Reduced norms and abelianizing
For a category \({\mathcal C}\) (in which there is a concept of ``abelian'' object, e.g., the category of groups) one can formulate the following interesting problem: given an object C of \({\mathcal C}\) one chooses the morphism k: \(C\to k(C)\) in \({\mathcal C}\) (named the ``abelianization'' of C) which is universal in the following sense: given any abelian object B in \({\mathcal C}\) and a morphism f: \(C\to B\), there exists a unique morphism \(f_{ab}: k(C)\to B\) in \({\mathcal C}\) such that \(f=f_{ab}\circ k\). In this context a question could be asked: Is the determinant det: \(M_ n(R)\to R\) (for a commutative (unital) ring R) the ``abelianization'' of \(M_ n(R)\) in the category of R-algebras and multiplicative polynomial laws? (This problem has a natural extension to R-Azumaya algebras and their reduced norms.) One can consider this question as an aspect of the problem: Is the determinant an abelianization? (It turns out that the problem is in fact to choose the correct category.) In this paper the above question is studied and a totally positive answer is obtained. Other interesting situations are also presented.
category of R-algebras and multiplicative polynomial laws
scientific article; zbMATH DE number 4173207
scientific article