Poset resolutions and lattice-linear monomial ideals. (English) Zbl 1203.13016
The present paper studies minimal free resolutions of monomial ideals in the polynomial ring on \(n\) variables with coefficients on a field. The motivation of the techniques presented in this paper comes from the lcm-lattice construction of V. Gasharov, I. Peeva and V. Welker [Math. Res. Lett. 6, No. 5-6, 521–532 (1999; Zbl 0970.13004)]. The author introduces the class of lattice-linear monomial ideals, for which he gives an explicit construction of their minimal free resolution using the lcm-lattice. This new class of ideals contains some previously studied subclasses, including monomial ideals with a linear free resolution see J. Herzog and T. Hibi [Nagoya Math. J. 153, 141–153 (1999; Zbl 0930.13018)] and Scarf ideals, see D. Bayer, I. Peeva and B. Sturmfels [Math. Res. Lett. 5, No.1-2, 31–46 (1998; Zbl 0909.13010)]. The main tool used in the paper to produce lattice-linear resolutions is a construction due to A. Tchernev that takes a finite poset and produces a sequence of vector spaces and linear maps. This construction is used to produce poset resolutions of monomial ideals and in this framework minimal free resolutions of lattice-linear ideals are described. Other examples of poset resolutions (e.g. Taylor and Eliahou-Kervaire resolutions) are also given.
Reviewer: Eduardo Saenz-de-Cabezon (Logroño)
MSC:
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
Software:
posetsReferences:
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