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In the following, I use the word "explicit" in the following sense: No choices of bases (of vector spaces or field extensions), non-principal ultrafilters or alike which exist only by Zorn's Lemma (or …

Let $V$ be a vector space over a field $K$. Call a linear map $F : V^* \to K$ representable if there is some $v \in V$ such that $F(w) = \langle w,v \rangle$ for all $w \in V^*$. Here, $\langle w,v \r …

Fix a $\mathbb{Q}$-algebra $R$. Let's call an endomorphism $f : M \to M$ of an $R$-module $M$ locally nilpotent if for every $m \in M$ there is some $n \in \mathbb{N}$ such that $f^n(m)=0$. Equivalent …

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 m …

When $q$ is a prime power, then on the one hand the $q$-binomial coefficient $\binom{n}{k}_q$ equals the number of $k$-dimensional subspaces of $\mathbb{F}_q^n$, and on the other hand it is the genera …