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## Separable algebras

Let $k$ be a commutative ring and let $A$ be a $k$-algebra. In this post we’ll investigate a condition on $A$ which generalizes the condition that $A$ is a finite separable field extension (in the case that $k$ is a field). It can be stated in many equivalent ways, as follows. Below, “bimodule” always means “bimodule over $k$.”

Definition-Theorem: The following conditions on $A$ are all equivalent, and all define what it means for $A$ to be a separable $k$-algebra:

1. $A$ is projective as an $(A, A)$-bimodule (equivalently, as a left $A \otimes_k A^{op}$-module).
2. The multiplication map $A \otimes_k A^{op} \ni (a, b) \xrightarrow{m} ab \in A$ has a section as an $(A, A)$-bimodule map.
3. $A$ admits a separability idempotent: an element $p \in A \otimes_k A^{op}$ such that $m(p) = 1$ and $ap = pa$ for all $a \in A$ (which implies that $p^2 = p$).

(Edit, 3/27/16: Previously this definition included a condition involving Hochschild cohomology, but it’s debatable whether what I had in mind is the correct definition of Hochschild cohomology unless $k$ is a field or $A$ is projective over $k$. It’s been removed since it plays no role in the post anyway.)

When $k$ is a field, this condition turns out to be a natural strengthening of the condition that $A$ is semisimple. In general, loosely speaking, a separable $k$-algebra is like a “bundle of semisimple algebras” over $\text{Spec } k$.

## Coalgebraic geometry

Previously we suggested that if we think of commutative algebras as secretly being functions on some sort of spaces, we should correspondingly think of cocommutative coalgebras as secretly being distributions on some sort of spaces. In this post we’ll describe what these spaces are in the language of algebraic geometry.

Let $D$ be a cocommutative coalgebra over a commutative ring $k$. If we want to make sense of $D$ as defining an algebro-geometric object, it needs to have a functor of points on commutative $k$-algebras. Here it is:

$\displaystyle D(-) : \text{CAlg}(k) \ni R \mapsto |D \otimes_k R| \in \text{Set}$.

In words, the functor of points of a cocommutative coalgebra $D$ sends a commutative $k$-algebra $R$ to the set $|D \otimes_k R|$ of setlike elements of $D \otimes_k R$. In the rest of this post we’ll work through some examples.

## Coalgebras of distributions

Mathematicians are very fond of thinking about algebras. In particular, it’s common to think of commutative algebras as consisting of functions of some sort on spaces of some sort.

Less commonly, mathematicians sometimes think about coalgebras. In general it seems that mathematicians find these harder to think about, although it’s sometimes unavoidable, e.g. when discussing Hopf algebras. The goal of this post is to describe how to begin thinking about cocommutative coalgebras as consisting of distributions of some sort on spaces of some sort.

## Lie algebras are groups

Once upon a time I imagine people were very happy to think of Lie algebras as “infinitesimal groups,” but presumably when infinitesimals fell out of favor this interpretation did too. In this post I’ll record an observation that can justify thinking of Lie algebras as groups in a strong sense: they are group objects in a certain category which can be interpreted as a category of “infinitesimal spaces.”

Below we work throughout over a field of characteristic zero.

For starters, the universal enveloping algebra functor $\mathfrak{g} \mapsto U(\mathfrak{g})$, which a priori takes values in algebras (it’s left adjoint to the forgetful functor from algebras to Lie algebras), in fact takes values in Hopf algebras. This upgraded functor continues to be a left adjoint, although the forgetful functor is less obvious. Given a Hopf algebra $H$, its primitive elements are those elements $x \in H$ satisfying

$\Delta x = x \otimes 1 + 1 \otimes x$

where $\Delta$ is the comultiplication. The primitive elements of a Hopf algebra form a Lie algebra, and this gives a forgetful functor from Hopf algebras to Lie algebras whose left adjoint is the upgraded universal enveloping algebra functor.

The key observation is that this upgraded functor $\mathfrak{g} \to U(\mathfrak{g})$ is fully faithful; that is, there is a natural bijection between Lie algebra homomorphisms $\mathfrak{g} \to \mathfrak{h}$ and Hopf algebra homomorphisms $U(\mathfrak{g}) \to U(\mathfrak{h})$. This is more or less equivalent to the claim that the natural inclusion $\mathfrak{g} \to U(\mathfrak{g})$ induces an isomorphism from $\mathfrak{g}$ to the Lie algebra of primitive elements of $U(\mathfrak{g})$, which can be proven using the PBW theorem.

Hence Lie algebras embed as a full subcategory of Hopf algebras; that is, they can be thought of as Hopf algebras satisfying certain properties, rather than having extra structure (in the nLab sense). What are these properties? For starters, they are all cocommutative. This is important because cocommutative Hopf algebras are group objects in the category of cocommutative coalgebras (this is not true with “cocommutative” dropped!), which in turn can be interpreted as a category of infinitesimal spaces. (For example, this category is cartesian closed, and in particular distributive.)

Hence Lie algebras are group objects in cocommutative coalgebras satisfying some property (for example, “conilpotence”; see Theorem 3.8.1 here).

## Dualizable objects (and morphisms)

Let $R$ be a ring. Previously we characterized the finitely presented projective (right) $R$-modules as the tiny objects in $\text{Mod}(R)$: the objects $P$ such that

$\displaystyle \text{Hom}(P, -) : \text{Mod}(R) \to \text{Ab}$

preserves colimits. We also highlighted the key role that these modules play in Morita theory.

