Hilbert Coefficients of Hilbert Filtrations

“…The coefficient e 0 (I) is called the multiplicity of I. P. Samuel [Sam51] showed existence of this polynomial for the I-adic filtration {I n } n∈Z . Many results about Hilbert polynomials for admissible filtrations were proved in [GR98] and [RV10].…”

Section: Introductionmentioning

confidence: 99%

Variations on the Grothendieck–Serre Formula for Hilbert Functions and Their Applications

Masuti

Sarkar

Verma

2016

Algebra and Its Applications

View full text Add to dashboard Cite

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 genaralisation for multi-graded filtrations.

show abstract

“…However, it can be done by the same argument as the proof of (s n−1 t) n−1 ∈ (s n , t n )(s n , s n−1 t, t n ) n−1 , and hence we have proved (2).…”

Section: Examplementioning

confidence: 79%

“…On the other hand, if A is an analytically unramified local ring, then { I n } n≥0 is a Hilbert filtration (cf. [2]), and so there exists a family { e i (I) } 0≤i≤d of integers such that…”

Section: Corollariesmentioning

confidence: 99%