Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear maps defined on $\prod_{i \in I} M_i$ (this exists by the usual construction). Thus there is a universal multilinear map $\otimes : \prod_{i \in I} M_i \to \bigotimes_{i \in I} M_i$. Some years ago, I wanted to examine this infinite tensor product, but in the literature I could not find anything going beyond some natural isomorphisms (e.g. associativity) or the submodule consisting of tensors which become eventually constant a specific element of $\prod_{i \in I} M_i$ which yields a colimit of finite tensor products (denoted $U_x$ below). In general, it seems to be quite hard to describe $\bigotimes_{i \in I} M_i$. For example, for a field $K$, $K \otimes_K \otimes_K ...$ has dimension $|K^\*|^{\aleph_0}$ (see below) and you cannot write down a basis, which might be scary when you see it the first time. The point is that multilinear relations cannot be applied infinitely many times at once: For example in $K \otimes_K \otimes_K ...$, we have $x_1 \otimes x_2 \otimes ... = y_1 \otimes y_2 \otimes ...$ if and only if $x_i = y_i$ for almost all $i$ and for the rest we have $\prod_i x_i = \prod_i y_i$.

2.2 Let $W_i$ be another family of vector spaces over a field $K$. Then there is a canonical map $\alpha : \bigotimes_{i \in I} Hom(V_i,W_i) \to Hom(\bigotimes_{i \in I} V_i, \bigotimes_{i \in I} W_i).$ Is $\alpha$ injective? This is known when $I$ is finite.

Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear maps defined on $\prod_{i \in I} M_i$ (this exists by the usual construction). Thus there is a universal multilinear map $\otimes : \prod_{i \in I} M_i \to \bigotimes_{i \in I} M_i$. Some years ago, I wanted to examine this infinite tensor product, but in the literature I could not find anything going beyond some natural isomorphisms (e.g. associativity) or the submodule consisting of tensors which become eventually constant a specific element of $\prod_{i \in I} M_i$ which yields a colimit of finite tensor products (denoted $U_x$ below). In general, it seems to be quite hard to describe $\bigotimes_{i \in I} M_i$. For example, for a field $K$, $K \otimes_K \otimes_K ...$ has dimension $|K^*|^{\aleph_0}$ (see below) and you cannot write down a basis, which might be scary when you see it the first time. The point is that multilinear relations cannot be applied infinitely many times at once: For example in $K \otimes_K \otimes_K ...$, we have $x_1 \otimes x_2 \otimes ... = y_1 \otimes y_2 \otimes ...$ if and only if $x_i = y_i$ for almost all $i$ and for the rest we have $\prod_i x_i = \prod_i y_i$.

2.2 Let $W_i$ be another family of vector spaces over a field $K$. Then there is a canonical map $\alpha : \bigotimes_{i \in I} \operatorname{Hom}(V_i,W_i) \to \operatorname{Hom}(\bigotimes_{i \in I} V_i, \bigotimes_{i \in I} W_i).$ Is $\alpha$ injective? This is known when $I$ is finite.

1.3 If $A$ is a field, and $M_i$ has basis $B_i$, then $B_x = \cup_{E \subseteq I} \bigotimes_{i \in E} B_i \otimes \otimes_{i \notin E} x_i$ is a basis of $U_x$ and thus $\cup_{x \in R} B_x$ is a basis of $\bigotimes_{i \in I} M_i$. According to this question, this has cardinality $\max(|X|,|I|,\max_i(\dim(M_i)))$.

If $A=A_i=K$ is a field with $U=K^x$, then there is a vector space isomorphism between $\bigotimes_{i \in I} K$ and the group algebra $K[U^I / U^{(I)}]$. A sufficient, not neccessary, condition for the existence of a $K$-algebra isomorphism is that $U^{(I)}$ is a direct summand of $U^I$, which is quite rare (see this question). Nevertheless, we can ask if these $K$-algebras isomorphic. In some sense I have proven this already locally (subalgebras given by finitely generated subgroups of the group $U^I / U^{(I)}$ are isomorphic, in a terribly uncanonical way). Many questions I'm currently posing here are addressed to this problem.

