Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.jalgebra.2021.04.022

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DOI： 10.1216/jca.2021.13.589

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DOI： 10.1080/00927872.2020.1726940

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DOI： 10.7146/math.scand.a-114906

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DOI： 10.1017/nmj.2017.10

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Academic Press Inc.  

An affine oriented matroid M is a combinatorial abstraction of an affine hyperplane arrangement. From M, Novik, Postnikov and Sturmfels [11] constructed a squarefree monomial ideal OM in a polynomial ring S˜ and got beautiful results. Developing their theory, we will show the following. (1) If S˜/OM is Cohen–Macaulay, then the bounded complex BM (a regular CW complex associated with M) is a contractible homology manifold with boundary. This is closely related to Dong's theorem ([5]), which used to be Zaslavsky's conjecture.(2) We give a characterization of M such that S˜/OM is Cohen–Macaulay, which states that the converse of [11, Corollary 2.6] is essentially true.

DOI： 10.1016/j.jcta.2018.01.004

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

Let k be an algebraically closed field of characteristic 0, and A = circle plus(i is an element of N) A(i) Cohen-Macaulay graded domain with A(0) = k. If A is semi-standard graded (i.e., A is finitely generated as a k[A(1)]-module), it has the h-vector (h(0), h(1),...,h(s)), which encodes the Hilbert function of A. From now on, assume that s = 2. It is known that if A is standard graded (i.e., A = k[A1]), then A is level. We will show that, in the semi-standard case, if A is not level, then h1 + 1 divides h2. Conversely, for any positive integers h and n, there is a non-level A with the h-vector (1, h, (h + 1)n). Moreover, such examples can be constructed as Ehrhart rings (equivalently, normal toric rings). (C) 2017 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.jpaa.2017.03.011

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TAYLOR & FRANCIS INC  

In this short note, we give a characterization of domains satisfying Serre's condition (R-1) in terms of their canonical modules. In the special case of toric rings, this generalizes a result of the second author [9] where the normality is described in terms of the shape of the canonical module.

DOI： 10.1080/00927872.2016.1172611

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

We exhibit a global bound for the Lyubeznik numbers of a ring of prime characteristic. In addition, we show that for a monomial ideal, the Lyubeznik numbers of the quotient rings of its radical and its polarization are the same. Furthermore, we present examples that show striking behavior of the Lyubeznik numbers under localization. We also show related results for generalizations of the Lyubeznik numbers. (C) 2015 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.jpaa.2015.03.010

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

We characterize the seminormality of an affine semigroup ring in terms of the dualizing complex, and the normality of a Cohen-Macaulay semigroup ring by the "shape" of the canonical module. We also characterize the seminormality of a toric face ring in terms of the dualizing complex. A toric face ring is a simultaneous generalization of Stanley-Reisner rings and affine semigroups. (C) 2014 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jalgebra.2014.11.013

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER-VERLAG ITALIA SRL  

We prove that the Eliahou-Kervaire resolution of a Cohen-Macaulay stable monomial is supported by a regular CW complex whose underlying space is a closed ball. We also show that the modified Eliahou-Kervaire resolutions of variants of a Borel fixed ideal (e.g., a squarefree strongly stable ideal) are supported by regular CW complexes, and their underlying spaces are closed balls in the Cohen-Macaulay case.

DOI： 10.1007/s13348-014-0104-0

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

In this paper, we introduce a new algebraic concept, which we call squarefree P-modules. This concept is inspired from Karu's proof of the non-negativity of the cd-indices of Gorenstein* posets, and supplies a way to study cd-indices from the viewpoint of commutative algebra. Indeed, by using the theory of squarefree P-modules, we give several new algebraic and combinatorial results on CW-posets. First, we define an analogue of the cd-index for any CW-poset and prove its non-negativity when a CW-poset is Cohen-Macaulay. This result proves that the h-vector of the barycentric subdivision of a Cohen Macaulay regular CW-complex is unimodal. Second, we prove that the Stanley-Reisner ring of the barycentric subdivision of an odd dimensional Cohen Macaulay polyhedral complex has the weak Lefschetz property. Third, we obtain sharp upper bounds of the cd-indices of Gorenstein* posets for a fixed rank generating function. (C) 2014 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.aim.2014.07.037

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Language：English   Publisher：Academic Press Inc.  

