[Paper Review] Non-exactness of direct products of quasi-coherent sheaves

Ryo Kanda|arXiv (Cornell University)|Oct 20, 2018

Algebraic structures and combinatorial models18 references3 citations

TL;DR

This paper proves that for a noetherian scheme with an ample family of invertible sheaves (i.e., divisorial schemes), the category of quasi-coherent sheaves has exact direct products if and only if the scheme is affine. The result generalizes known non-exactness results for projective lines and punctured spectra, using the Gabriel-Popescu embedding and Roos’ characterization of Grothendieck categories satisfying Ab6 and Ab4*. The key contribution is establishing the equivalence between exactness of direct products and affineness in this broad class of schemes.

ABSTRACT

For a noetherian scheme that has an ample family of invertible sheaves, we prove that direct products in the category of quasi-coherent sheaves are not exact unless the scheme is affine. This result can especially be applied to all quasi-projective schemes over commutative noetherian rings. The main tools of the proof are the Gabriel-Popescu embedding and Roos' characterization of Grothendieck categories satisfying Ab6 and Ab4*.

Motivation & Objective

• To determine when direct products are exact in the category of quasi-coherent sheaves on a noetherian scheme.
• To generalize known non-exactness results for projective lines and punctured spectra to a broader class of schemes.
• To establish a structural characterization of when QCoh(X) satisfies Grothendieck's Ab4* condition.
• To show that Ab4* exactness implies affineness for divisorial noetherian schemes, using category-theoretic tools.

Proposed method

• Utilizes the Gabriel-Popescu theorem to embed the category of quasi-coherent sheaves into a category of modules over a ring.
• Applies Roos’ characterization of Grothendieck categories satisfying both Ab6 and Ab4* to analyze exactness of products.
• Employs the notion of bilocalizing subcategories and quotient categories to decompose the module category and analyze subobjects.
• Uses the fact that a noetherian scheme with an ample family of invertible sheaves is divisorial, enabling the application of Serre’s criterion for affineness.
• Applies the fully faithful pullback functor along closed immersions to relate exactness in QCoh(X) to that in QCoh(Y) for closed subschemes Y.
• Leverages the equivalence between Ab4* and the existence of projective effacements in Grothendieck categories to link exactness to the presence of enough projectives.

Experimental results

Research questions

• RQ1 Under what conditions is the category of quasi-coherent sheaves on a noetherian scheme closed under exact direct products?
• RQ2 Is the exactness of direct products in QCoh(X) equivalent to the scheme X being affine for divisorial noetherian schemes?
• RQ3 Can the non-exactness of direct products on non-affine schemes be generalized beyond specific examples like the projective line or punctured spectra?
• RQ4 How do Grothendieck category axioms Ab4* and Ab6 interact in the context of quasi-coherent sheaves on schemes?
• RQ5 What role does the existence of an ample family of invertible sheaves play in determining the exactness of direct products in QCoh(X)?

Key findings

• For a divisorial noetherian scheme X, direct products in QCoh(X) are exact if and only if X is affine.
• The category QCoh(X) has enough projectives if and only if it satisfies Ab4*, establishing a strong link between homological properties and geometric structure.
• If X is a scheme containing a non-affine divisorial noetherian closed subscheme, then QCoh(X) does not satisfy Ab4*.
• The equivalence QCoh(X) ≅ Mod^I S via the Gabriel-Popescu embedding allows transfer of exactness properties from module categories to sheaf categories.
• The object O_X is projective in QCoh(X) if and only if X is affine, which implies vanishing of higher cohomology groups H^d(X, -) for d ≥ 1.
• Serre’s criterion for affineness is applied to conclude that vanishing of higher cohomology implies affineness, completing the proof of the main theorem.