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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)].

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