TITLE: Is $Tor_A(k,k)$ a bicommutative Hopf algebra?
QUESTION [7 upvotes]: Let $A$ be a commutative (or graded commutative) algebra over a field $k.$ In some sources, such as Mcleary's book on spectral sequences, Corollary 7.12, pg. 248, it is claimed that $\text{Tor}_A(k,k)$ is a bicommutative Hopf algebra. Is this a typo? Obviously, $\text{Tor}$ has a commutative algebra structure, but is it true that the coaddition is cocommutative?

REPLY [9 votes]: This is not true. Consider the algebra $A=T(V)/V^{\otimes 2}$, it is a commutative algebra whose augmentation ideal has zero multiplication. We have $\mathrm{Tor}_A(k,k)\cong T(V[1])$ with the shuffle product and deconcatenation coproduct, so the coproduct is very much noncommutative (it is the coproduct of the cofree conilpotent coalgebra co-generated by $V[1]$).