The relative Picard functor on schemes over a symmetric monoidal category. (English) Zbl 1222.14095
Let \(({\mathbf C},\otimes,1)\) be an abelian, closed symmetric monoidal category containing all small limits and colimits. Denote by \(\text{Comm}({\mathbf C})\) the category of commutative monoid objects in \({\mathbf C}\). Since \({\mathbf C}\) is a closed monoidal category, any commutative monoid \(A\in\text{Comm}({\mathbf C})\) defines another closed symmetric monoidal category \((A\text{-Mod},\otimes_A, A)\), namely the category of \(A\)-modules. Then \(\text{Aff}_{{\mathbf C}}:= \text{Comm}({\mathbf C})^{op}\) is called the category of affine schemes over \({\mathbf C}\). If \(A\) is an object of \(\text{Comm}({\mathbf C})\), then the corresponding object in \(\text{Aff}_{{\mathbf C}}\) will be denoted by \(\text{Spec}(A)\). This kind of generalized algebraic geometry (over an abelian, closed symmetric monoidal category) has been studied widely in the past, mainly in the context of ringed toposes, Grothendieck sites, and Tannaka categories.
Very recently, B. Toën and M. Vaquié [J. K-Theory 3, No. 3, 437–500 (2009; Zbl 1177.14022)] have introduced the notion of Zariski coverings in the category \(\text{Aff}_{{\mathbf C}}\), thereby determining a Grothendieck site such that the representable presheaves on \(\text{Aff}_{{\mathbf C}}\) are in fact sheaves. Furthermore, these authors defined the concept of a scheme over \({\mathbf C}\) essentially by using sheaves of sets on \(\text{Aff}_{{\mathbf C}}\) and suitable Zariski coverings of them. A scheme \(X\) over \({\mathbf C}\) comes equipped with a functorially defined structure sheaf \({\mathcal O}_X\), with values taken in \(\text{Comm}({\mathbf C})\).
In the paper under review, the author extends this approach by Toën and Vaquié by establishing the notion of relative Picard functor on the category of schemes over \({\mathbf C}\). To this end, quasi-coherent \({\mathcal O}_X\)-modules on a scheme \((X,{\mathcal O}_X)\) over \({\mathbf C}\) are suitably defined, and their basic functorial properties for a particular class of schemes, the so-called bicomplete schemes, are described in full detail. Then, for certain morphisms \(h: (X,{\mathcal O}_X)\to (S,{\mathcal O}_S)\) of bicomplete schemes over \({\mathbf C}\), a relative Picard functor \(\text{Pic}_{X/S}: (\text{Sch}_{{\mathbf C}}/S)^{op}\to Ab\) together with its associated sheaf \(\text{Pic}_{X/S}\) on \(\text{Sch}_{{\mathbf C}}/S\) is defined and analyzed, where \(\text{Sch}_{{\mathbf C}}/S\) stands for the subcategory of bicomplete schemes over a bicomplete base scheme \((S,{\mathcal O}_S)\). The author’s main result, in this context, is a theorem (Theorem 1.1.) stating the following:
(a) The relative Picard functor \(\text{Pic}_{X/S}\) defines a separated presheaf on \(\text{Sch}_{{\mathbf C}}/S\).
(b) Under the additional assumption that the structure morphism \(h: (X,{\mathcal O}_X)\to (S,{\mathcal O}_S)\) has a section \(g: (S,{\mathcal O}_S)\to(X,{\mathcal O}_X)\), the relative Picard functor \(\text{Pic}_{X/S}\) defines a sheaf on \(\text{Sch}_{{\mathbf C}}/S\).
This result generalizes a classical theorem on the relative Picard functor in the usual algebraic geometry of schemes over a fixed commutative ring. A lucid discussion of that classical theorem can be found in S. Kleiman’s masterly essay “The Picard Scheme” [in: Fundamental algebraic geometry: Grothendieck’s FGA explained. Mathematical Surveys and Monographs 123. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1085.14001)], where it appears as Theorem 9.2.5. in Section 2.
