Winfried Bruns And

We describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski’s theory of complete ideals and of representation theory. Let R be a commutative ring. The determinantal ideals attached to matrices with entries in R play ubiquitous roles in the study of the syzygies of R–modules. In this note, we describe some the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski’s theory of complete ideals and of representation theory, the results being sharper for rings containing the rationals. Let R be an integral domain (or a field) and φ : R → R anR-linear map of rank r. It is an easy exercise to show that the d-th symmetric power S(φ) : S(R) → S(R) has rank ( r+d−1 d ) . Let It(φ) denote the ideal generated by the minors (of a matrix representing φ). Since rankφ = max{r : Ir(φ) 6= 0}, one can immediately determine the radicals of the ideals It(S (φ)), namely Rad It(S (φ)) = Rad Ir(φ) if (