nLab Picard stack

under construction

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 $S\in \mathcal{B}$ any object, write $\mathcal{B}_{/S}$ for its slice (∞,1)-topos.

Here $\mathcal{B}$ contains a canonical group object $\mathbb{G}_m \in Grp(\mathcal{B})$, 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

$$\mathbb{G}_m \;\colon\; R \mapsto R^\times \,.$$

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

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

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

$$X \longrightarrow \mathbf{B}\mathbb{G}_m$$

in $\mathcal{B}_{/S}$ modulate $\mathbb{G}_m$-principal ∞-bundles on $X$, whose canonically associated ∞-bundles are algebraic $\mathbb{G}_a$-∞-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

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

is the Picard $\infty$-stack of $X$.

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

$$\mathbf{Pic}(X) \;\colon\; (S' \to S) \mapsto \mathcal{B}(S' \underset{S}{\times}X, \mathbf{B}\mathbb{G}_m) \,.$$

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 $\tau_0 \mathbf{Pic}(X)$ is, if it exists as a formal group scheme, the Artin-Mazur formal group $\Phi^1_X$.

Related concepts

• Brauer stack

References

• The Stacks Project, The Picard stack

• Jacob Lurie, section 8.2 of Derived algebraic geometry, PhD thesis, 2004 (pdf, web)

• Lettre de Grothendieck à Deligne, 1974 (pdf) (Edited by M. Künzer)