Is $A$ coflat over $A//B$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T01:20:51Z http://mathoverflow.net/feeds/question/84122 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84122/is-a-coflat-over-a-b Is $A$ coflat over $A//B$? Vitaly Lorman 2011-12-22T21:50:07Z 2011-12-23T12:48:24Z <p>Let $A$ be a Hopf algebra over a field $k$. Let $B$ be a normal subHopf algebra of $A$. Is $A$ coflat over $A//B$? An explanation would be greatly appreciated.</p> <p>(A novice to Hopf algebras, I am attempting to follow the computation of the homotopy of some Thom spectra in Kochman's book. Given $F$, an $A//B$-free coresolution of $k$, Kochman states that $F \Box_{A//B} A$ is an $A$-free coresolution of $k \Box_{A//B} A \cong B$. I don't see why $-\Box_{A//B} A$ preserves exactness.)</p> http://mathoverflow.net/questions/84122/is-a-coflat-over-a-b/84158#84158 Answer by Justin Noel for Is $A$ coflat over $A//B$? Justin Noel 2011-12-23T12:48:24Z 2011-12-23T12:48:24Z <p>I'm going to assume that your Hopf algebras are connected in which case this follows from Theorem 4.10 of Milnor-Moore (On the structure of Hopf-algebras). That result shows that $A\cong B\otimes A//B$ as a left $B$-module and right $A//B$-comodule. I should point out that this result is remarkably useful.</p> <p>This means $A$ is an extended $A//B$-comodule over a field and hence it is injective in the category of $A//B$-comodules. The fact that extended coalgebras are injective (when working over a field) is an easy exercise, but you can also find the result in the context of Hopf-algebroids as A1.2.2 'in Ravenel's Complex Cobordism and Stable Homotopy.'</p> <p>Since $A$ is injective the functor $-\square _{A//B} A$ is exact. </p>