[Paper Review] Pure exact structures and the pure derived category of a scheme

Sergio Estrada, James Gillespie+1 more|arXiv (Cornell University)|Aug 12, 2014

Algebraic structures and combinatorial models11 references4 citations

TL;DR

This paper introduces the $⋯$-pure derived category of a scheme using $λ$-purity in closed symmetric monoidal Grothendieck categories, constructing a cofibrantly generated, injective model structure on unbounded chain complexes whose trivial objects are $⋯$-pure acyclic complexes. The key contribution is a homotopy-theoretic realization of the pure derived category via $⋯$-pure injective model structures, with applications to quasi-coherent sheaves on quasi-separated schemes and equivalence between the pure derived category and the derived category of flat sheaves.

ABSTRACT

Let $\mathcal C$ be closed symmetric monoidal Grothendieck category. We define the pure derived category with respect to the monoidal structure via a relative injective model category structure on the category $\mathbf{C}(\mathcal C)$ of unbounded chain complexes in $\mathcal C$. We use $λ$-Purity techniques to get this. As application we define the stalkwise pure derived category of the category of quasi--coherent sheaves on a quasi-separated scheme. We also give a different approach by using the category of flat quasi--coherent sheaves.

Motivation & Objective

• To define a geometrically meaningful notion of purity for quasi-coherent sheaves on schemes using stalk-wise exactness, rather than categorical purity.
• To resolve the discrepancy between categorical purity and geometric purity in non-affine schemes, where categorical purity is strictly stronger.
• To construct a homotopy-theoretic model for the pure derived category using $⋯$-pure acyclic complexes and injective model structures.
• To show that the pure derived category of a locally finitely presented category is equivalent to the derived category of its flat modules.
• To extend the theory of pure derived categories to schemes by leveraging $λ$-purity and $⋯$-pure injective model structures in closed symmetric monoidal Grothendieck categories.

Proposed method

• Uses $λ$-purity theory in locally $λ$-presentable Grothendieck categories to define $⋯$-pure monomorphisms and exact sequences.
• Constructs a cofibrantly generated, injective model structure on the category of unbounded chain complexes $\mathbf{C}(\mathcal{C})$ in a closed symmetric monoidal Grothendieck category $\mathcal{C}$, with trivial objects being $⋯$-pure acyclic complexes.
• Defines the fibrant objects as those complexes with $⋯$-pure injective components and exactness under $\otimes$-functors with all $S \in \mathcal{C}$, generalizing DG-injective complexes.
• Applies Hovey's correspondence to establish the model structure from an injective cotorsion pair in the exact category $\mathbf{C}(\mathcal{C})_{⋯}$ of $⋯$-pure exact sequences.
• Uses deconstructibility of the category of flat quasi-coherent sheaves $\mathrm{Flat}(X)$ on a scheme $X$ to construct an injective model structure on $\mathbf{C}(\mathrm{Flat}(X))$.
• Shows that the homotopy category of this model structure is equivalent to the derived category $\mathcal{D}(\mathrm{Flat}(X))$, and that this is equivalent to the pure derived category $\mathcal{D}_{\otimes\text{-pur}}(\mathfrak{Qcoh}(X))$.

Experimental results

Research questions

• RQ1 How can purity be redefined in the context of quasi-coherent sheaves on non-affine schemes to reflect their local geometric structure?
• RQ2 What model category structure realizes the pure derived category of a scheme using geometric purity instead of categorical purity?
• RQ3 Is the pure derived category of a locally finitely presented category equivalent to the derived category of its flat modules?
• RQ4 Can the $⋯$-pure derived category be constructed via a cofibrantly generated, injective model structure on unbounded chain complexes in a closed symmetric monoidal Grothendieck category?
• RQ5 Does the derived category of flat quasi-coherent sheaves on a scheme $X$ coincide with the homotopy category of a suitable model structure on $\mathbf{C}(\mathrm{Flat}(X))$?

Key findings

• The $⋯$-pure derived category $\mathcal{D}_{⋯\text{-pur}}(\mathcal{C})$ is realized as the homotopy category of a cofibrantly generated, injective model structure on $\mathbf{C}(\mathcal{C})$, where trivial objects are $⋯$-pure acyclic complexes.
• The fibrant objects in this model structure are precisely the contractible complexes with $⋯$-pure injective components, generalizing DG-injective complexes.
• For a quasi-separated scheme $X$, the $⋯$-pure derived category of $\mathfrak{Qcoh}(X)$ is equivalent to the derived category of flat quasi-coherent sheaves $\mathcal{D}(\mathrm{Flat}(X))$, establishing a geometrically meaningful pure derived category.
• The injective model structure on $\mathbf{C}(\mathrm{Flat}(X))$ has fibrant objects that are dg-cotorsion complexes with flat components, and trivial objects are acyclic complexes with flat cycles.
• The homotopy category of the $⋯$-pure injective model structure on $\mathbf{C}(\mathcal{C})$ is equivalent to the pure derived category $\mathcal{D}_{\mathrm{pur}}(\mathcal{A})$ when $\mathcal{A}$ is locally finitely presented.
• The equivalence between $\mathcal{D}_{\mathrm{pur}}(\mathcal{A})$ and $\mathcal{D}(\mathrm{Flat}(A))$ is established via correspondence of injective cotorsion pairs and weak equivalences in the respective model structures.