Asymptotic behavior of dimensions of syzygies

Let R be a commutative noetherian local ring, and M a finitely generated R-module of infinite projective dimension. It is well-known that the depths of the syzygy modules of M eventually stabilize to the depth of R. In this paper, we investigate the conditions under which a similar statement can be made regarding dimension. In particular, we show that if R is equidimensional and the Betti numbers of M are eventually non-decreasing, then the dimension of any sufficiently high syzygy module of M coincides with the dimension of R.


Introduction
Throughout this paper R will denote a commutative noetherian local ring with identity element, unique maximal ideal m and residue field k := R/m. Let M be a finitely generated R-module. The ith Betti number of M is given by β i (M ) := dim k (Tor R i (k, M )). A minimal free resolution of M then has the form This question was also explored in the last section of [5]. In [5,Remark 5.2 (i)] it is noted that if R is unmixed and equidimensional, then (dim(Ω i (M ))) ∞ i=0 is constant for i ≫ 0. This is clear since the associated primes of any submodule of R βi(M) are also associated primes of R, and are therefore primes of maximal dimension by assumption.
It is worth noting that the asymptotic behavior of the depths of syzygy modules is known. Given M the sequence (depth(Ω i (M ))) ∞ i=0 is constant for all i ≫ 0. Let pd(M ) denote the projective dimension of M . In particular if pd(M ) = ∞, then depth(Ω n (M )) depth(R) for n max{0, depth(R) − depth(M )}, with at most one strict inequality at either n = 0 or n = depth(R) − depth(M ) + 1; see [9,Proposition 10] or [2,Proposition 1.2.8]. It follows therefore that if pd(M ) = ∞ and R is Cohen-Macaulay, then dim(Ω n (M )) = dim(R) for n ≫ 0.
All of our results are for modules whose Betti numbers are eventually nondecreasing. Therefore finding a proof for the following conjecture of L. Avramov would improve our results.

Conjecture 4.
[1] The Betti numbers of any finitely generated module over an arbitrary noetherian local ring are eventually non-decreasing.
There are a plethora of cases for which this conjecture is known to be true. J. Lescot [8,Corollaire 6.5] showed that over a Golod ring, which is not a hypersurface any finitely generated module of infinite projective dimension will have eventually increasing Betti numbers. Also L.-C. Sun [11,Corollary] showed that over rings of codepth less than or equal to three and Gorenstein rings of codepth four all finitely generated modules have eventually non-decreasing Betti numbers. Several other interesting cases are also proven in [3], [4] and [12].
Whenever the Betti numbers of a module are eventually strictly increasing it is known that the the dimension of a sufficiently high syzygy will have the dimension of the ring. This is clear from the next lemma, which is mentioned without proof in [5, Remark 5.2 (iii)].

