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Cellular resolutions of monomial modules. (English) Zbl 0909.13011
Let $$k$$ be a field, $$S=k[X_1, \dots, X_n]$$ and $$T=k[X_1^{\pm 1}, \dots, X_n^{\pm 1}]$$. A monomial module is an $$S$$-submodule $$M$$ of $$T$$ generated by monomials. The authors are interested in the case when $$M$$ is co-Artinian, i.e., it is generated by the set $$\min(M)$$ of its minimal monomials. Of special interest are the two cases when $$\min(M)$$ is finite (in this case, $$M$$ is isomorphic to a monomial ideal in $$S)$$ and when $$\min(M)$$ is a group under multiplication (then $$M$$ is the lattice module $$M_L=S \{X^a\mid a\in L\}$$ associated to some lattice $$L\subset \mathbb{Z}^n$$ such that $$L\cap \mathbb{N}^n= \{0\}).$$ The authors associate to any co-Artinian monomial module $$M$$ a closed unbounded $$n$$-dimensional convex polyhedron $$P_t$$ in $$\mathbb{R}^n$$ with vertices $$(t^{a_1}, \dots, t^{a_n})$$, $$X_1^{a_1} \dots X_n^{a_n} \in\min (M)$$, where $$t$$ is a sufficiently large real number. This polyhedron was introduced by I. Bárány, R. Howe and H. E. Scarf [Math. Program. 66 A, No. 3, 273-281 (1994; Zbl 0822.90104)]. The bounded faces of $$P_t$$ form a regular cell complex hull$$(M)$$. Homogenizing the differentials of the oriented chain complex of hull$$(M)$$ one obtains a $$\mathbb{Z}^n$$-graded complex of free $$S$$-modules $$F_{\text{hull}(M)}$$ which is a (non-minimal) free resolution of $$M$$ over $$S$$. In the case of lattice modules, relating $$M_L$$ to the lattice ideal $$I_L=(X^a- X^b\mid a-b\in L) \subset S$$, the authors obtain a free resolution of $$S/I_L$$. The hull resolution generalizes results obtained by D. Bayer, I. Peeva and B. Sturmfels for generic monomial ideals [Math. Res. Lett. 5, No. 1-2, 31-46 (1998; see the preceding review)] and for generic lattice ideals [J. Am. Math. Soc. (to appear)].

##### MSC:
 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 13D25 Complexes (MSC2000) 55U15 Chain complexes in algebraic topology
##### Keywords:
minimal monomials
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