We give a purely combinatorial characterization of complete Stanley-Reisner rings having a principally generated (equivalently, finitely generated) Cartier algebra. © 2013 Elsevier Inc.

DOI： 10.1016/j.jalgebra.2013.09.006

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

We construct an Eliahou-Kervaire-like minimal free resolution of the alternative polarization of a Borel fixed ideal I. It yields new descriptions of the minimal free resolutions of I itself and I (sq) , where (-) (sq) is the squarefree operation in the shifting theory. These resolutions are cellular, and the (common) supporting cell complex is given by discrete Morse theory. If I is generated in one degree, our description is equivalent to that of Nagel and Reiner.

DOI： 10.1007/s10801-012-0409-6

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：DUKE UNIV PRESS  

For a monomial ideal I of a polynomial ring S, a polarization of 1 is a square-free monomial ideal J of a larger polynomial ring (S) over tilde such that S/I is a quotient of (S) over tilde /J by a (linear) regular sequence. We show that a Borel fixed ideal admits a nonstandard polarization. For example, while the usual polarization sends xy(2) is an element of S to x(1)y(1)y(2) is an element of (S) over tilde, ours sends it to x(1)y(2)y(3). Using this idea, we recover/refine the results on square-free operation in the shifting theory of simplicial complexes. The present paper generalizes a result of Nagel and Reiner, although our approach is very different.

DOI： 10.1017/S0027763000022315

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TAYLOR & FRANCIS INC  

We define sliding functors, which are exact endofunctors of the category of multigraded modules over a polynomial ring. They preserve several invariants of modules, especially the (usual) depth and Stanley depth. In a similar way, we can also define the polarization functor. While this idea has appeared in papers of Bruns-Herzog and Sbarra, we give slightly different approach. Keeping these functors in mind, we treat simplicial spheres of Bier-Murai type.

DOI： 10.1080/00927872.2010.547540

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

Recently, G. Floystad studied higher Cohen-Macaulay property of certain finite regular cell complexes. In this paper, we partially extend his results to squarefree modules, toric face rings, and simplicial posets. For example, we show that if (the corresponding cell complex of) a simplicial poset is l-Cohen-Macaulay, then its codimension one skeleton is (l+ 1)-Cohen-Macaulay.

DOI： 10.1090/S0002-9939-2011-10734-2

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

A finite poset P is called simplicial if it has the smallest element (0) over cap, and every interval [(0) over cap, x] is a Boolean algebra. The face poset of a simplicial complex is a typical example. Generalizing the Stanley-Reisner ring of a simplicial complex, Stanley assigned the graded ring A(P) to P. This ring has been studied from both combinatorial and topological perspectives. In this paper, we will give a concise description of a dualizing complex of A(P), which has many applications. (C) 2011 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.jpaa.2011.02.009

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

We apply Miller's theory on multigraded modules over a polynomial ring to the study of the Stanley depth of these modules. Several tools for Stanley's conjecture are developed, and a few partial answers are given. For example, we show that taking the Alexander duality twice (but with different "centers") is useful for this subject. Generalizing a result of Apel, we prove that Stanley's conjecture holds for the quotient by a cogeneric monomial ideal. (C) 2011 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jalgebra.2011.05.028

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：DUKE UNIV PRESS  

A toric face ring, which generalizes both Stanley-Reisner rings and affine semigroup rings, is studied by Bruns, Romer and their coauthors recently. In this paper, under the "normality" assumption, we describe a dualizing complex of a toric face ring R in a very concise way. Since R. is not a graded ring in general, the proof is not straightforward. We also develop the square-free module theory over R, and show that; the Cohen-Macaulay, Buchsbaum, and Gorenstem* properties of R are topological properties of its associated cell complex.

DOI： 10.1017/S0027763000009806

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MATEMATISK INST  

Let A = circle plus(i is an element of N) A(i) bea Koszul algebra over a field K = A(0),and *mod A the category of finitely generated graded left A-modules. The linearity defect Id(A) (M) of M is an element of *mod A is an invariant defined by Herzog and Iyengar. An exterior algebra E is a Koszul algebra which is the Koszul dual of a polynomial ring. Eisenbud et al. showed that Id(E)(M) < infinity tor all M is an element of *mod E. Improving this, we show that the Koszul dual A(!) of a Koszul commutative algebra A satisfies the following.
LLet M is an element of *mod A(!). If {dim(K) M(i) [i is an element of Z} is bounded, then Id(A!)(M) < infinity.
If A is complete intersection, then reg(A!) (M) < infinity and Id(A!) (M) < infinity for all M is an element of *mod A(!).
If E =Lambda(y(1),..., y(n)) is an exterior algebra, then Id(E)(M) < e(n!)2((n-1)!) for M is an element of *mod E with c := max{dim(K) M(i) | i is an element of Z}.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TAYLOR & FRANCIS INC  

