $R$ and we use the notations $\mathcal{GI}$ $(\mathcal{GF}, \mathcal{GP})$ for the class of Gorenstein injective (flat, projective respectively) $R$-modules. We prove (Theorem 1) that the following are equivalent: 1. Every exact complex of injective modules is totally acyclic. 2. Every exact complex of Gorenstein injective modules is in $\widetilde{\mathcal{GI}}$. 3. Every complex in $dw(\mathcal{GI})$ is dg-Gorenstein injective. Theorem 2 shows that the analogue result for complexes of flat and Gorenstein flat modules also holds. We prove (Corollary 1) that, over a commutative noetherian ring $R$, the equivalent statements in Theorem 1 (as well as their counterparts from Theorem 2) hold if and only if the ring is Gorenstein. Thus we improve on a result of Iyengar's and Krause's; in [18] they proved that for a commutative noetherian ring $R$ with a dualizing complex, the class of exact complexes of injectives coincides with that of totally acyclic complexes of injectives if and only if $R$ is Gorenstein. We are able to remove the dualizing complex hypothesis. In the second part of the paper we focus on two sided noetherian rings that satisfy the Auslander condition. We prove (Theorem 6) that for such a ring $R$ that also has finite finitistic flat dimension, every complex of injective (left and respectively right) $R$-modules is totally acyclic if and only if $R$ is a Gorenstein ring

Urban Field Geology for K-8 Teachers

Geologists have long recognized the value of field trips, and the National Science Education Standards recommend them for K-8 science curricula. However, many teachers avoid running field trips. This article presents the results of a series of summer institutes for urban geoscience teachers in Milwaukee, Wisconsin, in which field and laboratory activities were developed in the context of local geology and field trip practices of "Teaming Up", reducing novelty space, and pre- and post-trip activities were encouraged. Teachers were also introduced to the use of the Action Planning model to assist in implementing curricular changes within their schools to encourage the use of field trips. The researchers determined that the teachers in these workshops increased their personal belief in their ability to teach earth science more effectivly. They also felt more comfortable with content material and found that action plans were an effective way to enact change regarding the role of field trips in their schools and in their earth science curricula. Educational levels: Graduate or professional

Canonical Filtrations of Gorenstein Injective Modules

The principle "Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra" is given in [3]. There is a remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So perhaps it is surprising that tensor products of Gorenstein injective modules over Gorenstein rings of finite Krull dimension are Gorenstein injective. Our main result is in support of the principle. Over commutative, noetherian rings injective modules have direct sum decompositions into indecomposable modules. We will show that Gorenstein injective modules over Gorenstein rings of finite Krull dimension have filtrations analogous to those provided by these decompositions. This result will then provide us with the tools to prove that all tensor products of Gorenstein injective modules over these rings are Gorenstein injective.Comment: 9 pages; It has been accepted for publication in Proceedings of the American Mathematical Societ

Pure injective and absolutely pure sheaves

We study two notions of purity in categories of sheaves: the categorical and the geometric. It is shown that pure injective envelopes exist in both cases under very general assumptions on the scheme. Finally we introduce the class of locally absolutely pure (quasi--coherent) sheaves, with respect to the geometrical purity, and characterize locally Noetherian closed subschemes of a projective scheme in terms of the new class.Comment: Updated version. To appear in Proc. Edinburgh Math. So

Transfinite tree quivers and their representations

The idea of "vertex at the infinity" naturally appears when studying indecomposable injective representations of tree quivers. In this paper we formalize this behavior and find the structure of all the indecomposable injective representations of a tree quiver of size an arbitrary cardinal $\kappa$. As a consequence the structure of injective representations of noetherian $\kappa$-trees is completely determined. In the second part we will consider the problem whether arbitrary trees are source injective representation quivers or not.Comment: to appear in Mathematica Scandinavic