nLab Picard stack



Complex geometry



Special and general types

Special notions


Extra structure



under construction



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.


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.


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.


Last revised on August 15, 2022 at 15:35:44. See the history of this page for a list of all contributions to it.