Let k be a field, let R = k[x 1 , . . . , x m ] be a polynomial ring with the standard Z m -grading (multigrading), let L be a Noetherian multigraded R-module, and let E Φ − → G → L → 0 be a finite free multigraded presentation of L over R. Given a choice S of a multihomogeneous basis of E, we construct an explicit canonical finite free multigraded resolution T • (Φ, S) of the R-module L. In the case of monomial ideals our construction recovers the Taylor resolution. A main ingredient of our work is a new linear algebra construction of independent interest, which produces from a representation φ over k of a matroid M a canonical finite complex of finite dimensional k-vector spaces T • (φ) that is a resolution of Ker φ. We also show that the length of T • (φ) and the dimensions of its components are combinatorial invariants of the matroid M, and are independent of the representation map φ.
Abstract. Let Q = k[x 1 , . . . , xn] be a polynomial ring over a field k with the standard N n -grading. Let φ be a morphism of finite free N n -graded Qmodules. We translate to this setting several notions and constructions that appear originally in the context of monomial ideals. First, using a modification of the Buchsbaum-Rim complex, we construct a canonical complex T•(φ) of finite free N n -graded Q-modules that generalizes Taylor's resolution. This complex provides a free resolution for the cokernel M of φ when φ satisfies certain rank criteria. We also introduce the Scarf complex of φ, and a notion of "generic" morphism. Our main result is that the Scarf complex of φ is a minimal free resolution of M when φ is minimal and generic. Finally, we introduce the LCM-lattice for φ and establish its significance in determining the minimal resolution of M .
We introduce to the context of multigraded modules the methods of modules over categories from algebraic topology and homotopy theory. We develop the basic theory quite generally, with a view toward future applications to a wide class of graded modules over graded rings in [TV]. The main application in this paper is to study the Betti poset B = B(I, k) of a monomial ideal I in the polynomial ring R = k[x 1 , . . . , xm] over a field k, which consists of all degrees in Z m of the homogeneous basis elements of the free modules in the minimal free Z m -graded resolution of I over R. We show that the order simplicial complex of B supports a free resolution of I over R. We give a formula for the Betti numbers of I in terms of Betti numbers of open intervals of B, and we show that the isomorphism class of B completely determines the structure of the minimal free resolution of I, thus generalizing with new proofs results of Gasharov, Peeva, and Welker [GPW99]. We also characterize the finite posets that are Betti posets of a monomial ideal.
Abstract. Let I be an ideal in a Noetherian commutative ring R with unit, let k ≥ 2 be an integer, and let α k : S k I −→ I k be the canonical surjective R-module homomorphism from the kth symmetric power of I to the kth power of I. When pd R I ≤ 1 or when I is a perfect Gorenstein ideal of grade 3, we provide a necessary and sufficient condition for α k to be an isomorphism in terms of upper bounds for the minimal number of generators of the localisations of I. When I = m is a maximal ideal of R we show that α k is an isomorphism if and only if R m is a regular local ring. In all three cases for I our results yield that if α k is an isomorphism, then α t is also an isomorphism for each 1 ≤ t ≤ k.
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