Proceedings of the 44th Symposium on Ring Theory and ...

and third question is also known to be affirmative in <strong>the</strong> group algebra case [12] but
almost nothing is known outside this case.
4. Growth <strong>of</strong> Betti numbers. The local case
Let us return to <strong>the</strong> situation where R = (R, m, k) is a local noe<strong>the</strong>rian k-algebra. The
following questions were among questions posed in <strong>the</strong> late 1970s and <strong>the</strong> early 1980s.
They are still open even in <strong>the</strong> commutative artinian case, and even if we also add <strong>the</strong>
selfinjective assumption.
Questions 23. Let R = (R, m, k) be a local noe<strong>the</strong>rian k-algebra. Let M be an indecomposable
finitely generated R-module <strong>of</strong> infinite projective dimension.
(1) Assume that M has complexity 1. Is <strong>the</strong> sequence <strong>of</strong> Betti numbers {β i (M)} i
eventually constant?
(2) Is <strong>the</strong> sequence <strong>of</strong> Betti numbers {β i (M)} i eventually nondecreasing?
The first question has an affirmative answer if R is a complete intersection ([10]). In <strong>the</strong>
radical square zero case <strong>the</strong> answer is also affirmative. We sketch <strong>the</strong> pro<strong>of</strong> below (see
also [15])
Proposition 24. Let R = (R, m, k) is a local artinian ring with m 2 = 0 and let M be a
finitely generated R-module with cx M = 1. Then <strong>the</strong> Betti numbers <strong>of</strong> M are eventually
constant.
Pro<strong>of</strong>. Let F be a finitely generated free R-module. We observe first that since m 2 = 0,
every submodule <strong>of</strong> mF is semisimple [3], so all <strong>the</strong> syzygies <strong>of</strong> M must be semisimple. Let
k denote <strong>the</strong> largest possible value <strong>of</strong> a Betti number <strong>of</strong> M and assume that it corresponds
to <strong>the</strong> i-th Betti number, that is β i (M) = k. This means that <strong>the</strong> i-th syzygy <strong>of</strong> M is a
direct sum <strong>of</strong> k simple modules, hence β i+1 (M) ≥ k. Our choice <strong>of</strong> k implies now that
β i+j (M) = k for all j ≥ 0 and <strong>the</strong> result follows.
□
Question 1 also has an affirmative answer in <strong>the</strong> case where R = (R, m, k) is a commutative
Gorenstein artinian ring with m 3 = 0, see [15]. Question 2 is also pretty much
unresolved. In <strong>the</strong> local commutative artinian case, Gasharov and Peeva have shown
([15]) that for a finitely generated module M, we have <strong>the</strong> following:
β i+1 (M) ≥ (2e − l(R) + h − 1)β i (M)
for large enough i. Here e = dim k m/m 2 , h is <strong>the</strong> Loewy length <strong>of</strong> R, and l(R) is <strong>the</strong>
length <strong>of</strong> R. They have also shown that if <strong>the</strong> constant 2e − l(R) + h − 1 ≥ 2, <strong>the</strong>n <strong>the</strong>
sequence <strong>of</strong> Betti numbers has exponential growth. However it is not hard to produce
examples <strong>of</strong> local commutative artinian rings where <strong>the</strong> constant 2e − l(R) + h − 1 is
a negative number. We also want to mention <strong>the</strong> following two results due to Ramras
[23, 24]:
Theorem 25. Let (R, m, k) be a regular local ring <strong>of</strong> dimension at least two, and let
S = R/m k for some k ≥ 2. Let M be a finitely generated non free S-module. Then, for
each i ≥ 1 we have βi+2(M) S > βi S (M).
□
and
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