Results
We denote the length of an R-module M by λ R (M ) or simply λ(M ) when the ring is unambiguous.
Lemma 5. Let R be a noetherian local ring and M a finitely generated R-module.
The following lemma is used in the proof of our main result, Theorem 8.
Lemma 6. Let R be a noetherian local ring and M a finitely generated R-module. For a given n ∈ N suppose that β 0 (M ) β 1 (M ) . . . β 2n−1 (M ) and that Supp(Ω 2n (M )) = Spec(R). Then we have the following: Proof. Choose p ∈ Min(R) Supp(Ω 2n (M )). Localizing part of a minimal free resolution of M at p, we get an exact sequence of finite-length R p -modules of the following form.
Since ϕ 0 is the zero map, M p = 0; hence p / ∈ Supp(M ). It follows that Let q ∈ Spec(R) Supp(Ω 2i (M )) for some i with 0 i n − 1. Localizing part of a minimal free resolution of M at q we obtain an exact sequence of the following form. (1) provides the reverse containment, (c) is now immediate.
We make the following fact explicit in order to clarify some of our argumentation. Proof. We may assume that pd(M ) = ∞. By replacing M by a sufficiently high syzygy, one may assume that β i+1 (M ) β i (M ) for all i 0. Assuming M was replaced by an even (odd) syzygy, if Supp(Ω 2i (M )) = Spec(R) for i ≫ 0, then all of the statements hold for even (odd) syzygies. Therefore we may suppose that there exist infinitely many i ∈ N such that Supp(Ω 2i (M )) = Spec(R).
Therefore it remains to show that Min(Ω 2i (M )) ⊆ Min(R) for i ≫ 0. Choose q ∈ Min(Ω 2m (M )). Let S := R q , M i := (Ω 2m+2i (M )) q for i 0 and n := qR q . Note that n is the maximal ideal for S. For all i 0 we obtain a commutative diagram of the form where the top row is exact and N i := Im(α i ). If the matrix A i defining the map α i : S bi → S bi has some entries which are units, then by Fact 7 we can reduce this sequence by taking away free summands; hence we may assume that A i has all of its entries in n.
Let H i n (−) denote the ith local cohomology functor with respect to n. For background on local cohomology see [7]. Since M i+1 has finite length H 0 n (M i+1 ) ∼ = M i+1 and H j n (M i+1 ) = 0 for all j > 0. From the long exact sequence of local cohomology modules associated to the short exact sequence we get an exact sequence Here we are defining γ i : M i → H 1 n (N i ) to be the map found in exact sequence (4). By the additivity of length we get the first and third steps in the next display from sequences (4) and (3) respectively.
Since the sequence (λ(M i )) ∞ i=0 is positive and non-increasing it is eventually constant. Choose ℓ ∈ N such that λ(M ℓ ) = λ(M ℓ+1 ). Then λ(Im(γ ℓ ))) = 0. Therefore γ ℓ is the zero map. From (4), it follows that H 1 n (ψ ℓ ) : H 1 n (N ℓ ) → H 1 n (S b ℓ ) is an isomorphism. We have shown that H j n (ψ ℓ ) and H j n (φ ℓ ) are isomorphisms for all j 1. Using the commutativity of (2) it follows that is an isomorphism for all j 1. Since H j n (−) is an S-linear functor the map H j n (α ℓ ) is defined by matrix multiplication from the matrix A ℓ applied to the components of H j n (S b ℓ ). Since A ℓ has entries in n it must kill socle elements of H j n (S b ℓ ). Therefore H j n (S b ℓ ) has no socle elements. Since H j n (S b ℓ ) is n-torsion it follows that H j n (S b ℓ ) = 0 for all j 1. By [7,Theorem 9.3] we get the second equality in the next display. dim(R q ) = dim(S) = sup{j| H j n (S) = 0} = 0 Thus q ∈ Min(R); hence Min(Ω 2i (M )) ⊆ Min(R) for all i ≫ 0, and (a) follows.
Corollary 9. Let R be a noetherian local ring and M a finitely generated Rmodule with eventually non-decreasing Betti numbers. Then (dim(Ω 2i (M ))) ∞ i=0 and (dim(Ω 2i+1 (M ))) ∞ i=0 are constant for i ≫ 0. If pd(M ) = ∞ then one sequence stabilizes to dim(R) and the other sequence stabilizes to dim(R/p) for some p ∈ Min(R).
Corollary 2 follows immediately. Note that if R is a domain or if dim(R) 1, then R is equidimensional; hence, one can apply Corollary 2.
Remark 10. It should be noted that [5,Remark 5.6] claims that using [5, Proposition 5.5] one can show that if R is equidimensional and Conjecture 4 is true, then dim(Ω n (M )) is constant for n ≫ 0. However, [5,Proposition 5.5] requires the assumption that dim(R) 2. Therefore although the conclusions of [5, Remark 5.6] are correct, the justification given for these conclusions is invalid. One should note the justification uses a localization argument, so it is invalid in every positive dimension, not just dimension 1.
We now turn our attention to determining how quickly (Supp(Ω 2i (M ))) ∞ i=0 stabilizes once the Betti numbers of M become non-decreasing. has non-zero finite length homology. By the New Intersection Theorem [10] it follows that dim(R p ) 1.