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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 $\text{Spec } R$).

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

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.

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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 $\mathbb{CP}^n$. When $n = 3$ 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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## Z[sqrt{-3}] is the Eisenstein integers glued together at two points

Today’s post is a record of a very small observation from my time at PROMYS this summer. Below, by $\text{Spec } R$ I mean a commutative ring $R$ regarded as an object in the opposite category $\text{CRing}^{op}$.

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## The Yoneda lemma I

For two categories $C, D$ let $D^C$ denote the functor category, whose objects are functors $C \to D$ and whose morphisms are natural transformations. For $C$ a locally small category, the Yoneda embedding is the functor $C \to \text{Set}^{C^{op}}$ sending an object $x \in C$ to the contravariant functor $\text{Hom}(-, x)$ and sending a morphism $x \to y$ to the natural transformation $\text{Hom}(-, x) \to \text{Hom}(-, y)$ given by composition. The goal of the next few posts is to discuss some standard properties of this embedding and try to gain some intuition about it.

Below, whenever we talk about the Yoneda lemma we implicitly restrict our attention to locally small categories.

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## Structures on hom-sets

Suppose I hand you a commutative ring $R$. I stipulate that you are only allowed to work in the language of the category of commutative rings; you can only refer to objects and morphisms. (That means you can’t refer directly to elements of $R$, and you also can’t refer directly to the multiplication or addition maps $R \times R \to R$, since these aren’t morphisms.) Geometrically, I might equivalently say that you are only allowed to work in the language of the category of affine schemes, since the two are dual. Can you recover $R$ as a set, and can you recover the ring operations on $R$?

The answer turns out to be yes. Today we’ll discuss how this works, and along the way we’ll run into some interesting ideas.

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## Ideals and the category of commutative rings

In this post I’d like to give a better (by which I mean category-theoretic) definition of the lattice of ideals than the standard one. We know that the lattice of ideals has meets and joins defined by intersection and sum, respectively, and that if a lattice is viewed as a category whose arrows are the order relation, then meet and join are the product and coproduct, respectively. But we also know that the lattice of radical ideals of a finitely-generated reduced integral $\mathbb{C}$-algebra $R$ is dual to the lattice of algebraic subsets of $\text{MaxSpec } R$ (and that the lattice of prime ideals is dual to the lattice of algebraic subvarieties), and there is a very general category-theoretic formalism for understanding subobjects in a category. It turns out that this formalism reproduces the lattice of ideals of an arbitrary commutative ring – as long as we run it in the opposite category $\text{CRing}^{op}$.

Edit, 2/9/10: The above claim is wrong. But let me tell you the construction I had in mind and you can judge whether it is more natural than the usual definition.

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