Coproduct of graded Algebras 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!

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.
