TENSOR POWERS OF SYMMETRICAL ALGEBRAS

We prove that over an integral domain a module is projective iff an ap propriate tensor power of its symmetric algebra is an integral domain. Further, we show that contracting parts of primary decompositions of the zero ideal in appropriate tensor powers of a symmetric algebra one obtains families of ideals canonically associated to a module, having the same radical as Fitting ideals. More precisely, we prove that tho se new ideals lie between the annihilators of exterior powers of the m odule and their radicals. An immediate consequence of our results is a way to recover the radicals of Fitting ideals of a module from the sy mmetric algebra of that module (with its grading forgotten).