Classifying finite localizations of quasi-coherent sheaves

📝 Abstract

Given a quasi-compact, quasi-separated scheme X, a bijection between the tensor localizing subcategories of finite type in Qcoh(X) and the set of all subsets $Y\subseteq X$ of the form $Y=\bigcup_{i\in\Omega}Y_i $, with $X\setminus Y_i$ quasi-compact and open for all $i\in\Omega $, is established. As an application, there is constructed an isomorphism of ringed spaces (X,O_X)–>(Spec(Qcoh(X)),O_{Qcoh(X)}), where $(Spec(Qcoh(X)),O_{Qcoh(X)})$ is a ringed space associated to the lattice of tensor localizing subcategories of finite type. Also, a bijective correspondence between the tensor thick subcategories of perfect complexes $\perf(X)$ and the tensor localizing subcategories of finite type in Qcoh(X) is established.

💡 Analysis

Given a quasi-compact, quasi-separated scheme X, a bijection between the tensor localizing subcategories of finite type in Qcoh(X) and the set of all subsets $Y\subseteq X$ of the form $Y=\bigcup_{i\in\Omega}Y_i $, with $X\setminus Y_i$ quasi-compact and open for all $i\in\Omega $, is established. As an application, there is constructed an isomorphism of ringed spaces (X,O_X)–>(Spec(Qcoh(X)),O_{Qcoh(X)}), where $(Spec(Qcoh(X)),O_{Qcoh(X)})$ is a ringed space associated to the lattice of tensor localizing subcategories of finite type. Also, a bijective correspondence between the tensor thick subcategories of perfect complexes $\perf(X)$ and the tensor localizing subcategories of finite type in Qcoh(X) is established.

📄 Content

In his celebrated work on abelian categories P. Gabriel [6] proved that for any noetherian scheme X the assignments

induce bijections between (1) the set of all tensor Serre subcategories of coh X , and (2) the set of all subsets U ⊆ X of the form U = S i∈Ω Y i where, for all i ∈ Ω, Y i has quasi-compact open complement X \Y i .

As a consequence of this result, X can be reconstructed from its abelian category, coh X , of coherent sheaves (see Buan-Krause-Solberg [4,Sec. 8]). Garkusha and Prest [8,9,10] have proved similar classification and reconstruction results for affine and projective schemes.

Given a quasi-compact, quasi-separated scheme X , let D per (X ) denote the derived category of perfect complexes. It comes equipped with a tensor product ⊗ := ⊗ L X . A thick triangulated subcategory T of D per (X ) is said to be a tensor subcategory if for every E ∈ D per (X ) and every object A ∈ T , the tensor product E ⊗ A also is in T . Thomason [26] establishes a classification similar to (1.1) for tensor thick subcategories of D per (X ) in terms of the topology of X . Hopkins and Neeman (see [15,22]) did the case where X is affine and noetherian.

Based on Thomason’s classification theorem, Balmer [1] reconstructs the noetherian scheme X from the tensor thick triangulated subcategories of D per (X ). This result has been generalized to quasi-compact, quasi-separated schemes by Buan-Krause-Solberg [4].

The main result of this paper is a generalization of the classification result by Garkusha and Prest [8,9,10] to schemes. Let X be a quasi-compact, quasiseparated scheme. Denote by Qcoh(X ) the category of quasi-coherent sheaves.

We say that a localizing subcategory S of Qcoh(X ) is of finite type if the canonical functor from the quotient category Qcoh(X )/S → Qcoh(X ) preserves directed sums.

Theorem (Classification). Let X be a quasi-compact, quasi-separated scheme. Then the maps

induce bijections between (1) the set of all subsets of the form V = S i∈Ω V i with quasi-compact open complement X \V i for all i ∈ Ω, (2) the set of all tensor localizing subcategories of finite type in Qcoh(X ).

As an application of the Classification Theorem, we show that there is a 1-1 correspondence between the tensor finite localizations in Qcoh(X ) and the tensor thick subcategories in D per (X ) (cf. [16,8,10]).

Theorem. Let X be a quasi-compact and quasi-separated scheme. The assignments

supp X (H n (E))} and S → {E ∈ D per (X ) | H n (E) ∈ S for all n ∈ Z} induce a bijection between (1) the set of all tensor thick subcategories of D per (X ), (2) the set of all tensor localizing subcategories of finite type in Qcoh(X ).

Another application of the Classification Theorem is the Recostruction Theorem. A common approach in non-commutative geometry is to study abelian or triangulated categories and to think of them as the replacement of an underlying scheme. This idea goes back to work of Grothendieck and Manin. The approach is justified by the fact that a noetherian scheme can be reconstructed from the abelian category of coherent sheaves (Gabriel [6]) or from the category of perfect complexes (Balmer [1]). Rosenberg [24] proved that a quasi-compact scheme X is reconstructed from its category of quasi-coherent sheaves.

In this paper we reconstruct a quasi-compact, quasi-separated scheme X from Qcoh(X ). Our approach, similar to that used in [8,9,10], is entirely different from Rosenberg’s [24] and less abstract.

Following Buan-Krause-Solberg [4] we consider the lattice L f.loc,⊗ (X ) of tensor localizing subcategories of finite type in Qcoh(X ) as well as its prime ideal spectrum Spec(Qcoh(X )). The space comes naturally equipped with a sheaf of commutative rings O Qcoh(X) . The following result says that the scheme (X , O X ) is isomorphic to (Spec(Qcoh(X )), O Qcoh(X) ).

Theorem (Reconstruction). Let X be a quasi-compact and quasi-separated scheme. Then there is a natural isomorphism of ringed spaces f : (X , O X ) ∼ -→ (Spec(Qcoh(X )), O Qcoh(X) ).

Other results presented here worth mentioning are the theorem classifying finite localizations in a locally finitely presented Grothendieck category C (Theorem 3.5) in terms of some topology on the injective spectrum Sp C , generalizing a result of Herzog [13] and Krause [19] for locally coherent Grothendieck categories, and the Classification and Reconstruction Theorems for coherent schemes.

The category Qcoh(X ) of quasi-coherent sheaves over a scheme X is a Grothendieck category (see [5]), so hence we can apply the general localization theory for Grothendieck categories which is of great utility in our analysis. For the convenience of the reader we shall recall some basic facts of this theory.

We say that a subcategory S of an abelian category C is a Serre subcategory if for any short exact sequence 0 → X → Y → Z → 0 in C an object Y ∈ S if and only if X , Z ∈ S. A Serre subcategory S of a Grothendieck category C is localizing if it is closed under taking

This content is AI-processed based on ArXiv data.