
> 
> **Question 1.** Is a correct proof of Leray's theorem (the one that says that
>  a connected graded Hopf algebra $H$ over a field of characteristic $0$ is
>  isomorphic as an algebra to the symmetric algebra $\operatorname\*{Sym}\left(
> H^{+}/\left( H^{+}\right) ^{2}\right) $, where $H^{+}=\operatorname\*{Ker}
> \epsilon$ is the kernel of the counit $\epsilon$ of $H$) contained in
>  any version of the famous article "On the structure of Hopf algebras" by
>  Milnor and Moore?
> 
> 
> **Question 2.** What are the currently existing sources for Leray's theorem?
> 
> 
> 




---


**Detailed version.** One of the earliest, and still most influential, papers
on Hopf algebras is the article "On the structure of Hopf algebras" by Milnor
and Moore. It exists in two versions:


[A] [John W. Milnor, John C. Moore, *On the structure of Hopf algebras
(preprint)*, reprinted in Milnor's Collected Papers](http://gen.lib.rus.ec/book/index.php?md5=CCB13231812534818220BEF2F01F13F1).


[B] [John W. Milnor, John C. Moore, *On the structure of Hopf algebras*, The
Annals of Mathematics, Second Series, Vol. 81, No. 2 (Mar., 1965), pp. 211--264](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf).


I am interested in the following theorem, which I shall call *Leray's theorem*
despite being utterly confused about its provenance (possibly Hopf, Samelson
and Borel have some claims here as well -- any help with the history?):



> 
> **Leray's theorem.** Let $\mathbf{k}$ be a commutative $\mathbb{Q}$-algebra.
>  Let $H$ be a graded Hopf algebra over $\mathbf{k}$ (in the modern meaning of
>  this word -- i.e., it has to have an associative multiplication $m$, a unit
>  $u$, a coassociative comultiplication $\Delta$, a counit $\epsilon$, and an
>  antipode $S$; and all these maps are graded). Assume that $H$ is connected
>  (i.e., the $0$-th graded component of $H$ is isomorphic to $\mathbf{k}$) and
>  commutative (i.e., all $a,b\in H$ satisfy $ab=ba$). Let $H^{+}
> =\operatorname\*{Ker}\epsilon$ be the kernel of the counit $\epsilon
> :H\rightarrow\mathbf{k}$ of $H$. Then, $H\cong\operatorname\*{Sym}\left(
> H^{+}/\left( H^{+}\right) ^{2}\right) $ as graded $\mathbf{k}$-algebras.
> 
> 
> 


From what I understand, Milnor and Moore claim to prove this theorem in the
particular case when $\mathbf{k}$ is a field. Specifically, it appears to
follow from Theorem 4.6 in [A] and from Theorem 7.5 in [B].


(Notice that the notations in [A] are different from mine. In particular,
associativity and coassociativity are not required by default from Hopf
algebras, but rather required only when explicitly stated. If they wouldn't
require $\mathbf{k}$ to be a field, their results would hence be more general
than what I call Leray's theorem. Also notice that they are using the
notations $I\left( H\right) $ for $H^{+}$, as well as $Q\left( H\right) $
for $\mathbf{k}\otimes\_{H}H^{+}\cong H^{+}/\left( H^{+}\right) ^{2}$, and
finally $A\left( V\right) $ for the symmetric algebra $\operatorname\*{Sym}
V$, at least when we are working with evenly graded algebras.)


What I don't understand is how they prove all this. Their proof relies on a
lemma (Lemma 4.3 in [A], resp. Proposition 4.17 in [B]), which states (at
least in the particular case we care about) that in the situation of Leray's
theorem, if we further assume that $\mathbf{k}$ is a field, the natural
morphism $\left\{ \text{primitive elements of }H\right\} \rightarrow
H^{+}/\left( H^{+}\right) ^{2}$ (which is well-defined since primitive
elements of $H$ always lie in $H^{+}$) is a monomorphism. In other words, the
only primitive element in $\left( H^{+}\right) ^{2}$ is $0$. The proof of
this lemma relies on some argument where $H$ is assumed to be finitely
generated, and then an induction is done on the number of generators,
presenting $H$ as a some-sort-of-extension of a Hopf subalgebra $A^{\prime}$
with one less generator by a quotient-of-sorts $A^{\prime\prime}$ with one
generator. ("Generator" always mean algebra generator.) This looks highly
suspicious to me, since I don't think that Hopf algebras can be deconstructed
in such a simple way. I also have never seen arguments like this ever after
Milnor's and Moore's work. Thus, the first question: Is it correct?


I am currently writing up my own proof of Leray's theorem, which uses the
Eulerian idempotent and is probably common knowledge in the Hopf-combinatorics
crowd. Thus, the second question: What should I be referencing?



Is Leray's theorem on commutative Hopf algebras proven in Milnor-Moore?
I forgot to give this question some closure. Basically, it was answered by @JohnPalmieri in his comment; let me just expand this answer.


