Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plucker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.

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Combinatorial Commutative Algebra

Basic notions Cohomology Resolutions and derived functors Limits Gradings, filtrations, and Grobner bases Complexes from a sequence of ring elements Local cohomology Auslander-Buchsbaum formula and global dimension Depth and cohomological dimension Cohen-Macaulay rings Gorenstein rings Connections with sheaf cohomology Projective varieties The Hartshorne-Lichtenbaum vanishing theorem Connectedness Polyhedral applications $D$-modules Local duality revisited De Rham cohomology Local cohomology over semigroup rings The Frobenius endomorphism Curious examples Algorithmic aspects of local cohomology Holonomic rank and hypergeometric systems Injective moduels and Matlis duality Bibliography Index.

Twenty-Four Hours of Local Cohomology

Given a lattice polytope Q ⊆ ℝ
 n
 , we define an affine scheme that reflects the possibilities of splitting Q into a Minkowski sum. Denoting by Y the toric Gorenstein singularity induced by Q, we construct a flat family over with Y as special fiber. In case Y has an isolated singularity, this family is versal.

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The versal deformation of an isolated toric Gorenstein singularity

We give a criterion for an arbitrary ring generated by monomials to be Cohen-Macaulay in terms of certain numerical and topological properties of the additive semigroup generated by the exponents of the monomials. As a consequence, the Cohen-Macaulayness of such a ring is dependent upon the characteristic of the ground field. Introduction. Let N denote the set of nonnegative integers. By an affine semigroup we mean a finitely generated submonoid S of the additive monoid N n, where n is some positive integer. Let k[S] denote the semigroup ring of S over a field k. Then one can identify k[S] with the subring of a polynomial ring k[t l , ... , tn] generated by the monomials IX = lil ••• t:n, x = (Xl"'" Xn) E S. Obviously, every subring of k[t l , ... , In] generated by a finite set of monomials is the semigroup ring of the affine semigroup in N n generated by the exponents of the monomials. So one has, up to isomorphisms, a one-to-one correspondence between affine semigroups and affine varieties which are given parametrically by finite sets of monomials. When illustrating problems of algebraic geometry one almost inevitably tends to choose varieties of this type. Even when dealing with a quite general variety, either its singularities or a certain blowup may well be defined in local coordinates by monomials [15, 19]. Moreover, one can also use rings generated by monomials to study solutions of linear equations in nonnegative integers or, equivalently, invariants of a torus acting linearly on a polynomial ring [11, 26]. Therefore, a criterion for such a ring to be Cohen-Macaulay in terms of the associated semigroup would be very useful. It should be mentioned that the first example of a non-Coh~n­ Macaulay domain (in modem language), given by F. S. Macaulay at the beginning of this century [17, p. 98], was the ring k[/t, Ith, Il/~, Ii] and that analyzing this example, Grobner [7] already posed the problem of classifying rings generated by monomials of the same degree with respect to their Cohen-Macaulayness. The first step toward such a criterion was taken by Hochster [11], who succeeded in characterizing normal rings generated by monomials in terms of the associated semigroups and showed that they are always Cohen-Macaulay. Although this result was motivated by a conjecture on rings of invariants of reductive linear algebraic groups, which was later settled [12], its proof deserved much attention. It suggested Received by the editors July 22, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 13RI0; Secondary 14M05. 145 ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 146 N. V. TRUNG AND L. T. HOA the use of topological techniques in studying rings generated by monomials. The next step was a criterion given by Goto et al. for the case that the ring has a system of parameters consisting of monomials [5]. These results inspired many other works and were re-proved many times by different techniques, such as rational resolution [15], Hilbert functions of graded algebras [25], homology of polyhedral complexes [13,26], and the Hodge algebra [10]. In both steps, one used the coincidence of an affine semigroup S with one of its extensions to indicate the Cohen-Macaulayness of k[S]. Inspired by this phenomenon, Goto and Watanabe [6] defined a suitable extension Sf of S (see below) and claimed that Sf = S is a necessary and sufficient condition for k[S] to be CohenMacaulay. Let Z and Q denote the sets of integers and rational numbers, respectively. Consider the elements of S as points in the space Qn. Let G denote the additive group in zn generated by S and put r = rankzG. Let Cs denote the convex rational polyhedral cone spanned by S in Qn. Then Cs is r-dimensional. Suppose that Fl , ... , Fm are the (r I)-dimensional faces of Cs. Let Sj denote the set of elements x E G such that x + yES for some element yES n F;, i = 1, ... , m. Then they define Sf = n?_lSj' In this paper, we shall see that the condition Sf = S is not sufficient for the Cohen-Macaulayness of k[S], and that one has to add some topological condition on the convex cone Cs to get a correct criterion. To formulate this we need some more notation. Let [1, m] denote the set of the integers 1, ... , m. For every subset J of [1, m], set GJ = n Sj \ U Sj' i~J JEJ and let 'lTJ be the simplicial complex of nonempty subsets I of J with the property n j E IS n F; *" (0). Note that one calls 'lTJ acyclic if the reduced homology group Hi'ITJ ; k) vanishes for all q ~ O. MAIN THEOREM. Let S be an arbitrary affine semigroup. Then k[S] is a CohenMacaulay (resp. Gorenstein) ring if and only if the following conditions are satisfied: (i) Sf = S (resp. there exists an element x E G such that every element of G[l,m] is the difference of x by some element of S). (ii) For every nonempty proper subset J of [1, m], GJ = 0 or 'lTJ is acyclic. It will follow from some property of the Cousin complex of k[S] and from an explicit description of