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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
##### Keywords:
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