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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!

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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.

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  • $\begingroup$ Thanks a lot for your answer! How is the addition of words defined and can you give details for the differential graded algebra case? $\endgroup$ – user818182 Jul 21 '16 at 21:31
  • $\begingroup$ @user818182 Take a basis (of elements of pure degree) for $A$ and $B$, then you can take the words in the elements of the bases as a basis for the vector space underlying the coproduct. Now use distributivity of the product to get the sum defined for arbitrary words. $\endgroup$ – Daniel Robert-Nicoud Jul 21 '16 at 21:35

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