In the exercise from chapter $2$ of the book, "Introduction to Commutative Algebra" by Atiyah & Macdonald, I understand that the highlighted $A$-algebra $B$ is the the direct limit but I want to ask the following questions:
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How can we say that the direct limit is the tensor product of the given family? Shouldn't one also prove the universal property of the tensor product here? If yes than how can we prove that universal property since here we have arbitrary family of $A$-algebras which may be infinite.
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How the arbitrary element of the $A$-algebra $B$ will look like? My intuition says: For each $\lambda$, fix an element $b_\lambda$ $\in$ $B_\lambda$, then, any arbitrary elements of $B$ is of the form $\otimes$$_\lambda$$x_\lambda$ where $x_\lambda$=$b_\lambda$ for all but finitely many values of $\lambda$. But the problem is that how the tensor map $\otimes$$_\lambda$$x_\lambda$ will be defined since $\lambda$ runs over an arbitrary family.
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Can one define the tensor products in the same way for directed systems of arbitrary families of, say $A$-modules or say vector spaces of a field? Will the direct limits exist in those cases?
Best Answer
Note that the tensor product of algebras is in fact the coproduct in the category of $A$-algebras. What AM defines there should therefore satisfy the universal property of an infinite coproduct. I am not sure about the universal property with respect to multilinear maps of the underlying modules though. I believe one can derive something like this, yet one has to ask oneself, if it would be worth the effort.
I recall doing this construction in some exercise quite a while ago and would prefer not to reconsider how the elements look like. So I am afraid I wont be of much use regarding your second question. Maybe someone else can help out there. Personally I strive to do as much as possible using universal properties only...