1.3 If $A$ is a field, and $M_i$ has basis $B_i$, then $B_x = \cup_{E \subseteq I} \bigotimes_{i \in E} B_i \otimes \otimes_{i \notin E} x_i$ is a basis of $U_x$ and thus $\cup_{x \in R} B_x$ is a basis of $\bigotimes_{i \in I} M_i$. According to this question, this has cardinality $\max(|X|,|I|,\max_i(\dim(M_i)))$.

If $A=A_i=K$ is a field with $U=K^x$, then there is a vector space isomorphism between $\bigotimes_{i \in I} K$ and the group algebra $K[U^I / U^{(I)}]$. A sufficient, not neccessary, condition for the existence of a $K$-algebra isomorphism is that $U^{(I)}$ is a direct summand of $U^I$, which is quite rare (see this question). Nevertheless, we can ask if these $K$-algebras isomorphic. In some sense I have proven this already locally (subalgebras given by finitely generated subgroups of the group $U^I / U^{(I)}$ are isomorphic, in a terribly uncanonical way). Many questions I'm currently posing here are addressed to this problem.

Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear maps defined on $\prod_{i \in I} M_i$ (this exists by the usual construction). Thus there is a universal multilinear map $\otimes : \prod_{i \in I} M_i \to \bigotimes_{i \in I} M_i$. Some years ago, I wanted to examine this infinite tensor product, but in the literature I could not find anything going beyond some natural isomorphisms (e.g. associativity) or the submodule consisting of tensors which become eventually constant a specific element of $\prod_{i \in I} M_i$ which yields a colimit of finite tensor products (denoted $U_x$ below). In general, it seems to be quite hard to describe $\bigotimes_{i \in I} M_i$. For example, for a field $K$, $K \otimes_K \otimes_K ...$ has dimension $|K^*|^{\aleph_0}$ (see below) and you cannot write down a basis, which might be scary when you see it the first time. Before posing my question, I provide some results.

If $A=A_i=K$ is a field with $U=K^x$, then there is a vector space isomorphism between $\bigotimes_{i \in I} K$ and the group algebra $K[U^I / U^{(I)}]$. A necessary condition for the existence of a $K$-algebra isomorphism is that $U^{(I)}$ is a direct summand of $U^I$, which is quite rare (see this question). Nevertheless, we can ask if these $K$-algebras isomorphic. In some sense I have proven this already locally (subalgebras given by finitely generated subgroups of the group $U^I / U^{(I)}$ are isomorphic, in a terribly uncanonical way). Many questions I'm currently posing here are addressed to this problem.

Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear maps defined on $\prod_{i \in I} M_i$ (this exists by the usual construction). Thus there is a universal multilinear map $\otimes : \prod_{i \in I} M_i \to \bigotimes_{i \in I} M_i$. Some years ago, I wanted to examine this infinite tensor product, but in the literature I could not find anything going beyond some natural isomorphisms (e.g. associativity) or the submodule consisting of tensors which become eventually constant a specific element of $\prod_{i \in I} M_i$ which yields a colimit of finite tensor products (denoted $U_x$ below). In general, it seems to be quite hard to describe $\bigotimes_{i \in I} M_i$. For example, for a field $K$, $K \otimes_K \otimes_K ...$ has dimension $|K^\*|^{\aleph_0}$ (see below) and you cannot write down a basis, which might be scary when you see it the first time. The point is that multilinear relations cannot be applied infinitely many times at once: For example in $K \otimes_K \otimes_K ...$, we have $x_1 \otimes x_2 \otimes ... = y_1 \otimes y_2 \otimes ...$ if and only if $x_i = y_i$ for almost all $i$ and for the rest we have $\prod_i x_i = \prod_i y_i$. 

Before posing my question, I provide some results.

If $A=A_i=K$ is a field with $U=K^x$, then there is a vector space isomorphism between $\bigotimes_{i \in I} K$ and the group algebra $K[U^I / U^{(I)}]$. A sufficient, not neccessary, condition for the existence of a $K$-algebra isomorphism is that $U^{(I)}$ is a direct summand of $U^I$, which is quite rare (see this question). Nevertheless, we can ask if these $K$-algebras isomorphic. In some sense I have proven this already locally (subalgebras given by finitely generated subgroups of the group $U^I / U^{(I)}$ are isomorphic, in a terribly uncanonical way). Many questions I'm currently posing here are addressed to this problem.