The proof of Proposition 4.17 in the Milnor/Moore paper [B] is correct (at least the part that I was having troubles with). The argument is rather surprising and seriously uses the assumption that the base field is a field! Here are the main ideas (of the proof of Proposition 4.17, not of the proof of Leray's theorem):


* We want to prove that if $A$ is a commutative connected graded Hopf algebra over a field $K$ of characteristic $0$, then the natural morphism $P\left(A\right) \to Q\left(A\right)$ (where, we recall, $P\left(A\right)$ denotes the primitive elements of $A$, whereas $Q\left(A\right) = \left(\ker \varepsilon\right) / \left(\ker \varepsilon\right)^2$) is injective. (This is not the whole statement of Proposition 4.17, but the only part I care about here.)
* We can show that $A$ is a direct limit of Hopf subalgebras that are finitely generated as $K$-algebras. (This is the first place where we use that $K$ is a field.) So we WLOG assume that $A$ is finitely generated as a $K$-algebra (by abstract nonsense).
* Thus, $A = K\left[x\_1, x\_2, \ldots, x\_{m+1}\right]$ for some homogeneous elements $x\_1, x\_2, \ldots, x\_{m+1}$ of $A$ having positive degree. Assume WLOG that $x\_{m+1}$ has the highest degree among these elements $x\_1, x\_2, \ldots, x\_{m+1}$.
* Now, the $K$-subalgebra $A' := K\left[x\_1, x\_2, \ldots, x\_m\right]$ of $A$ is a Hopf subalgebra. To see this, notice that the coproducts $\Delta x\_i$ for $i \leq m$ all belong to $A' \otimes A'$ (because $\Delta x\_i - x\_i \otimes 1 - 1 \otimes x\_i$ is a sum of tensors whose tensor factors have smaller degree than $x\_i$, which entails that these factors cannot use $x\_{m+1}$ because any appearance of $x\_{m+1}$ would raise their degree higher than it can go).
* Now, the ideal of $A$ generated by all $a' - \varepsilon\left(a'\right)$ for $a' \in A'$ is a Hopf ideal of $A$, and therefore the quotient $A''$ of $A$ by this ideal is also a commutative Hopf algebra, with only one generator (the projection of $x\_{m+1}$).
* By induction over the number of generators, we can assume that the proposition is already proven for $A'$ instead of $A$ (since $A'$ needs only $m$ generators $x\_1, x\_2, \ldots, x\_m$). Also, it is easy to show that the proposition holds for $A''$ instead of $A$ (as $A''$ has only one generator). Splicing this together via an exact-sequence argument yields the proposition for $A$.


I have not gone through the details, but this looks like a nice argument which I haven't seen elsewhere.


Meanwhile, I prove different versions of Leray's theorem in different ways (based on ideas by Patras, Loday, etc.):


* If $K$ is any commutative $\mathbb{Q}$-algebra (not necessarily a field!), and if $A$ is a commutative connected graded $K$-Hopf algebra, then $A \cong \operatorname{Sym}\left(Q\left(A\right)\right)$ as algebras (where $Q\left(A\right) = \left(\ker \varepsilon\right) / \left(\ker \varepsilon\right)^2$ as before). This is (currently) Theorem 1.7.29 (e) in [Darij Grinberg and Victor Reiner, *Hopf algebras in Combinatorics*](http://www.cip.ifi.lmu.de/~grinberg/algebra/HopfComb-sols.pdf) (this will be on the arXiv soon, once I've added a few more things). Note that this version of Leray's theorem does not require $K$ to be a field, but (unlike Theorem 7.5 in [B]) only provides *one* isomorphism $\operatorname{Sym}\left(Q\left(A\right)\right) \to A$, whereas Theorem 7.5 in [B] gives an infinite family thereof. Roughly speaking, the isomorphism it provides comes from an "exponential of the Eulerian idempotent". Note also that (unlike Milnor and Moore) I only consider evenly graded algebras (i.e., no superalgebras), and I require coassociativity.
* In Theorem 40.13 of [*On the logarithm of the identity on
connected filtered bialgebras*](http://www.cip.ifi.lmu.de/~grinberg/algebra/logid.pdf), I replicate the whole infinite family of isomorphisms from Theorem 7.5 of [B] (though still only for evenly graded algebras with coassociativity). The proof is a long and painful mess (although it doesn't help that I haven't tried much to streamline it, but written it up as it evolved). It seems that the generality of $K$ not being a field is cutting off many useful shortcuts here; the fieldness of $K$ is used in [B] many times (apart from the uses mentioned above, I think it is used in making $E^0\left(A\right)$ a Hopf algebra).


Note that I am still a bit skeptical about whether Theorem 7.5 in [B] really does not require the associativity and the coassociativity of $A$. I do not care much about this, but somehow it strikes me as more of a miracle than I would have expected -- then again, the Milnor/Moore paper has its share of miracles...


