Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I gave commutative Hopf algebroid its own entry. (There used to be allusions to this concept at Hopf algbroid and at Hopf algebroid over a commutative base).
Besides the definition, I added discussion of the commutative Hopf algebroids arising as generalized dual Steenrod algebras, copied over from the coresponding section at Adams spectral sequence.
Nice.
In comparison to the Hopf algebroids over possibly noncommutative base it is a rather simple structure to define, but still extremely important in applications.
I have started a section Homological algebra of Hopf comodules with statements and proof of all facts needed to identify the entries of the second page of the $E$-Adams spectral sequence with Ext-groups of Hopf comodules.
I made a search through pdf of the Ravenel’s book and have not found the phrase “Hopf comodule”. Do you mean simply left comodule over a Hopf algebroid ? Is so, please say just so as for a Hopf algebraist Hopf module is a quite different thing: a module which is in the same time a comodule and the two have certain compatibility. Now, Hopf comodule would point toward some sort of dual version of that (what is the same, but we could also dualize algebroids to coalgebroids). In any case for a Hopf algebraist it is incorrect and very confusing to bring Hopf (co)modules if you just mean comodules (even if not over Hopf algebra but over a Hopf algebroid). Similarly there is a category of Hopf bimodules, rather standard thing in Hopfgebra. I may be wrong, in algebraic topology there may be such terminology as you suggested, but my search through Ravenel’s book did not give it and I am highly suspicious that you invented the term by improvization. Am I wrong ?
Right, good point, I’ll change that.
I have added (here) statement and proof of this here:
If $(\Gamma,A)$ graded commutative Hopf algebra such that
the underlying algebra is free graded commutative;
$\eta \colon A \to \Gamma$ is a flat morphism;
$\Gamma$ is generated by primitive elements $\{x_i\}_{i\in I}$
then the Ext of $\Gamma$-comodules from $A$ and itself is
$Ext_\Gamma(A,A) \simeq A[\{x_i\}_{i \in I}] \,.$For completeness I have spelled out the proof that constructing co-free comodules is right adjoint to the forgetful functor here.
I think there was one axiom missing in the explicit description of commutative Hopf algebroids (this def.). I was following def. A1.1.1 in Ravenel 86, and there this axiom is missing, too.
Namely we need to demand that
$\Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R$;
and
$\Psi \circ \eta_L = (\eta_L \otimes_A id) \circ \eta_L$
This is the dual of the condition that composition of morphisms in a groupoid respects source and target.
Notably, it is precisely these two conditions which also imply that the ground ring $A$ of the commutative Hopf algebroid $\Gamma$ is canonically a left and right $\Gamma$-comodule. This is used extensively in Ravenel 86 (e.g. def. A1.2.11 and corollary A1.2.12) but is never stated.
But please anyone let me know if I am missing something.
Ah, Ravenel of course does say that the coproduct is an $A$-bimodule homomorphism, which comes down to the same. Okay, then there was only an axiom missing in the $n$Lab entry. Fixed now.
In def. A1.2.11 of “the” cobar complex in Ravenel 86, doesn’t that need the assumption that $\Gamma$ is actually a Hopf algebra over $A$, instead of a Hopf algebroid, hence that left and right units coincide? Because otherwise the “unit coideal” $\overline{\Gamma}$ (defined either as the cokernel of the left of the right unit) is not both a left and a right $A$-module at the same time, unless I am missing something.
(The analogous statement in prop. 5.2.1 in Kochman 96 is indeed made just for coalgebras.)
1 to 10 of 10