Is the colimit of finite tensor products a tensor product?

Question

Let $(R_\lambda)_{\lambda\in\Lambda}$ be a family of $A$-algebras. Atiyah & MacDonald defines the "tensor product" of the family as the direct limit of the tensor product of finite subfamilies. But one can perfectly well extend the definition of finite tensor product to an infinite one as the unique homomorphism extending the multilinear map. (See https://mathoverflow.net/q/11767/5292 for a detailed discussion about this topic.) Categorically speaking, A & M's version of tensor product should be called coproduct rather than tensor product. Or is it conventional in the literature to call this a tensor product? Or is there a legitimate reason to call this a tensor product? Or am I missing something?

Answer 1

You are right, it is the coproduct (in the commutative case). It is called a tensor product because it satisfies the usual laws from the finite tensor products. By the way, the tensor product of algebras does not coincide with the tensor product of modules - rather we endow the tensor product of the underlying modules with the structure of an algebra. The infinite tensor product of modules isn't really useful because even for free modules it is very large (see the MO discussion). Many common techniques from finite tensor products fail here. For this reason it isn't studied much in the literature. And I think this is also the reason why, for algebras, one calls the directed colimit of the finite tensor products just the tensor product (although it is just one direct summand of the whole tensor product of the underlying modules, generated by those tensors which are eventually $\dotsc \otimes 1 \otimes 1 \otimes \dotsc$).