AFRICA MATHEMATICS ANNALS

Finiteness criteria for Bigraded algebras,
Application to the Rees rings of a Bifiltration>

Saint Blanc Amani KOUA ¹⁾, Abdoulaye ASSANE ²⁾,
and Daouda SANGARE ³⁾

University of Nangui Abrogoua, UFR-SFA
Mathematics and Computer Science Laboratory
02 BP 801 Abidjan 02, Côte d’Ivoire

Mathematics Subject Classification: (2010) 13A02, 13A30, 16W50, 16W70.
Key words: Rees ring, Bigraded algebra, Bifiltration.

Abstract:

Classical finiteness criteria for ℤ-graded algebras have existed in the literature since a long time, in particular those established by D. Rees in [8]. In this note, we are interested in analogous finiteness criteria but for ℤ² bigraded rings that we will apply to the Rees rings of a bifiltration. As essential results, we will establish ℤ²-bigraded versions of finiteness criteria given for ℤ-graded rings by Bruns and Herzog [4].

First, we’ll give a brief review on graded rings and bigraded rings. We will then recall the finiteness criteria for graded algebras established by Bruns and Herzog [4], and apply them to characterize nœtherian filtrations. We provide the proofs for the reader’s convenience.

Next, we establish analogues of these finiteness criteria for ℤ²-bigraded rings, and then apply them to the Rees rings of a bifiltration. In the special case of crossed bifiltrations, these finiteness criteria will be extended by adding, among others, sufficient conditions on the Veronese subrings of Rees rings, as in the case of filtrations. This leads, under appropriate asspumptions to transfer theorems on the nœtherianity of cross products of filtrations and Veronese subrings of index (k,l) of the Rees rings of these cross-products.