Tight closure of finite length modules in graded rings

Abstract

In this article, we look at how the equivalence of tight closure and plus closure (or Frobenius closure) in the homogeneous m-coprimary case implies the same closure equivalence in the nonhomogeneous m-coprimary case in standard graded rings. Although our result does not depend upon dimension, the primary application is based on results known in dimension 2 due to the recent work of H. Brenner. We also demonstrate a connection between tight closure and the m-adic closure of modules extended to R+ or R∞. We finally show that unlike the Noetherian case, the injective hull of the residue field over R+ or R∞ contains elements that are not killed by any power of the maximal ideal of R. This fact presents an obstruction to one possible method of extending our main result to all modules.