[Paper Review] Generators and representability of functors in commutative and noncommutative geometry

Alexey Bondal, Michel Van den Bergh|ArXiv.org|Apr 17, 2002

Algebraic structures and combinatorial modelsMathematics10 references75 citations

TL;DR

This paper establishes a sufficient condition for triangulated categories in commutative and noncommutative algebraic geometry to be saturated—meaning all finite-type cohomological functors are representable—by proving that the existence of a strong generator in an Ext-finite, Karoubian triangulated category implies saturation. The key result shows that bounded derived categories of coherent sheaves on smooth proper varieties (commutative or noncommutative) are saturated due to the presence of strong generators.

ABSTRACT

We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, hence saturated. In contrast the similar category for a smooth compact analytic surface with no curves is not saturated.

Motivation & Objective

• To provide an intrinsic criterion for saturation in triangulated categories arising in algebraic geometry.
• To address the representability of finite-type cohomological functors in derived categories.
• To prove that smooth proper varieties (commutative and noncommutative) have saturated derived categories via strong generators.
• To generalize Theorem 1.1 on representability to noncommutative settings using graded rings and qgr categories.

Proposed method

• Introduce the concept of a strong generator in a triangulated category, defined as an object whose iterated extensions and direct summands generate the entire category within finitely many steps.
• Use n-resolutions of functors with respect to subcategories to approximate cohomological functors in the absence of homotopy limits.
• Adapt techniques from Brown representability, modifying them to work in the absence of infinite sums by using finite approximations via strong generators.
• Prove that classical generators in smooth schemes are also strong generators, linking geometric smoothness to finitary generation.
• Apply the criterion to show that the derived category of coherent sheaves on smooth proper varieties is saturated.
• Use results from Keller and DG/A∞-geometry to show that quasi-compact, quasi-separated schemes are affine in a derived sense.

Experimental results

Research questions

• RQ1 Under what conditions is a triangulated category saturated, i.e., all finite-type cohomological functors representable?
• RQ2 Does the existence of a strong generator in an Ext-finite, Karoubian triangulated category imply saturation?
• RQ3 Can the representability result for coherent sheaves on smooth projective varieties be extended to noncommutative settings?
• RQ4 Do smooth proper schemes in noncommutative geometry also have saturated derived categories?
• RQ5 What is the relationship between classical generators and strong generators in smooth schemes?

Key findings

• A triangulated category that is Ext-finite, Karoubian, and admits a strong generator is saturated.
• The bounded derived category of coherent sheaves on any smooth proper variety—commutative or noncommutative—has a strong generator and is therefore saturated.
• Every quasi-compact, quasi-separated scheme admits a classical generator, and in the smooth case, such a generator is also a strong generator.
• The derived category of perfect complexes on a possibly singular projective variety over a field has representable finite-type cohomological functors.
• There exist examples of smooth compact analytic surfaces with no curves whose derived category is not saturated, showing the necessity of geometric conditions.
• The category qgr(R) for a graded coherent ring R is shown to be saturated when R satisfies appropriate finiteness and generation conditions.