Coproduct of graded Algebras

Question

Can someone give me a construction of the coproduct of two (not necessarily commutative) GRADED algebras $A,B$ over a commutative ring $k$?

I haven't found it on the internet.

Thank you!

Answer 1

I imagine you want the coproduct in the category of associative graded algebras. Let $$A = \bigoplus_{n\in \mathbb{Z}}A_i,$$ and similarly for $B$. Then the coproduct is given, as a graded vector space, by the space of all words in $A$ and $B$, and the elements of pure degree are words where the letters have pure degree, and the degree is given by the sum of the degrees. For example, if $a_i\in A_i$ and $b_j\in B_j$ then the word $$a_0b_1a_{-1}b_2$$ is of pure degree $0+1+(-1)+2 = 2$. The associative product is of course given by concatenation of words (modulo the product on $A$ and $B$).

This definition extends easily to give the coproduct in the category of associative differential graded associative algebras.

Remark: The coproduct in the category of commutative (differential) graded algebras is much, much easier. The underlying vector space is simply the direct sum of graded vector spaces.