Separable algebras

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

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

1. is projective as an -bimodule (equivalently, as a left -module).
2. The multiplication map has a section as an -bimodule map.
3. admits a separability idempotent: an element such that and for all (which implies that ).

(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 is a field or is projective over . It’s been removed since it plays no role in the post anyway.)

When is a field, this condition turns out to be a natural strengthening of the condition that is semisimple. In general, loosely speaking, a separable -algebra is like a “bundle of semisimple algebras” over .

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

.

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

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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.

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Maximum entropy from Bayes’ theorem

The principle of maximum entropy asserts that when trying to determine an unknown probability distribution (for example, the distribution of possible results that occur when you toss a possibly unfair die), you should pick the distribution with maximum entropy consistent with your knowledge.

The goal of this post is to derive the principle of maximum entropy in the special case of probability distributions over finite sets from

• Bayes’ theorem and
• the principle of indifference: assign probability to each of possible outcomes if you have no additional knowledge. (The slogan in statistical mechanics is “all microstates are equally likely.”)

We’ll do this by deriving an arguably more fundamental principle of maximum relative entropy using only Bayes’ theorem.

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