Pseudo representations
Let \(A\) be a ring and \(R\) an \(A\)-algebra. A representation of \(R\) over \(A\) of dimension \(d\) is a homomorphism of \(A\)-algebras \(\rho:R\to M_d(A)\). A pseudo-representation of \(R\) over \(A\) of dimension \(d\) is a central \(A\)-linear form \(T:R\to A\), whose value at \(1_R\) is \(d\) and for which a certain fundamental identity is true. For example, the trace of a faithful representation is a pseudo-representation of the same dimension. The purpose of the article is to prove a converse proposition. Assume that \(A\) is a Henselian separable ring with residue field \(F\) and maximal ideal \(\mathfrak m\), and let \(T\) be a pseudo-representation of \(R\) over \(A\) of dimension \(d\). If there exists an absolutely irreducible representation \(\overline\rho:R\otimes_A F\to M_d(F)\) such that \(T\bmod{\mathfrak m}=\text{tr }\overline\rho\), then \(T\) is the trace of a faithful representation \(\rho:R\to M_d(A)\) which is unique up to conjugation and makes the diagram: \[ \begin{tikzcd} R \ar[r,"\rho"]\ar[d] & M_d(A)\ar[d]\\R\otimes_A F \ar[r,"\overline{\rho}"'] & M_d(F)\end{tikzcd} \] commute. Thanks to this, one can build explicitly the universal deformation ring of an absolutely irreducible representation over \(F\).
scientific article; zbMATH DE number 952631