Tight Closure of Finite Length Modules in Graded Rings

Abstract

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 non-homogeneous 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 results of H. Brenner. We also show that unlike the Noetherian case, the injective hull of the residue field over $R^{+}$ or $R^{\infty }$ contains elements that are not killed by any power of the maximal ideal of R.

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