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Let $f: H_1 \rightarrow H_2$ be a map of graded connected cocommutative Hopf algebras over a perfect field. Let $H \subset H_1$ be the Hopf algebra kernel of $f$, and let $I \subset H_1$ be the kernel of $f$, viewed as an algebra map. Let $\bar H$ be the positive dimensional part of $H$, and let $(\bar H) \subset H_1$ be the algebra ideal generated by $\bar H$. Clearly $(\bar H) \subseteq I$.

Questions: Does $(\bar H) = I$? It seems likely that this is a standard fact. If so, where is this in the literature?

(The hypotheses that the Hopf algebras are cocommutative, and having the field be perfect, just happen to hold in the situation I am considering, and perhaps are irrelevant.)

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    $\begingroup$ What is the "algebra kernel"? The largest subalgebra on which $f$ vanishes? $\endgroup$ – user43326 May 15 at 16:37
  • $\begingroup$ I mean that I is just the usual kernel: it is an ideal in the ring theoretic sense. $\endgroup$ – Nicholas Kuhn May 15 at 17:23
  • $\begingroup$ I guess this is equivalent to say that the cokernels in the category of the algebras and Hopf algebras are same (or, can we say that the forgetful functor is right exact?), which is another standard fact. But I don't seem to have seen this explicitly written. $\endgroup$ – user43326 May 15 at 18:01
  • $\begingroup$ What do you mean by "the positive dimensional part of $H$" ? $\endgroup$ – Konstantinos Kanakoglou May 15 at 18:10
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    $\begingroup$ Nicholas, i feel confused: why are $H$ and $I$ different? Am i missing something obvious? $\endgroup$ – Konstantinos Kanakoglou May 15 at 20:34
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The paper "A correspondence between bi-ideals and sub-Hopf algebras in cocommutative Hopf algebras" by K. Newman (J. Algebra, Volume 36, Issue 1, July 1975, Pages 1-15) may answer your question.

See also Susan Montgomery, "Hopf algebras and their actions on rings," in particular the question at the bottom of p. 36, Theorem 3.4.6, and ensuing discussion. (This is what led me to Newman's paper.) And as I said in a comment, Proposition 1.3 in Wilkerson's paper "The Cohomology Algebras of Finite Dimensional Hopf Algebras" may be relevant (and see also Theorem 4.9 in Milnor-Moore), although it is only stated for surjections of Hopf algebras. Wilkerson is the only one of these working in the graded connected setting, so if you have graded connected Hopf algebras which are not actually Hopf algebras if you forget the grading, you should take care with the other results.

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  • $\begingroup$ Gosh, this is exactly what I was looking for. I also found the bicommutative case, published by Takeuchi in 1972, and mentioned by Newman, though he doesn't seem to know it has been published. Wilkerson is later - 1981 - but points out that Milnor-Moore basically said what one needs in the graded setting back in 1965. $\endgroup$ – Nicholas Kuhn May 16 at 1:11
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-too long for a comment-
I am a little confused about the way terminology is used in the OP.
Maybe i'm missing the point; in case i do not, the closest result i know of -quite general and does not refer specifically to graded or connected or cocommutative case- is Lemma 16.0.2, p. 306, of Sweedler's book.
Copying verbatim:

Let $K$ and $L$ be hopf algebras and $\pi:K\to L$ a surjective hopf algebra map. Let $A=\{ g\in K / (Id\otimes\pi)\Delta(g)=g\otimes 1$. $A$ is a subalgebra of $K$ and $\varepsilon_K|A$ is an augmentation. The augmentation ideal $(\ker\varepsilon)\cap A$ is denoted $A^+$. If $A^+ K$ denotes the right ideal in $K$, generated by $A^+$, we have: $$\ker\pi=A^+K$$

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  • $\begingroup$ Thanks: this looks similar to the things in John Palmieri's answer. Newman's paper has Sweedler's book in the references, so ... hmm ... why did he need to write his paper at all? $\endgroup$ – Nicholas Kuhn May 16 at 1:13
  • $\begingroup$ By the way, my terminology - `connected, graded' - is as in the original study of Hopf algebras by Milnor and Moore from the mid 1960's. The homology of an H-space (and H is for Hopf!) with field coefficients is a graded cocommutative Hopf algebra. $\endgroup$ – Nicholas Kuhn May 16 at 1:20
  • $\begingroup$ @Nicholas, i did not knew Newmann's paper. It seems interesting and relevant indeed. Can't say more since i have not read it. What confused me in the OP (and still does) was not the terms "graded" and "connected". It was the expression "positive dimensional part of $H$" and the distinction between the "kernels" $H$ and $I$ - as i said in my comments above. $\endgroup$ – Konstantinos Kanakoglou May 16 at 16:07

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