x n be a polynomial ring, and let ω S be its canonical module. First, we will define squarefreeness for n -graded S-modules. A Stanley-Reisner ring k = S/I , its syzygy module Syz i k , and Ext i S k ω S are always squarefree. This notion will simplify some standard arguments in the Stanley-Reisner ring theory. Next, we will prove that the i-linear strand of the minimal free resolution of a Stanley-Reisner ideal I ⊂ S has the "same information" as the module structure of Ext∨ is the Alexander dual of . In particular, if k has a linear resolution, we can describe its minimal free resolution using the module structure of the canonical module of k ∨ , which is Cohen-Macaulay in this case. We can also give a new interpretation of a result of Herzog and co-workers, which states that k is sequentially Cohen-Macaulay if and only if I ∨ is componentwise linear.

In this paper, we will study the local cohomology modules HiI(S) of a polynomial ring S = k[x1, …, xn] with supports in a (radical) monomial ideal I. When S/I is a Cohen–Macaulay ring of dimension d (more generally, if Extn−d(S/I, [wfr ]S) is Cohen–Macaulay), we can ‘visualize’ a ℤn-graded minimal injective resolution of Hn−dI(S) using Stanley–Reisner's simplicial complex of I.

Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions encoded by simplicial complexes. There are numerous equivalent ways to say that a monomial ideal is generic or cogeneric. For a generic monomial ideal, the associated primes satisfy a saturated chain condition, and the Cohen-Macaulay property implies shellability for both the Scarf complex and the Stanley-Reisner complex. Reverse lexicographic initial ideals of generic lattice ideals are generic. Cohen-Macaulayness for cogeneric ideals is characterized combinatorially; in the cogeneric case, the Cohen-Macaulay type is greater than or equal to the number of irreducible components. Methods of proof include Alexander duality and Stanley's theory of local h-vectors.c 2000 Academic Press Genericity of Monomial IdealsThis paper is a study of genericity properties of monomial ideals, initiated by . We will often use results from prior papers on this subject, although we have tried to make the exposition as self-contained as possible. The interested reader is encouraged to consult ), Miller (1998 and Peeva and Sturmfels (1998b) for the background. While the present paper is theoretical rather than algorithmic, we expect that our results on genericity will play a role for future implementations in Gröbner basis systems. Let M be a monomial ideal minimally generated by monomials m 1 , . . . , m r in a polynomial ring S = k[x 1 , . . . , x n ] over a field k. For a subset σ ⊆ {1, . . . , r}, we set m σ := lcm(m i |i ∈ σ), and let a σ := deg m σ ∈ N n denote the exponent vector of m σ .an n , we set deg xi (x a ) := a i , and we call supp(x a ) := {i|a i = 0} ⊆ {1, . . . , n} the support of x a . We say a monomial m ∈ S strictly divides m ∈ S, if m divides m and supp(m /m) = supp(m ). The above definition of genericity is more inclusive than the one given by , which we propose to call "strongly generic". That is, we call a monomial ideal M strongly generic if no two distinct minimal generators m i and m j have the same positive

No abstract

A squarefree module over a polynomial ring S = k[x 1 , . . . , x n ] is a generalization of a Stanley-Reisner ring, and allows us to apply homological methods to the study of monomial ideals more systematically.The category Sq of squarefree modules is equivalent to the category of finitely generated left Λ-modules, where Λ is the incidence algebra of the Boolean lattice 2 {1,...,n} . The derived category D b (Sq) has two duality functors D and A. The functor D is a common one with H i (D(M • )) = Ext n+i S (M • , ω S ), while the Alexander duality functor A is rather combinatorial. We have a strange relation. This equivalence corresponds to the Koszul duality for Λ, which is a Koszul algebra with Λ ! ∼ = Λ. Our D and A are also related to the Bernstein-Gel'fand-Gel'fand correspondence.The local cohomology H i I∆ (S) at a Stanley-Reisner ideal I ∆ can be constructed from the squarefree module Ext i S (S/I ∆ , ω S ). We see that Hochster's formula on the Z n -graded Hilbert function of H i m (S/I ∆ ) is also a formula on the characteristic cycle of H n−i I∆ (S) as a module over the Weyl algebra A = k x 1 , . . . , x n , ∂ 1 , . . . , ∂ n (if char(k) = 0). 1991 Mathematics Subject Classification. Primary 13D25; Secondary 13D02, 13D45, 13F55, 13N10, 18E30.a natural way, see Proposition 3.2. On the other hand, extending an idea of Eagon-Reiner [8], Miller [15] and Römer [20] constructed the Alexander duality functor A on Sq.Since A is exact, we can regard it as a duality functor on D b (Sq).Using D b (Sq), we can get simple and systematic proofs of many results in [15,20,21,24,25]. Moreover, we prove a strange natural equivalencewhere T is the translation functor on D b (Sq).Let E = S * 1 be the exterior algebra. A squarefree module over E, which was defined by Römer [20], is also a natural concept. The category Sq E of squarefree E-modules is equivalent to Sq S in a natural way. A famous theorem of Bernstein-Gel'fand-Gel'fand [4] states that the bounded derived category of finitely generated Z-graded S-modules is equivalent to the bounded derived category of finitely generated Z-graded left E-modules. The functors defining this equivalence preserve the squarefreeness, and coincide with A • D and D • A in the squarefree case under the equivalence Sq S ∼ = Sq E . We have another relation to Koszul duality. The incidence algebra Λ of 2 {1,...,n} is a Koszul algebra whose quadratic dual Λ ! is isomorphic to Λ itself. The functors A • D and D • A give a non-trivial autoequivalence of D b (Sq). This equivalence corresponds to the Koszul dualityIn the last section, under the assumption that char(k) = 0, we study modules over the Weyl algebra k x 1 , . . . , x n , ∂ 1 , . . . , ∂ n associated to squarefree modules (e.g., the local cohomology module H i I ∆ (S)). Especially, we give the formula for their characteristic cycles.After I received the referee's report for the first version, I widely revised the paper. Among other things, Proposition 4.6 is a new result of the second version which was submitted in September 2001. ...