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A cohomological study of local rings of embedding codepth 3. (English) Zbl 1259.13010

Let \(A\) be a commutative noetherian local ring with maximal ideal \(\mathfrak m\) and residue field \(k\). Let \(c\) be the embedding codepth of \(A\), that is, the difference between \(\mu (\mathfrak m)\) (the minimum number of generators of \(\mathfrak m\)) and \(d:=\mathrm{depth}(A)\). Write the completion of \(A\) as \(R/I\) where \(R\) is a regular local ring of dimension \(\mu (\mathfrak m)\).
Rings with \(c \leq2\) are well understood. The case \(c=3\) has been studied in some papers in the last forty years. If \(F\) is a minimal free resolution (of length 3) of \(R/I\) over \(R\), it is easy to see that \(F \otimes_Rk\) has a (graded commutative) \(k\)-algebra structure (in our case \(c=3\), even \(F\) has a commutative differential graded algebra structure). This algebra \(F \otimes_Rk\) is (up to isomorphism) an invariant of \(A\), and the possible isomorphism classes were determined in [J. Weyman, J. Algebra 126, No. 1, 1–33 (1989; Zbl 0705.13008)] and [L. L. Avramov, A. R. Kustin and M. Miller, J. Algebra 118, No. 1, 162–204 (1988; Zbl 0648.13008)] allowing us to classify these local rings \(A\).
The paper under review also focuses in this case \(c=3\). First, a deeper study of this classification is made, obtaining new restrictions for these algebra structures. Second, the Bass numbers \(\mu_A^j:=\dim_k\mathrm{Ext}_A^j(k,A)\) are studied for these rings. Their Bass series are computed and it is obtained the following result that answers (always in the case \(c=3\)) some open questions: there exists a real number \(\gamma_A >1\) such that \(\mu_A^{d+i} \geq \gamma_A \mu_A^{d+i-1}\) for all \(i \geq 1\) (with one exception for i=2, an explicitly described class of rings, and the obvious exception of Gorenstein rings for i=1).
Reviewer’s remark: Question 3.8 has been recently considered in [L. W. Christensen and O. Veliche, “Local rings of embedding codepth 3. Examples”, arXiv:1209.4256, to appear in Algebr. Represent. Theory], giving in particular a negative answer to Conjecture 3.10.

MSC:

13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series

Software:

TorAlgebra
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References:

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