Let $R$ be a commutative ring and let $M$ be an $R$-module. In this article, we introduce the concept of the Zariski socles of submodules of $M$ and investigate their properties. Also we study modules with Noetherian second spectrum and obtain some related results.

Suppose X is a locally noetherian Deligne–Mumford stack. Definition 1.2 has an obvious variant X̂ét using the underlying smaller étale site Xét and the restriction Oc Xét of Oc X to this site. By [3, 12.7.4], the category of cartesian Oc X -modules on Xlis-ét is equivalent to the category of Oc Xét-modules on Xét: (1.1) ModXlis-ét,cart(Oc X ) ' ModXét(Oc Xét) Definition 1.3. Let X be a locally n...

A cohomological support, Supp A (M), is defined for finitely generated modules M over a left noetherian ring R, with respect to a ring A of central cohomology operations on the derived category of R-modules. It is proved that if the A-module Ext R (M, M) is noetherian and Ext R (M, R) = 0 for i ≫ 0, then every closed subset of Supp A (M) is the support of some finitely generated R-module. This ...

We generalize two major ways of obtaining derived equivalences, the tilting process by Happel, Reiten and Smalø and Happel’s Tilting Theorem, to the setting of finitely presented modules over right coherent rings. Moreover, we extend the characterization of quasi–tilted artin algebras as the almost hereditary ones to all right noetherian rings. We also give a streamlined and general presentatio...