Noncommutative Projective Geometry

Abstract

This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.

Noncommutative curves and noncommutative surfaces

“…However, with more complicated examples one can exhibit nastier behaviour. For example, the polynomial ring T = S[z] satisfies Γ h (QGr(T ), πT, (+1)) = T , but Theorem 2.1.6 must still fail, since it fails for the factor ring S. What now goes wrong is that the shift functor s = (+1) in QGr(T ) is not ample; specifically, Definition 2.1.4(ii) fails for the map T S (see [76,Theorem 2.10]). The condition χ 1 and the analogues χ n , obtained by replacing Ext 1 by Ext n in its definition, are not well understood, although they do hold for most of the natural classes of algebras.…”

“…However, notice that, although U/S is infinite dimensional, The intrinsic reason why this holds will appear in §5, but it is easy to prove directly [76,Theorem 2.3]. Another amusing property of this ring is that the presentation and Hilbert series of S are independent of the characteristic of k. Thus (2.4) implies that the noetherian property fails to be preserved under reduction modulo p for any prime p!…”

“…Thus, if M /M is finite dimensional, then dim k Ext Unfortunately, not all noetherian connected graded rings satisfy χ 1 . The basic counterexample [76] is provided by…”

Torsion classes of finite type and spectra

“…The non-commutative analog of the category QGr A plays a prominent role in "non-commutative projective geometry" (see, e.g., [1,16,19]). …”

Non-Commutative Localization in Algebra and Topology

“…For example, van Oystaeyen and his school ( [139,140,142]) consider a certain class of graded rings for which they can use localizations to define their version of a noncommutative Proj-functor. A more restricted class of graded rings providing examples very close in behaviour to commutative projective varieties is studied by Artin, Zhang, Stafford and others (see [5,127] and refs. therein).…”