Basic question about affine group schemes - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:28:14Z http://mathoverflow.net/feeds/question/116045 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116045/basic-question-about-affine-group-schemes Basic question about affine group schemes Catherine 2012-12-11T05:36:25Z 2012-12-11T05:43:59Z <p>I've been reading Waterhouse's book "Introduction to affine group schemes", in part to help prepare myself for an (oral) advanced topic exam in algebraic geometry. There is one exercise in chapter 1 that has been giving me trouble. Let $G$ be an affine group scheme with associated Hopf algebra $A$. The exercise says that I should prove the following Hopf-algebraic fact about $A$ by translating it to a basic fact about the group theory of $G$ : "The map $A \otimes A \rightarrow A \otimes A$ sending $a \otimes b$ to $(a \otimes 1)(\Delta(b))$ is an algebra isomorphism".</p> <p>The other parts of the exercise give Hopf-algebraic facts corresponding to really basic group theory facts, like $(x^{-1})^{-1} = x$ and $(xy)^{-1} = y^{-1} x^{-1}$ and $1^{-1} = 1$. However, I can't figure out which group-theoretic fact the above corresponds to. It almost seems like it is saying that there is some automorphism of the group corresponding to the above Hopf-algebra isomorphism; however, the only group automorphisms I know that exist in general are the inner ones, and those don't seem to do the job.</p> http://mathoverflow.net/questions/116045/basic-question-about-affine-group-schemes/116046#116046 Answer by Sasha for Basic question about affine group schemes Sasha 2012-12-11T05:43:40Z 2012-12-11T05:43:40Z <p>Consider the map of groups $G\times G \to G\times G$, $(g_1,g_2) \mapsto (g_1,g_1\cdot g_2)$. Clearly, it is an isomorphism, hence induces an isomorphism of the corresponding Hopf algebras, which is given by precisely the map you are asking about.</p> http://mathoverflow.net/questions/116045/basic-question-about-affine-group-schemes/116047#116047 Answer by darij grinberg for Basic question about affine group schemes darij grinberg 2012-12-11T05:43:59Z 2012-12-11T05:43:59Z <p>You are confusing algebra isomorphisms with Hopf algebra isomorphisms. The map $A\otimes A \to A\otimes A$ given by $a\otimes b\mapsto \left(a\otimes 1\right)\left(\Delta\left(b\right)\right)$ is an algebra isomorphism but not a Hopf algebra isomorphism in general. So it corresponds not to a group automorphism of $G\times G$, but to an automorphism of the affine scheme $G\times G$. This automorphism is the one that sends $\left(x,y\right)$ to $\left(x,xy\right)$ in terms of points.</p>