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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
##### Keywords:
Koszul module; linearity defect; absolutely Koszul
Full Text:
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