Coarsening of Graded Local Cohomology

Abstract

Some criteria for graded local cohomology to commute with coarsening functors are proven, and an example is given where graded local cohomology does not commute with coarsening.

Key Words:

• Coarsening
• Graded local cohomology

2010 Mathematics Subject Classification:

• Primary 13D45
• Secondary 13A02

ACKNOWLEDGEMENT

I am grateful to Markus Brodmann for his encouraging support during the writing of this article. I also thank the referee for his careful reading and the suggested improvements.

The author was supported by the Swiss National Science Foundation.

Notes

In fact, in loc. cit., it is supposed that the ungraded ring underlying R is Noetherian, but in this special situation this is by [7, A.II.3.5] equivalent to R being Noetherian (as a graded ring, i.e., every increasing sequence of graded ideals is stationary).

A G-graded R-module M is called small if commutes with direct sums.

Using [5, 3.1; 3.4] one sees that this proof also applies if every power of 𝔞 has a projective resolution with small components. But since a projective G-graded R-module is small if and only if it is of finite type ([1, II.1.2]), this yields no improvement.

In loc. cit. one makes use only of the ITI property, but not of Noetherianness.

Communicated by U. Walther.