On (De)homogenized Gröbner Bases

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Let K be a field and R=⊕pₚ an N-graded K-algebra, which has an SM K-basis (i.e. a skew multiplicative K-basis) such that R holds a Gröbner basis theory. It is proved that there is a one-to-one correspondence between the set of Gröbner bases in R and the set of dh-closed homogeneous Gröbner 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< X₁,...,Xₙ> and the free algebra K< X₁,...,Xₙ,T>. Moreover, it is shown that dh-closed graded ideals in R[t] and K< X₁,...,Xₙ, T> can be realized by dh-closed homogeneous Gröbner bases. The latter result indeed tells us that algebras defined by dh-homogeneous Gröbner bases can be studied as Rees algebras effectively via more simpler algebras as demonstrated in ([7], [8]).