Talk:Hopf algebroid

"the concept was introduced by J.-H. Lu in 1996 as a result on work on groupoids in Poisson geometry..."
Surely the important role of Hopf algebroids in homotopy theory ought to be discussed? Its introduction can be traced to at least J.F. Adams in the 70s. -- 18:46, 5 September 2016‎ User_talk:65.112.10.245


 * Hmm. A google search shows that in 1981, Haynes R. MILLER says that a "a Hopf algebroid as a cogroupoid object in the category of commutative graded algebras." and cites J.F. Adams, Lectures on generalized cohomology, Lecture Notes in Mathematics 99 (SpringerVerlag, New York, 1969) I-138. for that definition. But from what I can tell, the Adams article never uses the suffix -oid anywhere.  Whatever. It is hardly clear that whatever it is that is being defined here is the same thing as what is being defined by Miller.  Perhaps these are distantly related things with the same name? Or are they actually the same thing? 67.198.37.16 (talk) 03:31, 19 September 2016 (UTC)
 * Then I can find this statement: All of the amenable Hopf algebroids we know are in fact Adams Hopf algebroids, defined in GH00 but implicit in Adams blue book, Ada74 Section III 13 and then I see: Ada74 is JF Adams, "Stable Homotopy and generalized homology" 384 pages, 1974 Chicago Lectures in Mathematics. -- this suggests that an "Adams Hopf algebroid" is ... slightly different than "a cogroupoid object in the category of commutative graded algebras"? Then there's something that replaces the last words by "graded k-agebras"... it would seem that its a bit of a task to sort out what is defined where, how ....


 * As a general note, wikipedia is terrible at math history, so anything you or anyone can do to better record the flow of historical events would be great. 67.198.37.16 (talk) 03:53, 19 September 2016 (UTC)


 * Groupoids in algebraic $k$-schemes, hence in dual form, commutative Hopf $k$-algebroids were used in 1960s in algebraic geometry and Ravenel's book details many developments of commutative Hopf algebroids in algebraic topology from 1970s or so. But the noncommutative generalization over noncommutative base algebra was not discovered until Lu, and somewhat earlier, an equivalent category of objects but in a different formalism of $\times_A$-bialgebras by Takeuchi. I will edit. Zoran.skoda (talk) 20:41, 2 September 2017 (UTC)

Some good examples
There are some good examples of Hopf algebroids on page 25 of Motivic stable homotopy groups giving techniques for computing motivic cohomology of (truncated) motivic Brown-Peterson spectra. This motivates some of the change of rings theorems. Wundzer (talk) 23:33, 17 February 2021 (UTC)