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Linearity defects of modules over commutative rings. (English) Zbl 1184.13039
Let \((R, \mathfrak{m}, k)\) be a local ring. Any complex \(F\) of finitely generated free \(R\)-modules with \(\partial(F)\subseteq \mathfrak{m}F\) has a natural \(\mathfrak{m}\)-adic filtration associated graded complex with respect to it is denoted \(\mathrm{lin}^RF\), and is called the linear part of \(F\). Let \(M\) be a finitely generated \(R\)-module, and let \(F\) be its minimal free resolution. The linearity defect of \(M\) is denoted by \(\mathrm{lk}_RM\) and it is defined as \[ \mathrm{ld}_RM=\sup\{i\in\mathbb{Z}: H_i(\mathrm{lin}^RF)\neq 0\}. \] A finitely generated \(R\)-module \(M\) is called Koszul if \(\mathrm{ld}_RM=0\). The ring \(R\) is called Koszul if \(k\) is a Koszul module. We say that \(R\) is absolutely Koszul if every finitely generated \(R\)-module has finite linearity defect; equivalently, has a Koszul syzygy module.
One of the main result in the paper under review is the following:
Let \(R\to S\) be a surjective homomorphism of local rings such that the projective dimension of the \(\mathrm{gr}_{\mathfrak{m}}R\)-module \(\mathrm{gr}_{\mathfrak{m}}S\) is finite. If \(S\) is absolutely Koszul, then so is \(R\). Moreover, in this case, one has an inequality \[ \mathrm{gl\,ld}R\leq\mathrm{gl\,ld}S+\mathrm{proj.dim}_RS. \]
The authors give different applications of this result.

MSC:
13D02 Syzygies, resolutions, complexes and commutative rings
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