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## The Picard groups

Let $R$ be a commutative ring. From $R$ we can construct the category $R\text{-Mod}$ of $R$-modules, which becomes a symmetric monoidal category when equipped with the tensor product of $R$-modules. Now, whenever we have a monoidal operation (for example, the multiplication on a ring), it’s interesting to look at the invertible things with respect to that operation (for example, the group of units of a ring). This suggests the following definition.

Definition: The Picard group $\text{Pic}(R)$ of $R$ is the group of isomorphism classes of $R$-modules which are invertible with respect to the tensor product.

By invertible we mean the following: for $L \in \text{Pic}(R)$ there exists some $L^{-1}$ such that the tensor product $L \otimes_R L^{-1}$ is isomorphic to the identity for the tensor product, namely $R$.

In this post we’ll meander through some facts about this Picard group as well as several variants, all of which capture various notions of line bundle on various kinds of spaces (where the above definition captures the notion of a line bundle on the affine scheme $\text{Spec } R$).

Let $\Sigma_g$ be a closed orientable surface of genus $g$. (Below we will occasionally write $\Sigma$, omitting the genus.) Then its Euler characteristic $\chi(\Sigma_g) = 2 - 2g$ is even. In this post we will give five proofs of this fact that do not use the fact that we can directly compute the Euler characteristic to be $2 - 2g$, roughly in increasing order of sophistication. Along the way we’ll end up encountering or proving more general results that have other interesting applications.