Let $A$ be a $d$-dimensional algebra over a field $K$. One can naturally assign to $A$ the linear algebraic $K$-group $\mathbf{GL_1}(A)$ that represents the functor $B \mapsto (A \otimes_K B)^\times$ for $K$-algebras $B$. My question is:

Is the functor $A \mapsto \mathbf{GL_1}(A)$ from finite-dimensional $K$-algebras to linear algebraic $K$-groups fully faithful, i.e. is the natural map $$ \hom_{K-\text{Alg}}(A,B) \to \hom_{K-\text{Grp}}(\mathbf{GL_1}(A), \mathbf{GL_1}(B)) $$ for finite-dimensional $K$-algebras $A$ and $B$ a bijection?


The $K$-algebra $K$ doesn't have non-trivial endomorphisms as it is the inital object in $K-\text{Alg}$, but the linear algebraic $K$-group $\mathbf{GL_1}(K)$ does have non-trivial endomorphisms, for example $x \mapsto x^n$ for a fixed $n \in \Bbb N$, thus the functor is not full.

  • 2
    $\begingroup$ So the functor is not full, and consideration of $k[t]$ shows that it is not faithful either. $\endgroup$ – Kevin Carlson Oct 22 '18 at 17:18

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