#### Pregled bibliografske jedinice broj: 171175

## Noncommutative localization in noncommutative geometry

Noncommutative localization in noncommutative geometry // Noncommutative Localization in Algebra and Topology / Ranicki, Andrew (ur.).

London: Cambridge University Press, 2005. str. 220-313

**Naslov**

Noncommutative localization in noncommutative geometry

**Autori**

Škoda, Zoran

**Vrsta, podvrsta i kategorija rada**

Poglavlja u knjigama, pregledni

**Knjiga**

Noncommutative Localization in Algebra and Topology

**Urednik/ci**

Ranicki, Andrew

**Izdavač**

Cambridge University Press

**Grad**

London

**Godina**

2005

**Raspon stranica**

220-313

**ISBN**

052168160X

**Ključne riječi**

Noncommutative localization, noncommutative geometry

**Sažetak**

The aim of these notes is to collect and motivate the basic localization toolbox for the geometric study of ``spaces'' locally described by noncommutative rings and their categories of one-sided modules. We present the basics of Ore localization of rings and modules in great detail. Common practical techniques are studied as well. We also describe a counterexample to a folklore test principle for Ore sets. Localization in negatively filtered rings arising in deformation theory is presented. A new notion of the differential Ore condition is introduced in the study of localization of differential calculi. To aid the geometrical viewpoint, localization is studied with emphasis on descent formalism, flatness, abelian categories of quasicoherent sheaves and generalizations, and natural pairs of adjoint functors for sheaf and module categories. The key motivational theorems from the seminal works of Gabriel on localization, abelian categories and schemes are quoted without proof, as well as the related statements of Popescu, Eilenberg-Watts, Deligne and Rosenberg. Cohn universal localization does not have good flatness properties, but it is determined by the localization map already at the ring level, like the perfect localizations are. Cohn localization is here related to the quasideterminants of Gelfand and Retakh ; and this may help the understanding of both subjects.

**Izvorni jezik**

Engleski

**Znanstvena područja**

Matematika

**Napomena**

The web link is unofficial (temporary). The official version is only printed (Cambridge Univ. Press). ISBN-10: 052168160X, ISBN-13: 9780521681605

**POVEZANOST RADA**

**Projekt / tema**

0098003

**Ustanove**

Institut "Ruđer Bošković", Zagreb

**Autor s matičnim brojem:**

Zoran Škoda, (260060)