572

AL . ROSENBERG

GAFA

Proof : It is just the application of Proposition 6.5.2 in the case when S = 0.

D

6 . 5 . 3 The Zariski topology in the affine case. Let A be the category R—mod of left modules over an associative ring R. Recall that a closed subset W of a topological space X is noetherian if any family Vt of closed subsets of X such that X equals to Пг^п ^ contains a finite subfamily which has the same property. In other words, the closed set W is noetherian iff the open set X — W \s quasi-compact.

6 . 5 . 3 . 1 Proposition. A closed in the Zariski topology subset W is rian if and only if it coincides with Spec{R — mod \ a) for some finitely generated two-sided ideal a.

Proof : 1) By Proposition 6.4.1, any left closed subcategory of the category R — mod equals to iî — m.od \ a' for some two-sided ideal a'. The left closed subcategory R — mod | a is finite if and only if the ideal a is finitely generated as a two-sided ideal. Therefore, according to Corollary 6.5.2.1, the closed set Spec(i? — m.od \ a) is noetherian for any finitely generated two-sided ideal a.

2 ) Suppose now that the closed set V — Spec(i? — m,od \ a') is rian. The two-sided ideal a' is the supremum (union) of an inductive tem T{a') of its finitely generated two-sided subideals. This implies that Spec(iî — m,od \a') is the intersection of Spec(i? — m.od | a), where a runs through the set T{a'). Since the topological space Spec(iî — mod \ a') is noetherian, it coincides with Spec(iî — mod \ a) for some a G T{a'). и

6 . 5 . 3 . 2 Corollary. For any associative ring R, the topological space {SpecR — mod^ -2^), where ZT is the Zariski topology is quasi-compact and has a base of quasi-compact open subsets.

6 . 5 . 3 . 3 Remark: Proposition 6.5.3.1 has been obtained in [R2] (a detailed account is in [R3]) as a corollary of the following, much more subtle, fact:

The intersection of all ideals of the left spectrum of a ring R coincides with the biggest locally nilpotent ideal in R.

One of the consequences of this theorem is that the topological space Spec(i? — mod^ ZT) is quasi-homeomorphic to the Levitzki spectrum of R which is, by definition, the subspace of the prime spectrum, Speci? formed by all the prime ideals p in iî such that the quotient ring R/p has no nonzero locally nilpotent ideals.

Note that the Levitzki spectrum, L Spec Л, is a sober space; i.e. any irreducible closed subset of LSpeci? has unique generic point (Theorem 5.3 in [R3]). о