Growth of graded noetherian rings

Abstract

Abstract. We show that every graded locally finite right noetherian algebra has sub-exponential growth. As a consequence, every noetherian algebra with exponential growth has no finite dimensional filtration which leads to a right (or left) noetherian associated graded algebra. We also prove that every connected graded right noetherian algebra with finite global dimension has finite GK-dimension. Using this, we can classify all connected graded noetherian algebras of global dimension two. IntroductionThe study of graded algebras has gained prominence recently due to developments in the area of noncommutative projective geometry (for example, see [ATV]). In this article, we prove several results involving the growth of graded algebras, and give applications to the theory of regular algebras.Throughout, k is a field and all algebras and modules are over the base field k. The term 'algebra' will be used to denote an associative k-algebra with unity. Let R be an (ungraded) algebra and M a right R-module. The Gelfand-Kirillov dimension of M is defined bywhere V ⊂ R and N ⊂ M run over all finite dimensional subspaces. We say M has exponential growth ifwhere V and N are as above. Otherwise, we say M has sub-exponential growth. If GKdim(M ) < ∞, then M has sub-exponential growth, but the converse is not true in general (see [S2]). In this paper we are interested in N-graded algebrasIn particular, if A is a graded right noetherian algebra and A 0 is finite dimensional, then A is locally finite. Our main result is the following theorem.

Criteria for $\sigma $-ampleness

“…for m ≫ 0. Thus the associated twisted homogenous coordinate ring has exponential growth and hence is not (right or left) noetherian [SZ,Theorem 0.1]. So D cannot be right σ −1 -ample, by [AV,Theorem 1.4].…”

Regular algebras of dimension 4 and their A∞-Ext-algebras

“…(a) and (b) are proved in [SteZ,3.1.1]. Part (c) is given in the proof of [SteZ,3.1.4]. We repeat the main idea here.…”

“…By the claim proved in the last paragraph one sees that GKdim A > 2. Finally, the GK-dimension of a noetherian AS regular algebra is an integer [SteZ,2.4], so GKdim ≥ 3. Now we start to focus on AS regular algebras of global dimension 4.…”

“…Recall that the GK-dimension of A is equal to the order of the zero of p(t) at 1 [SteZ,2.2]. Clearly GKdim A ≥ 1, whence p(1) = 0.…”

Actions of maximal growth

“…• ϕ(g), for any x ∈ S and g ∈ F r . If we insist that the growth functions should be equivalent under semi-isomorphisms then, as we see, for instance, in [18], all functions with exponential growth fall into the same equivalence class. Using recent results in [8] and [9], we give in Subsection 6.2 examples where S and S are semi-isomorphic, the growth of S is maximal and the growth of S is not.…”