Let C subset of N-d be an affine semigroup, and R=K[C] its semigroup ring. This article is a collection of various results on C-graded R-modules M = circle plus(c is an element of C) M-c, especially, monomial ideals of R. For example, we show the following: If R is normal and I subset of R is a radical monomial ideal (i.e., R/I is a generalization of Stanley-Reisner rings), then the sequentially Cohen-Macaulay property of R/I is a topological property of the geometric realization of the cell complex associated with I. Moreover, we can give a squarefree modules/constructible sheaves version of this result. We also show that if R is normal and I subset of R is a Cohen-Macaulay monomial ideal, then root I is Cohen-Macaulay again.

DOI： 10.1080/00927870802104295

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

Let S = K [x(1), . . .(,) x(n)] be a polynomial ring over a field K, and E = boolean AND < y(1), . . ., y(n)> an exterior algebra. The linearity defect Id(E)(N) of a finitely generated graded E-module N measures how far N departs from "componentwise linear". It is known that Id(E)(N) < infinity for all N. But the value can be arbitrary large, while the similar invariant Id(S)(M) for an S-module M is always at most n. We will show that if I-Delta (resp. J(Delta)) is the squarefree monomial ideal of S (resp. E) corresponding to a simplicial complex Delta subset of 2({1, . . .,n}), then Id(E)(E/J(Delta)) = Id(S)(S/I-Delta). Moreover, except some extremal cases, Id(E)(E/J(Delta)) is a topological invariant of the geometric realization vertical bar Delta(boolean OR)vertical bar of the Alexander dual Delta(boolean OR) of Delta. We also show that, when n >= 4, Id(E)(E/J(Delta)) = n - 2 (this is the largest possible value) if and only if Delta is an n-gon. (c) 2007 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jalgebra.2007.02.049

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

Let E = K[y(1),..., y(n)] be the exterior algebra. The (cohomological) distinguished pairs of a graded E-module N describe the growth of a minimal graded injective resolution of N. Romer gave a duality theorem between the distinguished pairs of N and those of its dual N*. In this paper, we show that under Bernstein-Gel'fand-Gel'fand correspondence, his theorem is translated into a natural corollary of local duality for ( complexes of) graded S = K[x(1),..., x(n)]-modules. Using this idea, we also give a Z(n)-graded version of Romer's theorem.

DOI： 10.1007/s10468-006-9037-y

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

Let A be a noetherian AS-regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and gr A the category of finitely generated graded left A-modules. Following Jorgensen, we define the Castelnuovo-Mumford regularity reg(M-center dot) of a complex M-center dot epsilon D-b (gr A) in terms of the local cohomologies or the minimal projective resolution of MO. Let A! be the quadratic dual ring of A. For the Koszul duality functor G : D-b(gr A) -> D-b(gr A(!)), we have reg(M-center dot) = max{i vertical bar H-i(G(M-center dot)) not equal 0}. Using these concepts, we interpret results of Martinez-Villa and Zacharia concerning weakly Koszul modules (also called componentwise linear modules) over A. As an application, refining a result of Herzog and Romer, we show that if J is a monomial ideal of an exterior algebra E = Lambda < y(1),..., y(d)>, d >= 3, then the (d - 2)nd syzygy of E/J is weakly Koszul. (c) 2005 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.jpaa.2005.09.014

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Grant-in-Aid for Encouragement of Young Scientists

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MATH SOC JAPAN  

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 A-modules, where A 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(S)(n+i)(M-., omega(S)), while the Alexander duality functor A is rather combinatorial. We have a strange relation D o A o D o A o D o A congruent to T-2n, where T is the translation functor. The functors A o D and D o A give a non-trivial autoequivalence of D-b(Sq). This equivalence corresponds to the Koszul duality for Lambda, which is a Koszul algebra with Lambda(! congruent to) Lambda. Our D and A are also related to the Bemstein-Gel'fand-Gel'fand correspondence.
The local cohomology H-1Delta(i)(S) at a Stanley-Reisner ideal I-Delta can be constructed from the squarefree module Ext(S)(i)(S/I-Delta,omega(S)). We see that Hochster's formula on the Z(n)-graded Hilbert function of H-m(i)(S/I-Delta) is also a formula on the characteristic cycle of H-IDelta(n-1)(S) as a module over the Weyl algebra A = k<x(1),...,x(n), partial derivative(1),...,partial derivative(n)> (if char(k) = 0).