If $k$ is a commutative ring, then $\text{Mod}(k)$ has a natural symmetric monoidal structure which allows us to describe another finiteness condition called dualizability. Unlike tininess, dualizability makes no reference to colimits; instead, it is a purely equational definition involving the monoidal structure. The dualizable modules are again the finitely presented projective $k$-modules.

Dualizability implies that we can treat finitely presented projective $k$-modules like finite-dimensional vector spaces in many ways: for example, dualizability allows us to define the trace of an endomorphism. Moreover, since dualizability is defined using only a monoidal structure, it makes sense in very general settings, and we’ll look at some more exotic examples of dualizable objects as well.

Duals are also a special case of a 2-categorical notion of adjunction which, in the 2-category of categories, functors, and natural transformations, reproduces the usual notion of adjunction. In a suitable 2-category it will also reproduce another characterization of finitely presented projective modules, this time over noncommutative rings.

This post should, but will not, include diagrams, so pretend that I’ve inserted some string diagrams or globular diagrams where appropriate.

## Generators

Previously we proved a theorem due to Gabriel characterizing categories of modules as cocomplete abelian categories with a compact projective generator, where “generator” meant “every object is a colimit of finite direct sums of copies of the object.”

But we also used “generator” to mean “every object is a colimit of copies of the object,” and noted that these conditions are not equivalent: as this MO question discusses, the abelian group $\mathbb{Z}$ satisfies the first condition but not the second. More generally, as Mike Shulman explains here, there are in fact many inequivalent definitions of “generator” in category theory.

The goal of this post is to sort through a few of these definitions, which turn out to be totally ordered in strength, and find additional hypotheses under which they agree. As an application we’ll restate Gabriel’s theorem using weaker definitions of “generator” and give a more explicit description of all of the rings Morita equivalent to a given ring.

## The double commutant theorem

Let $A$ be an abelian group and $T = \{ T_i : A \to A \}$ be a collection of endomorphisms of $A$. The commutant $T'$ of $T$ is the set of all endomorphisms of $A$ commuting with every element of $T$; symbolically,

$\displaystyle T' = \{ S \in \text{End}(A) : TS = ST \}$.

The commutant of $T$ is equal to the commutant of the subring of $\text{End}(A)$ generated by the $T_i$, so we may assume without loss of generality that $T$ is already such a subring. In that case, $T'$ is just the ring of endomorphisms of $A$ as a left $T$-module. The use of the term commutant instead can be thought of as emphasizing the role of $A$ and de-emphasizing the role of $T$.

The assignment $T \mapsto T'$ is a contravariant Galois connection on the lattice of subsets of $\text{End}(A)$, so the double commutant $T \mapsto T''$ may be thought of as a closure operator. Today we will prove a basic but important theorem about this operator.

## Non-unital rings

(This post was originally intended to go up immediately after the sequence on Gelfand duality.)

A rng (“ring without the i”) or non-unital ring is a semigroup object in $\text{Ab}$. Equivalently, it is an abelian group $A$ together with an associative bilinear map $m : A \otimes A \to A$ (which is not required to have an identity). This is what some authors mean when they say “ring,” but this does not appear to be standard. A morphism between rngs is an abelian group homomorphism which preserves multiplication (and need not preserve a multiplicative identity even if it exists); this defines the category $\text{Rng}$ of rngs (to be distinguished from the category $\text{Ring}$ of rings).

Until recently, I was not comfortable with non-unital rings. If we think of rings either algebraically as endomorphisms of abelian groups or geometrically as rings of functions on spaces, then there does not seem to be any reason to exclude the identity endomorphism resp. the identity function on a space. As for morphisms which don’t preserve identities, if $X \to Y$ is any map between spaces of some kind, then the identity function $Y \to F$ ($F$ is, say, a field) is sent to the identity function $X \to F$, so not preserving identities when they exist seems unnatural.

However, not requiring or preserving identities turns out to be natural in the theory of C*-algebras; in the commutative case, it corresponds roughly to thinking about locally compact Hausdorff spaces rather than just compact Hausdorff spaces. In this post we will discuss rngs generally, including a discussion of the geometric picture of commutative rngs, to get more comfortable with them. It turns out that we can study rngs by formally adjoining multiplicative identities to them. This is an algebraic version of taking the one-point compactification, and it allows us to extend Gelfand duality, in a suitable sense, to locally compact Hausdorff spaces (see this math.SE question for the precise statement, which we will not discuss here).

## Regular and effective monomorphisms and epimorphisms

Previously we observed that although monomorphisms tended to give expected generalizations of injective function in many categories, epimorphisms sometimes weren’t the expected generalization of surjective functions. We also discussed split epimorphisms, but where the definition of an epimorphism is too permissive to agree with the surjective morphisms in familiar concrete categories, the definition of a split epimorphism is too restrictive.

In this post we will discuss two other intermediate notions of epimorphism. (These all give dual notions of monomorphisms, but their epimorphic variants are more interesting as a possible solution to the above problem.) There are yet others, but these two appear to be the most relevant in the context of abelian categories.

## A meditation on semiadditive categories

The goal of today’s post is to introduce and discuss semiadditive categories. Roughly speaking, these are categories in which one can add both objects and morphisms. Prominent examples include the abelian categories appearing in homological algebra, such as categories of sheaves and modules and categories of chain complexes.

Semiadditive categories display some interesting categorical features, such as the prominence of pairs of universal properties and the surprising ways in which commutative monoid structures arise, which seem to be underemphasized in textbook treatments and which I would like to emphasize here. I would also like to emphasize that their most important properties are unrelated to the ability to subtract morphisms which is provided in an additive category.

In this post, for convenience all categories will be locally small (that is, $\text{Set}$-enriched).