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Minimal free resolutions and asymptotic behavior of multigraded regularity. (English) Zbl 1184.13038
The asymptotic behaviour of Castelnuovo-Mumford regularity of powers of ideals has been investigated intensively by many researchers recently. Usually one considers graded ideals (or modules) over the standard \(\mathbb{Z}\)-graded polynomial ring \(K[X_1,\dots,X_n]\) over a field \(K\).
In the the standard \(\mathbb{Z}^n\)-graded situation one must distinguish various types of regularity: the authors study the \(\mathbb{Z}^n\)-valued resolution regularity. Roughly speaking, it bounds the components of the multigraded shifts in the resolution. The main result is that this regularity is a linear function of \(k\) for modules \(I^kM\) where \(I\) is a multigraded ideal and \(M\) is a multigraded module. The main tools are a suitable variant of filter-regular sequences and Koszul homology.

MSC:
13D02 Syzygies, resolutions, complexes and commutative rings
13D45 Local cohomology and commutative rings
Software:
CoCoA; Macaulay2
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References:
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