DOI： 10.2969/jmsj/1191418707

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：INT PRESS BOSTON  

A few years ago, I defined a squarefree module over a polynomial ring S = k[x(1),...,x(n)] generalizing the Stanley-Reisner ring k[Delta] = S/I-Delta of a simplicial complex A subset of 2({1,...,n}). This notion is very useful in the Stanley-Reisner ring theory. In this paper, from a squarefree S-module M, we construct the k-sheaf M+ on an (n - 1) simplex B which is the geometric realization of 2({1,...,n}). For example, k[Delta](+) is (the direct image to B of) the constant sheaf on the geometric realization \Delta\ subset of B. We have H-i(B, M+) congruent to [H-m(i+1)(M)](0) for all i greater than or equal to 1. The Poincare-Verdier duality for sheaves M+ on B corresponds to the local duality for squarefree modules over S. For example, if \Delta\ is a manifold, then k[Delta] is a Buchsbaum ring and its canonical module K-k[Delta] is a squarefree module which gives the orientation sheaf of I A I with the coefficients in k.

DOI： 10.4310/MRL.2003.v10.n5.a7

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC  

We study the local cohomology modules H-l Sigma(i)(k[Delta]) of the Stanley-Reisner ring k[A] of a simplicial complex A with support in the ideal I-Sigma subset of k[Delta] corresponding to a subcomplex Sigma subset of Delta. We give a combinatorial topological formula for the multigraded Hilbert series, and in the case where the ambient complex is Gorenstein. compare this with a second combinatorial formula that generalizes results of Mustata and Terai. The agreement between these two formulae is seen to be a disguised form of Alexander duality. Other results include a comparison of the local cohomology with certain Ext modules, results about when it is concentrated in a single homological degree, and combinatorial topological interpretations of some vanishing theorems. (C) 2001 Academic Press.

DOI： 10.1006/jabr.2001.8932

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：CAMBRIDGE UNIV PRESS  

In this paper, we will study the local cohomology modules H-I(i)(S) of a polynomial ring S = k[x(1),..., x(n)] with supports in a (radical) monomial ideal I. When S/I is a Cohen-Macaulay ring of dimension d (more generally, if Ext(s)(n-d)(S/I, omega (s)) is Cohen- Macaulay), we can 'visualize' a Z(n)-graded minimal injective resolution of H-I(n-d)(S) using Stanley-Reisner's simplicial complex of I.

DOI： 10.1017/S030500410100514X

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

Recently, the local cohomology module H-1' (S) of a polynomial ring S with supports in a monomial ideal I has been studied by several authors. In the present paper, we will extend these results to a normal Gorenstein semigroup ring R =k[x(c)/c is an element of C] of C subset of Z(d). More precisely, we will study the local cohomology modules H-I(i)(R) with supports in monomial ideals I, and their injective resolutions. Roughly speaking, we will see that they only depend on the combinatorial properties of the face lattice of a polytope associated to R. Hence, if R is simplicial, it behaves just like a polynomial ring in our context. For example, the Bass numbers of H-I (i)(R) are always finite in the simplicial case. If R is not simplicial, this is not true as a famous example of Hartshorne shows. (C) 2001 Elsevier Science B.V. All rights reserved.

DOI： 10.1016/S0022-4049(00)00095-5

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS LTD  

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.

DOI： 10.1006/jsco.1999.0290

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC  

Let S = k[x(1),..., x(n)] be a polynomial ring, and let omega(s) be its canonical module. First, we will define squarefreeness for N-n-graded S-modules. A Stanley-Reisner ring k[Delta] = S/I-Delta, its syzygy module Syz(i)(k[Delta]), and Ex(s)(i)(k[Delta], omega(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-Delta subset of S has the "same information" as the module structure of Ext(s)(i)(k[Delta(v)], omega(s)), where Delta(v) is the Alexander dual of Delta. In particular, if k[Delta] has a linear resolution, we can describe its minimal free resolution using the module structure of the canonical module of k[Delta(v)], 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[Delta] is sequentially Cohen-Macaulay if and only if I(Delta)v is componentwise linear. (C) 2000 Academic Press.

