nLab
Picard stack

Contents

Context

Complex geometry

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

under construction

Contents

Idea

Fully generally one might call any Picard ∞-groupoid equipped with the structure of an ∞-stack a Picard ∞-stack. But as with Picard groups themselves, this fully general concept is typically considered in the special case of Picard ∞-groupoids of ∞-line bundles over a given space in algebraic geometry(E-∞ geometry. That is what we discuss here: moduli ∞-stacks of multiplicative group-principal ∞-bundles.

Definition

General abstract

For some algebraic site/(∞,1)-site such as the étale site or the étale (∞,1)-site, write \mathcal{B} for the (∞,1)-topos of (∞,1)-sheaves over that site. For SS\in \mathcal{B} any object, write /S\mathcal{B}_{/S} for its slice (∞,1)-topos.

Here \mathcal{B} contains a canonical group object 𝔾 mGrp()\mathbb{G}_m \in Grp(\mathcal{B}), the absolute multiplicative group given as an (∞,1)-presheaf by the assignment which sends any commutative ring/E-∞ ring to its group of units/∞-group of units

𝔾 m:RR ×. \mathbb{G}_m \;\colon\; R \mapsto R^\times \,.

The inverse image of 𝔾 m\mathbb{G}_m under base change along S*S \to \ast we will still denote by 𝔾 mGrp( /S)\mathbb{G}_m \in Grp(\mathcal{B}_{/S}).

Write B𝔾 m\mathbf{B}\mathbb{G}_m for the delooping of 𝔾 m\mathbb{G}_m.

For X /SX \in \mathcal{B}_{/S} any object, then morphisms

XB𝔾 m X \longrightarrow \mathbf{B}\mathbb{G}_m

in /S\mathcal{B}_{/S} modulate 𝔾 m\mathbb{G}_m-principal ∞-bundles on XX, whose canonically associated ∞-bundles are algebraic 𝔾 a\mathbb{G}_a-∞-line bundles. (…) (Notice that by the Koszul-Malgrange theorem these are often better thought of as line bundles with flat holomorphic connection…)

The internal hom/mapping stack

Pic(X)[X,B𝔾 m] /𝒮 \mathbf{Pic}(X) \coloneqq [X,\mathbf{B}\mathbb{G}_m] \in \mathcal{B}_{/\mathcal{S}}

is the Picard \infty-stack of XX.

Unwinding the definitions, this is the (∞,1)-presheaf which sends SSS'\to S to the ∞-groupoid of ∞-line bundle on the (∞,1)-fiber product with XSX \to S:

Pic(X):(SS)(S×SX,B𝔾 m). \mathbf{Pic}(X) \;\colon\; (S' \to S) \mapsto \mathcal{B}(S' \underset{S}{\times}X, \mathbf{B}\mathbb{G}_m) \,.

In essentially this form the definition is indicated for instance in (Lurie 04, section 8.2).

In good cases its 0-truncation is a scheme, in which case it is called the Picard scheme.

More concrete realization

See at Picard Scheme – Picard stack.

Properties

Differentiation / deformation theory

The Lie differentiation of τ 0Pic(X)\tau_0 \mathbf{Pic}(X) is, if it exists as a formal group scheme, the Artin-Mazur formal group Φ X 1\Phi^1_X.

References

Last revised on May 30, 2018 at 11:39:30. See the history of this page for a list of all contributions to it.