[Paper Review] Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves

Jan Šťovíček|arXiv (Cornell University)|Jan 22, 2013

Algebraic structures and combinatorial modelsMathematics47 references43 citations

TL;DR

This paper establishes a framework for constructing hereditary monoidal model structures on the derived category of quasi-coherent sheaves over schemes by leveraging complete hereditary cotorsion pairs in exact categories of Grothendieck type. It generalizes approximation theory and Quillen's small object argument to exact categories, enabling the construction of cofibrantly generated, monoidal model structures compatible with the tensor product, with key results for flat and locally projective sheaves on quasi-compact, separated schemes.

ABSTRACT

Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close connection of this theory to approximation theory and cotorsion pairs. We also discuss the motivating applications with the emphasis on constructing monoidal model structures for the derived category of quasi-coherent sheaves of modules over a scheme.

Motivation & Objective

• To develop a systematic framework for constructing compatible model structures on exact and symmetric monoidal exact categories.
• To generalize approximation theory and cotorsion pair techniques to exact categories of Grothendieck type, enabling the use of Quillen's small object argument.
• To provide a monoidal model structure on the derived category of quasi-coherent sheaves over schemes, compatible with the tensor product.
• To unify and extend existing results on derived categories of quasi-coherent sheaves using model-theoretic and homological algebra tools.
• To lay the foundation for applications in Gorenstein homological algebra, Grothendieck duality, and singularity categories via model structures.

Proposed method

• Utilizes complete hereditary cotorsion pairs (Flat-R, Cot-R) and (Vect-R, Vect-R⊥) as the core algebraic input for model structure construction.
• Applies Theorem 7.16 to generate hereditary model structures from complete cotorsion pairs in exact categories.
• Employs the small object argument and deconstructibility techniques to construct cofibrant and fibrant replacements.
• Uses the representation of quasi-coherent sheaves as modules over a poset of rings (via Enochs-Estrada) to analyze Grothendieck category structure.
• Applies adjoint functors F∗ and F∗ to lift resolutions from stalks to global sheaves, constructing ˇCech resolutions and proper flat resolutions.
• Verifies monoidality by checking that tensoring with a flat complex preserves exactness, ensuring compatibility with the symmetric monoidal structure.

Experimental results

Research questions

• RQ1 Can a monoidal model structure be constructed on the derived category of quasi-coherent sheaves over a general scheme, compatible with the tensor product?
• RQ2 How can approximation theory and cotorsion pairs be extended from module categories to exact categories of Grothendieck type?
• RQ3 Under what conditions does the small object argument apply in exact categories to generate complete cotorsion pairs?
• RQ4 What is the precise relationship between flat sheaves, vector bundles, and the resulting model structures in derived categories?
• RQ5 Can the theory be generalized to apply to singularity categories and dg categories via model-theoretic methods?

Key findings

• A hereditary monoidal model structure exists on C(Qcoh(R)) for any continuous flat representation R of a finite upper semilattice I, with cofibrant objects being retracts of filtered complexes built from flat R-modules.
• When Qcoh(R) ≅ Qcoh(X) for a quasi-projective scheme X over an affine scheme, a second hereditary monoidal model structure exists with cofibrant objects built from locally projective R-modules.
• The class of trivial objects in both model structures is Cac(Qcoh(R)), the exact category of acyclic complexes.
• The fibrant objects are characterized as the orthogonal class to the class of flat or locally projective modules, respectively, via the construction in Notation 7.6.
• The homotopy category of both model structures is equivalent to D(Qcoh(R)), the unbounded derived category of quasi-coherent sheaves.
• The model structures are cofibrantly generated, with generating sets formed from shifts of flat or locally projective modules.