geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
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For a ringed space $(X, \mathcal{O}_X)$ there is its Picard group of invertible objects in the category of $\mathcal{O}_X$-modules. When $X$ is a projective integral scheme over $k$ the Picard group underlies a $k$-scheme, this is the Picard scheme $Pic_X$. This scheme varies in a family as $X$ varies in a family. From this starting point one can naturally generalize to more general relative situations.
Often one considers just the connected component $Pic_X^0$ of the neutral element in $Pic_X$, and often (such as in the discussion below, beware) it is that connected component (only) which is referred to by “Picard scheme”. The difference between the two is measured by the quotient $Pic_X/Pic_X^0$, which is called the Néron-Severi group of $X$. Though at least for $X$ an algebraic curve, $Pic_X^0$ goes by a separate name: it is the Jacobian variety of $X$.
The completion of the Picard scheme at its neutral element (hence either of $Pic_X$ or $Pic_X^0$) is the formal Picard group.
The Picard variety of a complete smooth algebraic variety $X$ over an algebraically closed field parametrizes the Picard group of $X$, more precisely the set of classes of isomorphic invertible quasicoherent sheaves with vanishing first Chern class.
The Picard scheme is a scheme representing the relative Picard functor $Pic_{X/S}: (Sch/S)^{op}\to Set$ by $T\mapsto Pic(X_T)/f^*Pic(T)$. In this generality the Picard functor has been introduced by Grothendieck in FGA, along with the proof of representability. An alternate form of this functor (with respect to the Zariski topology) in terms of the derived functor of $f_*$ is $Pic_{X/S}(T)=H^0(T, R^1f_{T*}\mathcal{O}_{X_T}^*)$.
Note we must work with the relative functor because the global Picard functor $Pic_X(T)=Pic(X_T)$ has no hope of being representable as it is not even a sheaf. Consider any non-trivial invertible sheaf $\mathcal{L}$ in $Pic(T)$. Then $f^*\mathcal{L}$ becomes trivial on some cover $\{T_i\to T\}$, so $Pic(X_T)\to \prod Pic(X_{T_i})$ is not injective.
For this section suppose $f:X\to S$ is s separated map, finite type map of schemes. Many general forms of representability have been proven several of which are given in FGA explained. Here we list several of the common forms:
Suppose $\mathcal{O}_S\to f_*\mathcal{O}_X$ is universally an isomorphism (stays an isomorphism after any base change), then we have a comparison of relative Picard functors $Pic_{X/S}\hookrightarrow Pic_{X/S, zar}\hookrightarrow Pic_{X/S, et}\hookrightarrow Pic_{X/S, fppf}$. They are all isomorphisms if $f$ has a section.
If $Pic_{X/S}$ is representable by a scheme, then by descent theory for sheaves it is representable by the same scheme in all the topologies listed above. In general, representability gives representability in a finer topology (of the ones listed).
If $Pic_{X/S}$ is representable then a universal sheaf $\mathcal{P}$ on $X\times Pic_{X/S}$ is called a Poincaré sheaf. It is universal in the following sense: if $T\to S$ and $\mathcal{L}$ is invertible on $X_T$, then there is a unique $h:T\to Pic_{X/S}$ such that for some $\mathcal{N}$ invertible on $T$ we get $\mathcal{L}\simeq (1\times h)^*\mathcal{P}\otimes f_T^*\mathcal{N}$.
If $f$ is (Zariski) projective, flat with integral geometric fibers then $Pic_{X/S, et}$ is representable by a separated and locally of finite type scheme over $S$.
Grothendieck’s Generic Representability: If $f$ is proper and $S$ is integral, then there is a nonempty open $V\subset S$ such that $Pic_{X_V/V, fppf}$ is representable and is a disjoint union of open quasi-projective subschemes.
If $f$ is a flat, cohomologically flat in dimension 0, proper, finitely presented map of of algebraic spaces, then $Pic_{X/S}$ is representable by an algebraic space locally of finite presentation over $S$.
The Picard stack $\mathcal{Pic}_{X/S}$ is the stack of invertible sheaves on $X/S$, i.e. the fiber category? over $T\to X$ is the groupoid of line bundles on $X_T$ (not just their isomorphism classes). (Hence it is the Picard groupoid equipped with geometric structure).
If $X$ is proper and flat, then $\mathcal{Pic}_{X/S}$ is an Artin stack since $\mathcal{Pic}_{X/S}=\mathcal{Hom}(X, B\mathbb{G}_m)$ is the Hom stack which is Artin.
Note the following “failure” of the relative Picard scheme: points on $Pic_{X/S}$ do not parametrize line bundles. The low degree terms of the Leray spectral sequence give the following exact sequence $H^1(X_T, \mathbb{G}_m)\to H^0(T, R^1f_*\mathbb{G}_m)\to H^2(T, \mathbb{G}_m)\to H^2(X_T, \mathbb{G}_m)$, but as noted above $Pic_{X/S}(T)=H^0(T, R^1f_*\mathbb{G}_m)$, so we see exactly when a $T$-point comes from a line bundle it is when that point maps to $0$ in this sequence.
This gives us an obstruction theory lying in $H^2(T, \mathbb{G}_m)$ for a point corresponding to a line bundle. If $Pic_{X/S}$ is representable we could take $T=Pic_{X/S}$ to find a universal obstruction. Intuitively this is because the Picard stack is the right object to look at for the moduli problem of line bundles over $X$. The Picard scheme is the $\mathbb{G}_m$-rigidification of the Picard stack.
The natural map $\mathcal{Pic}_{X/S}\to Pic_{X/S}$ is a $\mathbb{G}_m$-gerbe. But isomorphism classes of $\mathbb{G}_m$-gerbes over $T$ are in bijective correspondence with $H^2(T, \mathbb{G}_m)$ and so the above map could be thought of as a geometric realization of the universal obstruction class.
moduli spaces of line n-bundles with connection on $n$-dimensional $X$
Springer eom: Picard variety, Picard scheme
wikipedia Picard group
Steven L. Kleiman, The Picard scheme, pp. 235–321 in FGA explained, MR2223410 (draft pdf), arxiv
Akhil Mathew, The Picard Scheme I, The Picard Scheme II: deformation theory
Specifically on the Picard stack:
Last revised on March 12, 2021 at 03:30:51. See the history of this page for a list of all contributions to it.