# Tensor product of d.g-algebras

I'd like to prove that the tensor product functor $- \otimes Y$, where $Y$ is a d.g-algebra over a field of characteristic 0, preserves finite products of d.g-algebras. This statement is in a paper by Bousfield and Gugenheim whose title is "On PL de-Rham theory and rational homotopy type".

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What have you tried so far, where do you get stuck ? –  tj_ Dec 22 '13 at 22:59
Do you assume something like the dga is in degree >=0?(or <=0 depending on conventions). At first glance that seems like a necessary condition to make something like this true. –  user36931 Dec 23 '13 at 8:19
user36931: Thank you for your advice. Here, we assume that the characteristic of a field is 0. –  Takashi Dec 23 '13 at 16:49
tj_:I made a stupid mistake when I wrote down explicitly the unique map induced by the universality of the given finite limit. But I still can't understand that the tensor functor preserves arbitrary limits if $Y$ is of finite type, i.e., $Y^{n}$ is finite dimensional for all $n \in \mathbb{N}_{>=0}$. What seems to be the problem with the case where Y is not an finite dimensional? –  Takashi Dec 24 '13 at 7:49