The n-Category Café

Lieven wrote:

structure theorem for semi-simple $k$-algebras (aka Weddenburn-Artin): is a direct sum of matrix algebras over division algebras with centers a finite dimensional field extension of $k$.

structure theorem for separable $k$-algebras: is a direct sum of matrix algebras over division algebras with centers a finite dimensional SEPARABLE field extension of $k$.

Thanks! I guess you’re assuming that all algebras here are finite-dimensional? I’m not usually nitpicky, but I am feeling that way today.

In categorical terms, studying the monoidal cat of commutative separable $k$-algebras is the same as studying the etale site of $k$.

Hey! Stop peeking into my brain!

Where’s a nice reference for this?

Also: I guess you’re again assuming these algebras are finite-dimensional?

if you want to extend this to noncommutative separable k-algebras, you’ll have to know something about the Brauer group of fields (they classify the division algebras having center that field).

I like Brauer groups because they arise as $\pi_1$ of some very nice homotopy 3-types, or weak 3-groupoids if you prefer. However, I am only interested in commutative algebras today, for the reason you guessed.

Nakayama-Eilenberg only says that separable implies symmetric Frobenius, not the other way around.

I’ve given the reference to DeMeyer-Ingraham II.2 before.

Here’s another : Knus-Ojanguren ‘theorie de la descente et algebres d’Azumaya’ Springer Lecture Notes in mathematics 389 (1974). section III.3 Thm 3.1 gives the structure result for sepable algebras over a field as i stated it above.

David wrote:

John wrote:

“A theorem of Eilenberg and Nakayama says an algebra A is separable iff it can be given the structure of a symmetric Frobenius algebra.”

Something is wrong here. Take $A$ to be the commutative ring $k[t]/t^2$, and $\epsilon(a+b t)=b$. This seems to satisfy all of the axioms of a symmetric Frobenius algebra, according to either $n$-Lab or Wikipedia, and it is very far from semi-simple/separable.

Whoops, you’re right. I’ll delete that false claim in my blog entry.

I think the right statement should be that a finite dimension $k$-algebra $A$ is separable if and only if its trace map makes it into a Frobenius algebra. By “trace”, I mean to embed $A$ into the algebra of all $k$-linear endomorphisms of $A$, by the left action of $A$ on itself, and take trace in that setting.

I don’t think that’s right. As I mentioned in my comment to Todd, Aguiar says an algebra is ‘strongly separable’ when its trace map makes it into a Frobenius algebra. Being strongly separable implies being separable, but not conversely. He gives a counterexample: the algebra of $n \times n$ matrices over $F_p$ is separable but not strongly separable when $n$ is divisible by $p$. The trace is zero in this case!

We’ll get this stuff ironed out eventually…

Todd wrote:

I’m feeling a little lazy to check this out carefully myself, but I seem to recall that another characterization of $A$ being separable is that $A$, regarded as a $A \otimes_k A^{op}$-module via the algebra map $m: A \otimes_k A^{op} \to A$, is projective.

I think that’s right, though I haven’t gone through the proof. If your computer is smart enough to read gzipped postscript files, this paper is a pleasant way to learn a bit more these issues:

• Marcelo Aguiar, A note on strongly separable algebras, Boletín de la Academia Nacional de Ciencias (Córdoba, Argentina), special issue in honor of Orlando Villamayor, 65 (2000) 51-60.

Otherwise you can get it here.

For starters, apparently $A$ is projective as a module of

$$A^e = A \otimes_k A^{op}$$

iff you can split the $A^e$-module map

$$m : A^e \to A$$

given by

$$m( a \otimes b) = a b$$

If you can split this map, $A$ is a direct summand of a free module, so it’s projective. Why is the converse true? Aguiar points us to chapter 10 of Pierce’s book Associative Algebras.

What is the connection between this and the separability condition of a symmetric Frobenius algebra? Is it something like $m \circ \delta$ is a nonzero scalar multiple of the identity on $A$, where $\delta$ is the comultiplication?

Rosebrugh, Sabadini and Walters use ‘separable’ for a Frobenius algebra with $m \circ \delta = 1$. But the word ‘separable’ means something quite different for associative algebras, and the Oxford group — for example, Jamie Vicary — has persuaded me that we should instead use the term ‘special’ for a Frobenius algebra with $m \circ \delta = 1$. See the followups to week268, where we had quite a long chat about the terminological morass!

Anyway, here’s what I quickly glean from Aguiar. Suppose we have a separable algebra, and suppose we pick a splitting of the $A^e$-module morphism

$$m : A^e \to A$$

Then the image of $1 \in A$ under this splitting is an idempotent $p \in A^e$. People call it a separability idempotent.

If we can choose the splitting so that $p$ is symmetric:

$$p = \sum_i a_i \otimes b_i = \sum_i b_i \otimes a_i$$

then people say $A$ is strongly separable.

You might prefer to think of the splitting as an extra structure, but most algebraists use ‘strong separability’ to mean a property: the existence of a splitting for which $p$ is symmetric. And there’s a reason why…

Every finite-dimensional algebra $A$ comes with a bilinear form

$$(a, b) \mapsto tr(L_a L_b)$$

where $L_a : A \to A$ means ‘left multiplication by $a$. And, it turns out that an algebra $A$ is strongly separable iff this bilinear form is nondegenerate, in the sense of giving an isomorphism between $A$ and $A^*$ — see Theorem 3.1 in Aguiar’s note.

Furthermore, the above bilinear form is nondegenerate iff $A$ can be made into a special Frobenius algebra. That’s because a Frobenius algebra is special ($m \circ \delta = 1$) iff its counit is given by

$$\epsilon(a) = tr(L_a)$$

and this counit only works if the above bilinear form is nondegenerate!

In short, an algebra is ‘strongly separable’ if there exists a symmetric separability idempotent, but we can also describe it as the property that

$$\epsilon(a) = tr(L_a)$$

serves as a counit making $A$ into a special Frobenius algebra.