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# The growth of graded noetherian algebras can not be exponential

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The growth of graded noetherian algebras can not be exponential Michael Bulois helped by Laurent Rigal 30 janvier 2006 Introduction The object of this document is to study the proof of the first theorem prensented in the article of Stephenson and Zang [SZ]. In the whole document, we will denote by k a field. In the second part A = ? i?N Ai will be a graded k-algebra with unity and M = ? i?N Mi a right graded A-module. 1 Few notions of growth The aim of this section is to present some notions of growth, in particular the one used in [SZ], to compare them and to present some of theirs basic properties which will be helpfull in order to understand some assumptions made in the main theorem's proof. While not precised, A will be a k-algebra and M a right A-module. 1.1 Gelfand-Kirillov dimension and SZ-exponential growth Definition 1.1. – Let A be a k-algebra and V a finite dimensional k-vectorial sub- space of A. We define dV ? NN by dV (n) =| n∑ i=0 V i| for any integer n ; where |E | denotes the dimension over k for any k-vectorial space E. – Now assume that M is a right module over A and N a k-subspace of M .

• growth propertie

• concerning modules

• xs ?

• zero dn's

• finitely generated

• exponential growth

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##### Exponential growth

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The growth of graded noetherian algebras can not be exponential
Michael Buloishelped byLaurent Rigal 30 janvier 2006
Introduction
The object of this document is to study the proof of the ﬁrst theorem prensented in the article of Stephenson and Zang [SZ]. In the whole document, we will denote byka ﬁeld. M M In the second partA=Aiwill be a gradedk-algebra with unity andM=Mia iNiN right gradedA-module.
1 Fewnotions of growth The aim of this section is to present some notions of growth, in particular the one used in [SZ], to compare them and to present some of theirs basic properties which will be helpfull in order to understand some assumptions made in the main theorem’s proof. While not precised,Awill be ak-algebra andMa rightA-module.
1.1 Gelfand-Kirillovdimension and SZ-exponential growth Deﬁnition 1.1.– LetAbe ak-algebra andVa ﬁnite dimensionalk-vectorial sub-n X Ni space ofA. We deﬁnedVNbydV(n) =|V|for any integern; where|E| i=0 denotes the dimension overkfor anyk-vectorial spaceE. – Nowassume thatMis a right module overAandNak-subspace ofM. We deﬁne n X i dN,V(n) =|N V|for any integern. i=0 Let us remark that for any non empty subspaceVAandNM, both of ﬁnite ? dimension,dVanddN,Vare incresing functions taking value inN. The object of the following deﬁnitions is to study the growth of these functions. N + Deﬁnition 1.2.– Letfbe a function ofN. We deﬁneγ(f)Ras logf(n) γ(f) = limsup logf(n) = limsup. n n+n+logn
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