Variations on the Grothendieck-Serre Formula for Hilbert functions and their applications
Shreedevi K. Masuti, Parangama Sarkar, J. K. Verma
TL;DR
This paper explores variations of the Grothendieck-Serre Formula for multi-graded algebras, providing proofs, applications to Hilbert functions, and discussing related conjectures and theorems in algebraic geometry.
Contribution
It offers new proofs and generalizations of key formulas and theorems in multi-graded algebra, including applications to Hilbert functions and partial solutions to conjectures.
Findings
Derived formulas of Sally for postulation numbers
Provided an alternate proof of Huneke-Ooishi Theorem
Discussed partial solutions to Itoh's conjecture
Abstract
In this expository paper we present proofs of Grothendieck-Serre Formula for multi-graded algebras and Rees algebras for admissible multi-graded filtrations. As applications, we derive formulas of Sally for postulation number of admissible filtrations and Hilbert coefficients. We also discuss a partial solution of Itoh's conjecture by Kummini and Masuti. We present an alternate proof of Huneke-Ooishi Theorem and a generalisation for multi-graded filtrations.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Commutative Algebra and Its Applications · Advanced Topics in Algebra