Ziegler spectra of serial rings

Ziegler Spectra of Valuation Rings

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Serial Rings

A module is called uniseriat if it has a unique composition series of finite length. A ring (always with 1) is called serial if its right and left free modules are direct sums of uniserial modules. Nakayama, who called these rings generalized uniserial rings, proved [21, Theorem 171 that every finitely generated module over a serial ring is a direct sum of uniserial modules. In section one we g...

The Structure of Serial Rings

A serial ring (generalized uniserial in the terminology of Nakayama) is one whose left and right free modules are direct sums of modules with unique finite composition series (uniserial modules.) This paper presents a module-theoretic discussion of the structure of serial rings, and some onesided characterizations of certain kinds of serial rings. As an application of the structure theory, an e...

The Ziegler and Zariski spectra of some domestic string algebras

Spectra, Spectra, Spectra – Tensor Triangular Spectra versus Zariski Spectra of Endomorphism Rings

We construct a natural continuous map from the triangular spectrum of a tensor triangulated category to the algebraic Zariski spectrum of the endomorphism ring of its unit object. We also consider graded and twisted versions of this construction. We prove that these maps are quite often surjective but far from injective in general. For instance, the stable homotopy category of finite spectra ha...