Let $V$ be a projective (or affine) variety. Does there exist a bijective correspondence between group structures on $V$ and Hopf algebra structures on the coordinate ring of $V$?
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asked May 5, 2010 at 20:50
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$\begingroup$ I'd say affine algebraic groups over $k$ are essentially the same as (fin gen commutative) Hopf $k$-algebras. $\endgroup$– QfwfqCommented May 5, 2010 at 22:01
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$\begingroup$ What do you mean by the coordinate ring of a projective variety? If you mean the global sections of the structure sheaf, this is typically just the constant functions. If you mean "the ring that it's Proj of" then aren't there normally more than one? So in the projective case I can't make sense of the question. $\endgroup$– Kevin BuzzardCommented May 5, 2010 at 22:13
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$\begingroup$ I suppose I mean when its coordinate ring when we consider it as a real affine variety - so scrap projective from the question and just consider affine. $\endgroup$– Aston SmytheCommented May 5, 2010 at 22:39
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$\begingroup$ As far as I can tell, the method of passing from a complex projective variety to an real affine variety (described at mathoverflow.net/questions/6332 ) is rather violent and very far from canonical. $\endgroup$– S. Carnahan ♦Commented May 6, 2010 at 0:23
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2 Answers
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As unknown(google) mentioned in the comments, there is an antiequivalence of categories between Hopf algebras (over some field k) and affine group schemes (over the spectrum of k), given by taking Spec in one direction, and global sections in the other. This yields a bijection between Hopf algebra structures on the coordinate ring of an affine scheme V and group structures on V.
Projective group varieties are abelian varieties, and do not bear any relation to Hopf algebras (as far as I know). Over the complex numbers, there is a transcendental description of abelian varieties using lattices in a complex vector space that admit a certain sesquilinear form.
answered May 5, 2010 at 22:46
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In the projective case, i.e. abelian varieties, this question is addressed in Mumford's On the equations defining abelian varieties. I, Invent. math. 1, 287--354, 1966. In §3, "the addition formula", the aim is to describe the group law, given a projective embedding, explicitly.
The starting point is exactly that the homogeneous coordinate ring is not a Hopf algebra, because a line bundle $L$ (defining the embedding) on $X$ does not pull back to $p_1^*L\otimes p_2^*L$ under the group law $X\times X\to X$. (This is what one would need to have induced comultiplication maps $H^0(X,L^n) \to H^0(X\times X, p_1^*L^n\otimes p_2^*L^n)\cong H^0(X, L^n)\otimes H^0(X, L^n)$.)
Instead Mumford finds that one can describe $X\times X\to X\times X$, $(x,y)\mapsto (x+y,x-y)$ explicitly in terms of the homogeneous coordinate ring.
