The Picard groups

Let be a commutative ring. From we can construct the category of -modules, which becomes a symmetric monoidal category when equipped with the tensor product of -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 of is the group of isomorphism classes of -modules which are invertible with respect to the tensor product.

By invertible we mean the following: for there exists some such that the tensor product is isomorphic to the identity for the tensor product, namely .

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 ).

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Five proofs that the Euler characteristic of a closed orientable surface is even

Let be a closed orientable surface of genus . (Below we will occasionally write , omitting the genus.) Then its Euler characteristic 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 , roughly in increasing order of sophistication. Along the way we’ll end up encountering or proving more general results that have other interesting applications.

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Hypersurfaces, 4-manifolds, and characteristic classes

In this post we’ll compute the (topological) cohomology of smooth projective (complex) hypersurfaces in . When the resulting complex surfaces give nice examples of 4-manifolds, and we’ll make use of various facts about 4-manifold topology to try to say more in this case; in particular we’ll be able to compute, in a fairly indirect way, the ring structure on cohomology. This answers a question raised by Akhil Mathew in this blog post.

Our route towards this result will turn out to pass through all of the most common types of characteristic classes: we’ll invoke, in order, Euler classes, Chern classes, Pontryagin classes, Wu classes, and Stiefel-Whitney classes.

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