Algebra over a commutative ring

An algebra over a commutative ring $R$ is a ring $A$ with a specified unital ring homomorphism $R\to A$ whose image commutes with every element of $A$. Scalar multiplication is $r\cdot a=\phi (r)a$. For commutative $A$, the commutation requirement is automatic.

An algebra over a commutative ring $A$ is finitely generated if it is a quotient ring of a polynomial ring $A[t_1,\dots ,t_n]$ in finitely many variables. Equivalently, finitely many elements generate it under addition, multiplication, and the action of $A$. The induced map of affine schemes $\mathrm{Spec}B\to \mathrm{Spec}A$ is a morphism of finite type exactly when this holds. Finite generation as an algebra does not require finite generation as an $A$-module.

An algebra over a commutative ring that is also a coalgebra, whose comultiplication and counit preserve the multiplication and unit. Equivalently it is a bimonoid in $k$-modules.

A bialgebra equipped with a coquasitriangular structure, namely a convolution-invertible scalar form $r:H\otimes H\to k$ satisfying $r(ab,c)=\sum r(a,c_{(1)})r(b,c_{(2)})$, $r(a,bc)=\sum r(a_{(1)},c)r(a_{(2)},b)$ and $\sum r(a_{(1)},b_{(1)})a_{(2)}{b}_{(2)}=\sum b_{(1)}{a}_{(1)}r(a_{(2)},b_{(2)})$, with unit normalization. The induced right-comodule braiding is $v\otimes w\mapsto \sum w_{(0)}\otimes v_{(0)}r(v_{(1)},w_{(1)})$.

A normalized scalar map $\gamma :H\otimes H\to k$ satisfying $\sum \gamma (a_{(1)},b_{(1)})\gamma (a_{(2)}{b}_{(2)},c)=\sum \gamma (b_{(1)},c_{(1)})\gamma (a,b_{(2)}{c}_{(2)})$, and $\gamma (1,a)=\epsilon (a)=\gamma (a,1)$. Some conventions require convolution invertibility; that extra hypothesis distinguishes strong from lax tensor transformations.

With the convention $m\ast \gamma =\gamma \ast n$ and invertible bialgebra scalar cocycle $\gamma$ for $n$, the same coalgebra has $n=\bar{\gamma }\ast m\ast \gamma$, or equivalently $m=\gamma \ast n\ast \bar{\gamma }$. Thus $n(a,b)=\sum \bar{\gamma }(a_{(1)},b_{(1)})m(a_{(2)},b_{(2)})\gamma (a_{(3)},b_{(3)})$. Specifying the equation prevents opposite twist conventions from being conflated.

The tensor of left modules over a bialgebra has action $h(u\otimes v)=\sum (h_{(1)}u)\otimes (h_{(2)}v)$. The unit module $k$ has action through the counit.