On (De)homogenized Gr\"obner Bases

TL;DR

This paper establishes a correspondence between Gr"obner bases in graded algebras and their homogenized counterparts in polynomial and free algebras, facilitating the study of dh-closed ideals and Rees algebras.

Contribution

It proves a one-to-one correspondence between Gr"obner bases in graded algebras and dh-closed homogeneous Gr"obner bases in polynomial and free algebras, enabling simplified analysis.

Findings

Correspondence between Gr"obner bases in R and dh-closed homogeneous Gr"obner bases in R[t]

Similar correspondence in free algebras K< X_1,...,X_n> and K< X_1,...,X_n,T>

dh-closed graded ideals can be realized by dh-closed homogeneous Gr"obner bases

Abstract

Let $K$ be a field and $R=\oplus_{p\in\mathbb{N}}R_p$ an $\mathbb{N}$-graded $K$-algebra, which has an SM $K$-basis (i.e. a skew multiplicative $K$-basis) such that $R$ holds a Gr\"obner basis theory. It is proved that there is a one-to-one correspondence between the set of Gr\"obner bases in $R$ and the set of dh-closed homogeneous Gr\"obner bases in the polynomial algebra $R[t]$; and that the similar result holds true if $R$ and $R[t]$ are replaced respectively by the free algebra $K< X_1,...,X_n>$ and the free algebra $K< X_1,...,X_n,T>$. Moreover, it is shown that dh-closed graded ideals in $R[t]$ and $K< X_1,...,X_n, T>$ can be realized by dh-closed homogeneous Gr\"obner bases. The latter result indeed tells us that algebras defined by dh-homogeneous Gr\"obner bases can be studied as Rees algebras effectively via more simpler algebras as demonstrated in ([7], [8]).