Tensor functors between categories of quasi-coherent sheaves

Brandenburg, Martin; Chirvăsitu, Alexandru

doi:10.1016/j.jalgebra.2013.09.050

Cited by 19 publications

(26 citation statements)

References 16 publications

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“…Brandenburg and Chirvasitu [BC12] have shown the following: Theorem 3.1. For qcqs schemes S and X, one has Hom(S, X) ≃ Fun L ⊗ (QCoh(X), QCoh(S)).…”

Section: The Case Of Schemes Revisitedmentioning

confidence: 92%

Algebraization and Tannaka duality

Bhatt

2016

Cambridge Journal of Mathematics

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INTRODUCTIONOur goal in this paper is to identify certain naturally occurring colimits of schemes and algebraic spaces. The statements, which are essentially algebraization results for maps between schemes and algebraic spaces, are elementary and explicit. However, our techniques are indirect: we use (and prove) some new Tannaka duality theorems for maps of algebraic spaces. Our approach to these theorems relies on a systematic deployment of perfect complexes (ergo, we use some derived algebraic geometry) instead of ample line bundles or vector bundles. Consequently, the Tannaka duality results we obtain have fewer, and much weaker, finiteness constraints than some of the existing ones: we only insist that our algebraic spaces be quasi-compact and quasi-separated (qcqs), and do not require any quasi-projectivity or noetherian hypotheses. All rings are assumed to be commutative.1.1. Algebraization of jets. The first colimit we identify is that of an affine (adic) formal scheme.Theorem 1.1. If A is a ring which is I-adically complete for some ideal I, and X is a qcqs algebraic space, then X(A) ≃ lim X(A/I n ) via the natural map.An equivalent formulation is: if A = lim A/I n , then Spec(A) is a colimit of the diagram {Spec(A/I n )} in the category of qcqs algebraic spaces. Theorem 1.1 is straightforward to prove if A/I is local; its content becomes apparent only when Spec(A/I) has some non-trivial global geometry. Note also that there are no noetherian assumptions on any object in sight, so the ideal I might not be finitely generated. In fact, the result extends to more general topological rings A that arise naturally in p-adic geometry (see Remark 4.3). This answers a question asked by Drinfeld, and has the following representability consequence in the theory of arc spaces, which was our original motivation for pursuing Theorem 1.1. Corollary 1.2. If X is a qcqs algebraic space, then the "formal arc" functor Arcs X (R) := X(R t ) is an fpqc sheaf on the category of rings, and is identified with the functor R → lim X(R[t]/(t n )).As the functor Arcs X is almost never locally finitely presented (even for X an algebraic variety), one cannot reduce Corollary 1.2 to the corresponding assertion on the category of noetherian rings (which is easier to prove). This corollary answers a question raised in [NS10, §2] and pointed out to us by Nicaise. The following feature of the proof of Theorem 1.1 seems noteworthy: given a compatible system {ǫ n : Spec(A/I n ) → X} ∈ lim X(A/I n ), we construct an algebraization ǫ : Spec(A) → X without ever musing about points of Spec(A) \ Spec(A/I). Algebraization of products.The second result deals with products, rather than cofiltered inverse limits, of rings; this question was brought to our attention by Poonen.An equivalent formulation is: the scheme Spec( i A i ) is a coproduct of {Spec(A i )} in the category of qcqs algebraic spaces. Note that some finiteness hypothesis on X is necessary: the (typically non-quasicompact) scheme ⊔ i Spec(A i ) is a coproduct of {Spec(A i )} in the cat...

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“…Brandenburg and Chirvasitu [BC12] have shown the following: Theorem 3.1. For qcqs schemes S and X, one has Hom(S, X) ≃ Fun L ⊗ (QCoh(X), QCoh(S)).…”

Section: The Case Of Schemes Revisitedmentioning

confidence: 92%

Algebraization and Tannaka duality

Bhatt

2016

Cambridge Journal of Mathematics

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“…Proof. -According to [8,Proposition 2.2.3], given a commutative kalgebra A and an arbitrary tensor category C (not necessarily satisfying End C (1) ∼ = k) there is an equivalence of categories between the category of k-linear tensor functors Mod(A) → C and that of k-algebra homomorphisms A → End C (1). But for k = A = End C (1) there is only one k-algebra homomorphism k → End C (1), namely the identity morphism.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

A general theory of André’s solution algebras

Nagy¹,

Szamuely²

2021

Annales De l'Institut Fourier

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“…for any pair of quasi-compact quasi-separated K-schemes X, Y . This was already proven in [BC14] without the fp-condition. In particular, the category Hom c⊗fp/K (Qcoh(X), Qcoh(Y )) is essentially discrete (i.e.…”

Section: Local Description Of Tensor Functorsmentioning

confidence: 52%

“…For example, if X is a quasi-compact quasi-separated scheme and Y is an arbitrary scheme, then any cocontinuous tensor functor F : Qcoh(X) → Qcoh(Y ) is induced by a unique morphism f : Y → X via pullback [BC14]; similar results hold for well-behaved algebraic spaces and algebraic stacks and are usually referred to as Tannaka reconstruction theorems [Lur05, Lur11, Sch12, FI13, Ton14, Bha16, BHL17, HR19]. Moreover, geometric properties of f , such as being affine or projective, can be formulated in terms of F [Bra14,Sch18].…”

Section: Introductionmentioning

confidence: 99%

Localizations of tensor categories and fiber products of schemes

Brandenburg

2020

Preprint

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We prove that the tensor category of quasi-coherent modules Qcoh(X ×S Y ) on a fiber product of quasi-compact quasi-separated schemes is the bicategorical pushout of Qcoh(X) and Qcoh(Y ) over Qcoh(S) in the 2-category of cocomplete linear tensor categories. In particular, Qcoh(X × Y ) is the bicategorical coproduct of Qcoh(X) and Qcoh(Y ). For this we introduce idals, which can be seen as non-embedded ideals, and use them to study localizations of cocomplete tensor categories in general.

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Tensor functors between categories of quasi-coherent sheaves

Cited by 19 publications

References 16 publications

Algebraization and Tannaka duality

Algebraization and Tannaka duality

A general theory of André’s solution algebras

Localizations of tensor categories and fiber products of schemes

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