[Paper Review] Model category structures arising from Drinfeld vector bundles

Sergio Estrada, Pedro A. Guil Asensio+2 more|ArXiv.org|Jun 29, 2009

Algebraic structures and combinatorial models10 references4 citations

TL;DR

This paper introduces a general construction of monoidal model category structures on unbounded chain complexes of quasi-coherent sheaves on semi-separated schemes, using compatible filtrations of module sections across open affines. It extends previous results on flat and vector bundle model structures without requiring closure under direct limits, and proves that unrestricted flat Mittag-Leffler sheaves do not generally yield such structures.

ABSTRACT

We present a general construction of model category structures on the category $\mathbb{C}(\mathfrak{Qco}(X))$ of unbounded chain complexes of quasi-coherent sheaves on a semi-separated scheme $X$. The construction is based on making compatible the filtrations of individual modules of sections at open affine subsets of $X$. It does not require closure under direct limits as previous methods. We apply it to describe the derived category $\mathbb D (\mathfrak{Qco}(X))$ via various model structures on $\mathbb{C}(\mathgrak{Qco}(X))$. As particular instances, we recover recent results on the flat model structure for quasi-coherent sheaves. Our approach also includes the case of (infinite-dimensional) vector bundles, and of restricted flat Mittag-Leffler quasi-coherent sheaves, as introduced by Drinfeld. Finally, we prove that the unrestricted case does not induce a model category structure as above in general.

Motivation & Objective

• To construct monoidal model category structures on unbounded chain complexes of quasi-coherent sheaves over semi-separated schemes.
• To generalize existing flat and vector bundle model structures without assuming closure under direct limits.
• To characterize when the unrestricted class of flat Mittag-Leffler quasi-coherent sheaves induces a model structure.
• To establish a framework for describing the derived category via various compatible model structures.
• To prove that the unrestricted flat Mittag-Leffler class fails to yield a model structure in general.

Proposed method

• Construct model structures on $\mathbb{C}(\mathfrak{Qco}(X))$ using compatible filtrations of sections over open affine subsets of a semi-separated scheme $X$.
• Define cofibrations as monomorphisms with cokernels in $\mathit{dg}\,\widetilde{\mathcal{C}}$, and fibrations as epimorphisms with kernels in $\mathit{dg}\,\widetilde{\mathcal{C}^\perp}$, for suitable classes $\mathcal{C}$.
• Ensure monoidality by requiring that the class $S_v$ of sections at each open affine $v$ is closed under tensor products and consists of flat modules over $\mathscr{R}(v)$.
• Use a new filtration technique that avoids reliance on direct limit closure, enabling broader applicability than prior methods.
• Apply the construction to recover known results on flat and vector bundle model structures as special cases.
• Prove non-existence of model structures for unrestricted flat Mittag-Leffler sheaves via homological obstruction arguments involving $\aleph_1$-free groups.

Experimental results

Research questions

• RQ1 Can model category structures on $\mathbb{C}(\mathfrak{Qco}(X))$ be constructed without assuming closure under direct limits?
• RQ2 Under what conditions does the class of Drinfeld vector bundles induce a monoidal model structure on $\mathbb{C}(\mathfrak{Qco}(X))$?
• RQ3 Does the unrestricted class of flat Mittag-Leffler quasi-coherent sheaves admit a model category structure compatible with the abelian structure?
• RQ4 How do the derived category $\mathbb{D}(\mathfrak{Qco}(X))$ and its morphisms relate to various model structures on $\mathbb{C}(\mathfrak{Qco}(X))$?
• RQ5 What homological obstructions prevent the unrestricted flat Mittag-Leffler class from inducing a cofibrantly generated model structure?

Key findings

• A general construction of model category structures on $\mathbb{C}(\mathfrak{Qco}(X))$ is established for semi-separated schemes, using compatible filtrations of sections without requiring closure under direct limits.
• The construction recovers the monoidal flat model structure on $\mathbb{C}(\mathfrak{Qco}(X))$ for quasi-compact, semi-separated schemes as a special case.
• The class of infinite-dimensional vector bundles on $X$ induces a monoidal model structure on $\mathbb{C}(\mathfrak{Qco}(X))$ when $X$ admits enough such bundles.
• The unrestricted class of flat Mittag-Leffler quasi-coherent sheaves does not induce a model category structure on $\mathbb{C}(\mathfrak{Qco}(X))$ in general.
• The class of all $\aleph_1$-free abelian groups does not admit a precovering class in $\mathrm{Mod}$-$\mathbb{Z}$, implying it cannot induce a cofibrantly generated model structure.
• A contradiction is derived by showing that $\mathrm{Hom}_{\mathbb{Z}}(G_{\mu^+}, B) = 0$ for a constructed $\aleph_1$-free group $G_{\mu^+}$ of infinite rank, contradicting the existence of a $\mathcal{D}$-precover of $\mathbb{Q}$.