all local cohomology modules of k[Sf] in terms of GJ and 'lTJ • We will also give some simple methods for checking the above conditions. As a consequence, one immediately gets the abovementioned results of Hochster and Goto et al. Another application is a criterion for the Rees algebra (blowing-up) of a ring generated by monomials to be Cohen-Macaulay (resp. Gorenstein). In particular, as in the work of Reisner on polynomial rings modulo ideals generated by square-free monomials [20] where a similar link to topology is given, we will show that the Cohen-Macaulayness of k[S] is dependent upon the characteristic of the License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use AFFINE SEMIGROUPS AND COHEN-MACAULAY RINGS 147 field k. Moreover, using the main theorem, we have been able to solve Grabner's problem for some particular cases [31] (cf. [21, 29, and 30)). We would like to mention that Stanley [26] also obtained a similar result on modules associated with solutions of systems of linear diophantine equations, which overlaps with ours only in Hochster's normal case. This paper is organized in five sections. §1 gives a counterexample to the result of Goto and Watanabe via some consideration on the Rees algebras of affine semigroup rings. In §2, S' will be related to the Cousin complex of k[S] in order to show that S' = S if k[S] is a Cohen-Macaulay ring. §3 deals with the local cohomology modules of k[S']. Criteria for k[S] to be Cohen-Macaulay (resp. Gorenstein) are given in §4. There we will also deal with the Buchsbaumness of k[S]. The aim of §5 is to construct an affine semigroup ring whose local cohomology modules are just the reduced homology groups of a given finite simplicial complex. All notations introduced above will be used throughout. Moreover, if A and B are subsets of zn, G(A) denotes the additive group generated by A in zn, and A ± B is the set of elements a ± b with a E A and b E B. If x, y, ... are elements of zn, we will denote their components by Xi'Yi"'" i = 1, ... , n, respectively. For unexplained notations and standard facts in commutative algebra, algebraic topology, and local cohomology, we refer the reader to [18, 24, and 8]. ACKNOWLEDGMENT. The authors would like to thank S. Ikeda for pointing out that our earlier conclusion on the Cohen-Macaulayness of the Rees algebras of Cohen-Macaulay rings generated by monomials is false (see §1). This led us to check the result of [6, II]. Thanks are also due to S. Goto for encouraging our study, and to L. Robbiano for some useful suggestions. 1. Counterexamples to the result of Goto and Watanabe. Let S be an arbitrary affine semigroup in N n. Set S = {x E G; PX E S for some p > O}, S(i) = {x E S; Xi = A}, i = 1, ... ,n. Then we call S standard if the following conditions are satisfied: (1) S = G n N n , (2) S(i) =F S(j) for i =F j, (3) rankZG(S(i») = r 1, i = 1, ... , n. Geometrically, these conditions mean that Cs has exactly n (r 1 )-dimensional faces lying on the hyperplane Xi = 0, i = 1, ... , n. In this case, we may assume that S(i) = S n F; and Si = S S(i)' Goto and Watanabe [6, Theorem 3.3.3] claimed that if S is standard, then k[S] is a Cohen-Macaulay ring if and only if S' = S. We shall see that this is false. First, we have to remark that every affine semigroup can be transformed isomorphically onto a standard one by the following technique which is due to Hochster [11, p. 323]. HOCHSTER'S TRANSFORMATION. Let W denote the vector space generated by S in Qn. Then one can find m linear functionals 11"'" 1m from W to Q corresponding with Fl , ..• , Fm such that Cs = {x E W; li{x} ~ Oforalli}. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 148 N. V. TRUNG AND L. T. HOA See also [15, p. 6]. Let T denote the linear transformation which sends every element x E W to the element (ll(X), ... , Im(x» E Qm. By replacing Ii by a suitable positive integer multiple, one can assume that T(S) ~ N m. Hochster has shown that T(S) is isomorphic to S and that T(S) = G(T(S)) n N m. Obviously, T also induces an isomorphism between the semigroups S n F; and T(S){iJ' i = 1, ... , m. Since S n F; =1= S n Fj for i =1= j, and rankzG(S n F;) = r 1, we can conclude that T(S) is a standard affine semigroup in N m. Moreover, since T(S)i = T(S) T(S)(i) = T(S S n F;) = T(Si)' we also have T(S)' = T(S'). According to this transformation, one can omit the assumption on the standardness of S in the claim of Goto and Watanabe. The counterexample will also be constructed in the nonstandard case by using the following observation. Let A: Qn -+ Q be a linear functional such that A(S) ~ N, and, if xES and A(X) = 0, then x = O. Then we call the affine semigroup E>,.:= {(x, p) E N n+1 ; xES and A(X) > p} a blowing-up extension of S. The name stems from the 

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Affine semigroups and Cohen-Macaulay rings generated by monomials

Introduction. There is a well-known relationship between periodic continued fractions and 2-dimensional cusp singularities. (See, for instance, [10], [11].) Let π: U→V be the minimal resolution of a 2dimensional cusp singularity (V,p). Then the exceptional set X=π-1(p) is either a cycle of s rational curves with self-intersection numbers a1, a2, ..., as≦-2 at least one of which is strictly smaller than -2(s≧2), or a rational curve with a node and with a self-intersection number a<0. Then we can associate to it the periodic continued fraction

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Higher-dimensional analogues of periodic continued fractions and cusp singularities

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Torus embeddings and dualizing complexes

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An elliptic surface covered by Mumford's fake projective plane

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Completion of real fans and Zariski-Riemann spaces

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Torus embeddings and tangent complexes

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Masa Nori Ishida

Papers

Torus embeddings and dualizing complexes

An elliptic surface covered by Mumford's fake projective plane

Completion of real fans and Zariski-Riemann spaces

Torus embeddings and tangent complexes

Tsuchihashi's cusp singularities are Buchsbaum singularities