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As is well known, using the Hilbert Nullstellensatz (and a more recent result of Cartier) one can show that commutative finitely generated Hopf algebras over $\mathbb{C}$ are equivalent to algebraic ...

Background: Let $H$ be a finitely generated commutative Hopf $k$-algebra, where $k$ is a field of non-zero characteristic. For $$ \widehat{H} := \text{Alg}_k\{H; k\}, $$ we recall (see Abe Chapter 4 ...

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 $V$?