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ABSTRACT
The Frobenius depth denoted by F-depth is defined by Hartshorne-Speiser in 1977 and later by Lyubeznik in 2006, in a different way, for rings of positive characteristic. The first aim of the present paper is to compare the F-depth with formal grade and reprove some results of Lyubeznik using formal local cohomology. Then the endomorphism rings of local cohomology modules will be considered. As an application, we reprove the results due to Huneke–Koh in positive characteristic and Lyubeznik in characteristic zero on the annihilators of local cohomology modules.
Acknowledgments
I am deeply grateful to the referee for his/her careful reading of the manuscript, appropriate and constructive suggestions to improve the paper. This project is based on a question asked by Professor Josep Àlvarez-Montaner on the comparison of F-depth and fgrade when the author was visiting the Universitat Politècnica de Catalunya, in September 2013. Hereby, I would like to express my thanks to the Universitat Politècnica and specially, to Josep, for their support and warm hospitality. I am also strongly indebted with Josep for several fruitful discussions. I would like to mention that when the author was finishing this project, he started a joint project with Alberto F. Boix and gave a characteristic free proof of the result of Huneke–Koh and Lyubeznik on the annihilator of local cohomology modules.