[Paper Review] A Note on Support in Triangulated Categories

Aaron Bergman|ArXiv.org|Apr 24, 2008

Algebraic structures and combinatorial models17 references3 citations

TL;DR

This paper introduces a new notion of compact support in triangulated categories, defined by finite-dimensional total hom-spaces from compact objects. It establishes that compactly supported objects in derived categories of coherent sheaves or modules correspond precisely to those with proper support or finite-dimensional cohomology, respectively, and applies this to derived Morita equivalence and D-brane physics at singularities.

ABSTRACT

In this note, I define a notion of a compactly supported object in a triangulated category. I prove a number of propositions relating this to traditional notions of support and give an application to the theory of derived Morita equivalence. I also discuss a connection to supersymmetric gauge theories arising from D-branes at a singularity.

Motivation & Objective

• To define and study compactly supported objects in triangulated categories with small coproducts.
• To relate this new notion of compact support to classical notions of support in algebraic geometry and representation theory.
• To establish equivalences between compactly supported subcategories and those with proper support or finite-dimensional cohomology.
• To apply the results to derived Morita equivalence and to supersymmetric gauge theories arising from D-branes at singularities.
• To provide a homological criterion for proper support using Ext-finite conditions on structure sheaves of curves.

Proposed method

• Define compactly supported objects via the finiteness of ∑ᵢ dim Hom(ℱ, 𝒪[i]) for all compact ℱ.
• Prove that the full subcategory of compactly supported objects is triangulated using the long exact sequence of Hom-functors.
• Use the hypercohomology spectral sequence and truncation techniques to relate compact support to cohomological finiteness.
• Apply the valuative criterion for properness to show that finite Ext-dimensions from structure sheaves of curves imply proper support.
• Leverage smoothness of varieties to ensure perfect complexes are bounded, enabling comparison with classical support theory.
• Use spectral sequences and long exact sequences to reduce the problem to cohomology modules and sheaf Ext groups.

Experimental results

Research questions

• RQ1 When does a bounded complex in D^b(Coh(X)) have compact support in the derived category?
• RQ2 What is the precise relationship between compact support and proper support of cohomology sheaves in smooth noetherian varieties?
• RQ3 How does compact support in D(A-Mod) relate to finite-dimensional cohomology modules of complexes?
• RQ4 Can the compactly supported subcategory be characterized in terms of Ext-finiteness from structure sheaves of curves?
• RQ5 What are the implications of compact support for derived Morita equivalence and D-brane configurations at singularities?

Key findings

• The subcategory of compactly supported objects in a triangulated category is triangulated and closed under shifts and extensions.
• For a noetherian variety X, a bounded complex of coherent sheaves is compactly supported if and only if all its cohomology sheaves have proper support.
• For an algebra A over a field, a bounded complex of A-modules is compactly supported if and only if all its cohomology modules are finite-dimensional.
• Finite-dimensional Ext groups from structure sheaves of reduced curves to a coherent sheaf 𝒪_C imply that the sheaf has proper support.
• In smooth noetherian varieties, the compactly supported subcategory of D^c(Coh(X)) is equivalent to the full subcategory of complexes with cohomology sheaves of proper support.
• The compact support condition is equivalent to the finiteness of ∑ᵢ dim Hom(𝒪_C, 𝒪_C[i]) for all reduced curves C in the support, under smoothness assumptions.