Matrix factorizations, exterior powers, and complete intersections

Abstract

Let R be a commutative ring with multiplicative identity, let Rn be a direct sum of n copies of R and let R, be the ring of n x n matrices over R. We begin with the investigation of conditions on the ring R which will imply that A, ,..., A, E R, have a common right factor having determinant r, a non-zerodivisor. Set 2f = xi=, R,A, , and define the norm of 9l by N(2l) = Ca.~ (det A)R. An obvious necessary condition for the existence of such a factor is that N(%) C rR. If the condition N(%) C rR (resp. N(‘%) = rR) suffices to imply the existence of such a factor, R, is said to have norm-induced factorization (resp. weak norminduced factorization). Initially, our interest in matrix factorization stemmed from the Morita correspondence between submodules of R” and left ideals in R, . A basic observation is the equivalence of the two conditions: “for PC Rn, A”P N R implies P N R”” and “N(N) N R implies % is a principal left ideal” (see Corollary 1, Section 2). Each of these conditions in turn are equivalent to weak norminduced factorization in R, (Proposition 1). If this factorization condition is present in R% for all n > 1, then the Grothendieck group SK,,(R) = 0 and stably free projective modules are free modules ([2, Prop. 3.7, Chap. 91 and Corollary 1, Section 2). It is also possible to conclude, in the case the zero divisors of R are contained in a finite union of prime ideals, that finitely generated maximal ideals locally having two generators must in fact be generated by two elements (Theorem 3). Thus weak norm-induced factorization in R, (n = 2 suffices) and R regular imply that maximal ideals of height 2 are complete intersections. If R is regular of dimension at most 2 then finitely generated projectives over the polynomial ring R[x] are extended from R (see [23]). We have then an elementary proof of the fact that if R is a regular local ring of dimension at most 2 then maximal ideals in R[x] are complete intersections (Corollary 1 of Theorem 3).