Dgdrxrt

I’m currently reading Sweedler’s Hopf Algebras, but am confused by the finite dimensionality in the following theorem:

Theorem 5.1.8 A finite dimensional Hopf algebra $H$ is semi-simple as an algebra if and only if$epsilon(int) neq 0$.

Sweedler seems to use the finite dimensionality of $H$ only for the implication
$$
text$H$ semisimple
implies
epsilon( textstyleint ) neq 0
$$
by using that $dim int = 1$, which seems unnecessary to me.

Question: Why does Sweedler requires $H$ to be finite dimensional?

Sweedler’s argumentation is as follows:

If $H$ is semi-simple, then there is a left ideal $I$ such that
$$
H = operatornameKer epsilon oplus I.
$$
For $x in operatornameKer epsilon$ and $y in I$ we have $xy in operatornameKer epsilon cap I$.
Hence $xy = 0$.
Then for any $h in H$,
$$
h = (h - epsilon(h)1) + epsilon(h)1,
$$
so $hy = epsilon(h)y$, since $(h - epsilon(h) 1) in operatornameKer epsilon$.
Thus $I subseteq int$, a $1$-dimensional space, hence $I = int$.
Since $H = operatornameKer epsilon oplus I$ we conclude $epsilon(int) neq 0$.

It seems to me that the inclusion $I subseteq int$ sufficies because then $epsilon(textstyle int) supseteq epsilon(I) neq 0$.

Remark: If I’m not mistaken then it should also follow from a later exercise (any Hopf algebra that contains a nonzero finite dimensional one-sided ideal is already finite dimensional itself) that the above theorem also holds for any infinite dimensional Hopf algebra $H$: By considering the above one-dimensional ideal $I$ we see that $H$ cannot be semisimple. On the other hand $int = 0$ (and hence $epsilon(int) = 0$) because for every nonzero $x in int$ the one-dimensional span $kx$ would be a left ideal.
But this doesn’t explain why Sweedler restricts the above theorem to finite dimensional Hopf algebras in the first place.

You don't need that hypothesis in the proof but you don't earn nothing by removing it, as a Hopf algebra satisfying the above conditions is automatically finite-dimensional. You may in fact prove the following result.

Theorem (Maschke Theorem for Hopf algebras) For a Hopf
algebra over a field $Bbbk $ the following assertions are equivalent.

1. 
   $H$ is semisimple as a ring.


2. There exists $tin int_H^l$ such that $%
   varepsilon left( tright) =1$.


3. 
   $H$ is separable as an algebra.

Proof:
To prove that $left( 1right) $ implies $left( 2right) $ consider the left $H$-linear morphism $varepsilon :Hrightarrow Bbbk $. Since $H$ is semisimple and $varepsilon $ is surjective, it admits a left $H$-linear section $sigma :Bbbk rightarrow H$. Set $t:=sigma left( 1_Bbbkright) $ and observe that for every $hin H$ we have $ht=hsigma left(1_Bbbk right) =sigma left( hcdot 1_Bbbk right) =sigma left(varepsilon left( hright) 1_Bbbk right) =varepsilon left( hright) t$
and that $varepsilon left( tright) =varepsilon left( sigma left(1_Bbbk right) right) =1_Bbbk $.

To prove that $left( 2right) $ implies $left( 3right) $ consider the Casimir element $e=sum t_left( 1right) otimes Sleft( t_left( 2right) right) $. Of course, $sum t_left( 1right) Sleft( t_left( 2right) right) =varepsilon left( tright) 1_B=1_B $, whence $e$ is a separability idempotent.

Finally, to prove that $left( 3right) $ implies $left( 1right) $ pick any surjective morphism of left $H$-modules $pi:Mrightarrow N$. Since it is in particular of $Bbbk $-vector spaces it admits a $Bbbk $-linear section $sigma :Nrightarrow M$. Of course, $sigma $ is not $H$-linear in general, but we may consider $tau:Nrightarrow M:nlongmapsto sum e^prime sigma left( e^prime prime
nright) $ where $e=sum e'otimes e''$ is the separability idempotent. This is $H$-linear because $sum e^prime sigma left(e^prime prime hnright) =sum he^prime sigma left( e^prime primenright) $ for every $hin H$ and it is still a section since $pi left(tau left( nright) right) =sum pi left( e^prime sigma left(e^prime prime nright) right) =sum e^prime pi left( sigma left(e^prime prime nright) right) =sum e^prime e^prime prime n=n$
for every $nin N$. $square$

Now the point is that a separable $Bbbk$-algebra is always finite-dimensional.