DOI： 10.1006/jabr.1999.8130

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

Let M = (m(1), ..., m(r)) be a monomial ideal of S = k[x(1), ..., x(n)]. Bayer-Peeva-Sturmfels studied a subcomplex F-Delta of the Taylor resolution, defined by a simplicial complex Delta subset of 2(r). They proved that if M is generic (i.e., no variable x(i) appears with the same non-zero exponent in two distinct monomials which are minimal generators), then F-Delta M is the minimal free resolution of S/M, where Delta(M) is the Scarf complex of M.
In this paper, we prove the following: for a generic (in the above sense) monomial ideal M and each integer depth S/M less than or equal to i < dim S/M, there is an embedded prime P is an element of Ass(S/M) of dim S/P = i. Thus a generic monomial ideal with no embedded primes is Cohen-Macaulay (in this case, Delta(M) is shellable). We also study a non-generic monomial ideal M whose minimal free resolution is F Delta for some Delta. In particular, we prove that if all associated primes of M have the same height, then M is Cohen-Macaulay and Delta is pure and strongly connected.

DOI： 10.1090/S0002-9939-99-04947-3

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Language：English   Publishing type：Research paper (international conference proceedings)   Publisher：MARCEL DEKKER  

Grant-in-Aid for Encouragement of Young Scientists

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC JNL-COMP SUBSCRIPTIONS  

DOI： 10.1006/jabr.1996.6952

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer New York  

DOI： 10.1007/PL00004331

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

We classify a reduced, irreducible and non-degenerate curve C subset of P-r such that its general hyperplane section C boolean AND H is arithmetically Gorenstein, but C itself is not. These curves are contained in surface scrolls and are closely related to Castelnuovo theory on curves in projective space.

DOI： 10.1090/S0002-9939-96-03163-2

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

This paper gives a new postulation of the Hilbert function of a Cohen-Macaulay homogeneous domain.
If A is a Cohen-Macaulay homogeneous algebra over a field k, there are positive integers h(0), h1,...,h(s) satisfying Sigma(i greater than or equal to 0) dim(k) A(i) lambda(i) = (h(0) + h(1) lambda + ... + h(s) lambda(s))/(1 - lambda)(d), where d is the Krull dimension of A. We call the vector (h(0), h(1),...,h(s)) the h-vector of A.
Let A be a Cohen-Macaulay homogeneous domain over C with the h-vector (h(0), h(1),...,h(s)). It is well known that h(i) greater than or equal to h(1) for all 2 less than or equal to i less than or equal to s - 1. We will show that if the equality holds for some 2 less than or equal to i less than or equal to s - 2 then h(1) = h(2) = ... = h(s-1) and h(s) less than or equal to h(1) (when h(s) greater than or equal to 2, the condition h(s-1) = h(1) also implies the same assertion). To prove this result, we will modify Castelnuovo's argument in his study on curves of maximal genus.

DOI： 10.1016/0022-4049(94)00139-1

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC JNL-COMP SUBSCRIPTIONS  

DOI： 10.1006/jabr.1994.1346

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Venue：倉敷シーサイドホテル  

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Venue：静岡大学  

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Venue：東京大学  

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Venue：University of Hawaii at Manoa  

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Venue：東京工業大学  

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Venue：岡山大学  

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Venue：Okayama University of Science  

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Venue：信州大学  

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Venue：Research Institute for Mathematical Sciences, Kyoto University  

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Venue：International productivity center, Shonan international village  

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Venue：関西大学  

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Venue：京都産業大学  

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Venue：岡山大学  

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Venue：International productivity center, Shonan international village  

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Venue：岡山大学  

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Venue：岡山大学  

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Venue：宮島コーラルホテル  

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Venue：学習院大学  

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Venue：下関市生涯学習プラザ  

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Venue：International Productivity Center (Hayama, Kanagawa )  

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Venue：Kyushu University  

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Venue：Research Institute for Mathematical Sciences (Kyoto University)  

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Venue：Quy Nhon University (Vietnam)  

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Venue：Research Institute for Mathematical Sciences, Kyoto University  

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Venue：International productivity center,Shonan international village  

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Venue：Hanoi Institute of Mathematics  

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Venue：Saga University  

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Venue：The University of New South Wales  

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