As the author points out, there is some recent similar work on the relative Picard functor on algebraic stacks by S. Brochard [Math. Ann. 343, No. 3, 541–602 (2009; Zbl 1165.14023)].
Very recently, B. Toën and M. Vaquié [J. K-Theory 3, No. 3, 437–500 (2009; Zbl 1177.14022)] have introduced the notion of Zariski coverings in the category \(\text{Aff}_{{\mathbf C}}\), thereby determining a Grothendieck site such that the representable presheaves on \(\text{Aff}_{{\mathbf C}}\) are in fact sheaves. Furthermore, these authors defined the concept of a scheme over \({\mathbf C}\) essentially by using sheaves of sets on \(\text{Aff}_{{\mathbf C}}\) and suitable Zariski coverings of them. A scheme \(X\) over \({\mathbf C}\) comes equipped with a functorially defined structure sheaf \({\mathcal O}_X\), with values taken in \(\text{Comm}({\mathbf C})\).
In the paper under review, the author extends this approach by Toën and Vaquié by establishing the notion of relative Picard functor on the category of schemes over \({\mathbf C}\). To this end, quasi-coherent \({\mathcal O}_X\)-modules on a scheme \((X,{\mathcal O}_X)\) over \({\mathbf C}\) are suitably defined, and their basic functorial properties for a particular class of schemes, the so-called bicomplete schemes, are described in full detail. Then, for certain morphisms \(h: (X,{\mathcal O}_X)\to (S,{\mathcal O}_S)\) of bicomplete schemes over \({\mathbf C}\), a relative Picard functor \(\text{Pic}_{X/S}: (\text{Sch}_{{\mathbf C}}/S)^{op}\to Ab\) together with its associated sheaf \(\text{Pic}_{X/S}\) on \(\text{Sch}_{{\mathbf C}}/S\) is defined and analyzed, where \(\text{Sch}_{{\mathbf C}}/S\) stands for the subcategory of bicomplete schemes over a bicomplete base scheme \((S,{\mathcal O}_S)\). The author’s main result, in this context, is a theorem (Theorem 1.1.) stating the following:
(a) The relative Picard functor \(\text{Pic}_{X/S}\) defines a separated presheaf on \(\text{Sch}_{{\mathbf C}}/S\).
(b) Under the additional assumption that the structure morphism \(h: (X,{\mathcal O}_X)\to (S,{\mathcal O}_S)\) has a section \(g: (S,{\mathcal O}_S)\to(X,{\mathcal O}_X)\), the relative Picard functor \(\text{Pic}_{X/S}\) defines a sheaf on \(\text{Sch}_{{\mathbf C}}/S\).
This result generalizes a classical theorem on the relative Picard functor in the usual algebraic geometry of schemes over a fixed commutative ring. A lucid discussion of that classical theorem can be found in S. Kleiman’s masterly essay “The Picard Scheme” [in: Fundamental algebraic geometry: Grothendieck’s FGA explained. Mathematical Surveys and Monographs 123. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1085.14001)], where it appears as Theorem 9.2.5. in Section 2.
As the author points out, there is some recent similar work on the relative Picard functor on algebraic stacks by S. Brochard [Math. Ann. 343, No. 3, 541–602 (2009; Zbl 1165.14023)].
Reviewer: Werner Kleinert (Berlin)
MSC:
| 14K30 | Picard schemes, higher Jacobians |
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 14A20 | Generalizations (algebraic spaces, stacks) |
| 14C22 | Picard groups |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 18D15 | Closed categories (closed monoidal and Cartesian closed categories, etc.) |
Keywords:
generalization of schemes; symmetric monoidal categories; Grothendieck sites; sheaves; quasi-coherent sheaves; Picard groups; relative Picard functorReferences:
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