Noncommutative curves and noncommutative surfaces

Abstract

Abstract. In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories.Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic, growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has led to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing up and down, has been developed.

Noncommutative Proj and coherent algebras

“…The category of (nonnegatively) graded algebras is a subcategory of the category of Z-algebras: to every graded algebra A = ⊕ n≥0 A n one can associate a Z-algebra A Z = ⊕A ij with A ij = A j−i . As was observed in [11], sec. 11.1, the AZ-theorem can be extended to the case when an ample sequence of objects does not have the form (σ n (O), n ∈ Z) for some object O and some autoequivalence σ, by working with Z-algebras.…”

A remark on closed noncommutative subspaces

“…Following Rosenberg [3], Van den Bergh [8], and others (the reader is referred to [6] for an overview), the fundamental objects of study in noncommutative algebraic geometry are Grothendieck categories, interpreted as categories of sheaves on (not explicitly defined) noncommutative schemes. Key to this approach is the notion of a closed subspace, investigated in detail, for example, in [1; 3; 4; 5; 8].…”

Noncommutative two-tori with real multiplication as noncommutative projective varieties

“…In other words, the category of vector bundles on T θ should have a nontrivial Morita autoequivalence. Note however, that there exists a generalization of the standard approach to noncommutative projective schemes in which graded algebras are replaced by more general objects called Z-algebras (see [4], [19]). If one allows these more general noncommutative "Z-projective schemes" then the condition that T θ has real multiplication becomes unnecessary.…”