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Multigraded regularity, $$a^{*}$$-invariant and the minimal free resolution. (English) Zbl 1142.13010
The notion of Castelnuovo–Mumford regularity of $$\mathbb{Z}$$-graded modules has been extended to multigraded modules over $${\mathbb Z}^k$$-graded rings in two ways: as the multigraded regularity (a set defined by the vanishing of multigraded pieces of local cohomology modules) by D. Maclagan and G. Smith [J. Reine Angew. Math. 571, 179–212 (2004; Zbl 1062.13004)] and as the resolution regularity vector, defined by the multi-degrees in a minimal free resolution of the module by J. Sidman and A. van Tuyl [Beitr. Algebra Geom. 47, No. 1, 67–87 (2006; Zbl 1095.13012)].
This paper investigates the relationship between these two notions and provides a lower and an upper bound containment for multigraded regularity in terms of the resolution regularity vector and other $$\mathbb{Z}$$-graded invariants. The notion of the $$a^{*}$$-invariant is generalized to multigraded modules in different ways: as a set and a vector and is used to examine the vanishing of multigraded pieces of local cohomology modules with respect to different multigraded ideals.

##### MSC:
 13D02 Syzygies, resolutions, complexes and commutative rings 13D45 Local cohomology and commutative rings
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