Bialgebras, counits and antipodes over a commutative ring

Let $R$ be a commutative ring. A bialgebra over $R$ is a unital associative $R$-algebra $(A,m,\eta )$ (Algebras over a commutative ring, central structure maps, and algebra homomorphisms) together with $R$-algebra homomorphisms $\mathrm{\Delta }:A\to A{\otimes }_{R}A$ (the coproduct) and $\epsilon :A\to R$ (the counit) such that $\mathrm{\Delta }$ is coassociative and the counit axioms hold:

$(\mathrm{\Delta }\otimes {\mathrm{id}}_A)\mathrm{\Delta }=({\mathrm{id}}_A\otimes \mathrm{\Delta })\mathrm{\Delta },(\epsilon \otimes {\mathrm{id}}_A)\mathrm{\Delta }={\mathrm{id}}_A=({\mathrm{id}}_A\otimes \epsilon )\mathrm{\Delta }\mathrm{.}$

The counit equations use the canonical identifications $R{\otimes }_{R}A\cong A\cong A{\otimes }_{R}R$. The target $A{\otimes }_{R}A$ has the $R$-algebra structure of The tensor product of $R$-algebras has multiplication $(a\otimes b)(a^{\mathrm{\prime }}\otimes b^{\mathrm{\prime }})=a{a}^{\mathrm{\prime }}\otimes b{b}^{\mathrm{\prime }}$.

A Hopf algebra over $R$ is a bialgebra $(A,m,\eta ,\mathrm{\Delta },\epsilon )$ together with an $R$-linear antipode $S:A\to A$ satisfying both convolution-inverse equations

$m(S\otimes {\mathrm{id}}_A)\mathrm{\Delta }=\eta \circ \epsilon =m({\mathrm{id}}_A\otimes S)\mathrm{\Delta },$

where $(\eta \circ \epsilon )(a)=\epsilon (a)1_A$. We do not assume that $S$ is invertible, involutive or multiplicative. Throughout, $\mathrm{\Delta }$ and $\epsilon$ are algebra homomorphisms, and the displayed order of the antipode factors is our convention.