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Multigraded regularity: syzygies and fat points. (English) Zbl 1095.13012
Castelnuovo-Mumford regularity is an important tool in the study of projective schemes and generally, from a more algebraic point of view, in studying finitely generated modules over graded rings. In this paper the authors consider a more general case, namely multigraded modules over $${\mathbb Z}^k$$-graded rings; geometrically those are of interest when studying subschemes of $${\mathbb P}^{n_1}\times ... \times {\mathbb P}^{n_k}$$. After giving a definition of regularity which is equivalent to the one in [D. MacLagan, G. G. Smith, J. Reine Angew. Math. 571, 179–212 (2004; Zbl 1062.13004)], and whose results are considered here in comparison with the results obtained in this paper, the authors use it to get information on the Hilbert function and ideal resolution of schemes of fat points.

##### MSC:
 13D02 Syzygies, resolutions, complexes and commutative rings 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 14F17 Vanishing theorems in algebraic geometry
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