Picard group

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Symmetric monoidal categories

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For a line bundle on $X$ with the tensor product.
- $\operatorname{Pic}\operatorname{Spec}R = \operatorname{Cl} (R)$ is the class group for a Dedekind domain
- Globalizes the notion of a number field.?
Alternatively, the Picard group can be defined as the sheaf cohomology \begin{align*}H^{1} (X,{\mathcal {O}}_{X}^{\times}).\end{align*}
Fits into a SES \begin{align*} 0\to \operatorname{Pic}^0(V) \to\operatorname{Pic}(V) \to {\operatorname{NS}}(V) \to 0 \end{align*} where ${\operatorname{NS}}$ is the Neron Severi group.
- This may require that $V$ is a Jacobian? Or something special happens when it is? #unanswered_questions

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Links to this page

spectral Brauer group

As a delooping of the Picard:
invertible sheaves

Tags: #AG Refs: Picard group
derived stacks

Picard stack
Talbot Talk 1

See the Picard group.
Jacobian

For $C$ a nonsingular algebraic curve, $\operatorname{Jac}(C)$ is the connected component of the identity in the Picard group $\operatorname{Pic}(C)$, i.e. the moduli space of degree 0 line bundles on $C$. Over ${\mathbf{C}}$, can be realized as $\operatorname{Jac}(C) \cong H^0(X; \Omega^1_{C}) {}^{ \vee }/ H^1(X; {\mathcal{O}}_X)$, where the embedding $H^1\hookrightarrow H^0$ uses theta functions.
Chow ring

When $X$ is a variety, ${\operatorname{CH}}_{\dim X - 1} X \cong \operatorname{Div} \operatorname{Cl} (X)$, the divisor class group. If $X\in {\mathsf{sm}}{\mathsf{Var}}_{/ {k}} $, then ${\operatorname{CH}}_{\dim X - 1}X \cong \operatorname{Pic}(X)$, the Picard group.