We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of... more We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of the simplicial affine semigroups. In this way, we characterize the Cohen-Macaulay type of the simplicial affine semigroup ring K[S]. We define the type of S, type(S), in terms of some Apéry sets of S and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when K[S] is Cohen-Macaulay. If K[S] is a d-dimensional Cohen-Macaulay ring of embedding dimension at most d + 2, then type(S) ≤ 2. Otherwise, type(S) might be arbitrary large and it has no upper bound in terms of the embedding dimension. Finally, we present a generating set for the conductor of S as an ideal of its normalization.
Numerical semigroup rings are investigated from the relative viewpoint. It is known that algebrai... more Numerical semigroup rings are investigated from the relative viewpoint. It is known that algebraic properties such as singularities of a numerical semigroup ring are properties of a flat numerical semigroup algebra. In this paper, we show that arithmetic and set-theoretic properties of a numerical semigroup ring are properties of an equi-gcd numerical semigroup algebra.
Let H ⊂ N d be a normal simplicial affine semigroup, R = K[H] its semigroup ring over the field K... more Let H ⊂ N d be a normal simplicial affine semigroup, R = K[H] its semigroup ring over the field K and ω R its canonical module, which is identified with an ideal in R. The Ulrich elements for H are those h in H such that for the multiplication map by x h from R into ω R , the cokernel is an Ulrich module. We say that the ring R is almost Gorenstein if Ulrich elements exist in H. We provide algebraic and combinatorial criteria to test the Ulrich property for arbitrary elements in H. In particular, unless R is a regular ring, we show that Ulrich elements are located among the exponents of the minimal monomial generators of ω R. Assume d = 2 and a 1 , a 2 are the vectors with coprime integer entries on the extremal rays of the cone C over H. Then any b in the Hilbert basis B H , different from a 1 , a 2 , is an Ulrich element in H if and only if c 1 + c 2 ∈ (a 1 + H) ∪ (a 2 + H) ∪ (b + H) for all c 1 , c 2 ∈ B H. For the bottom element of H, which is defined as the smallest among the exponents of the monomials in ω R ordered componentwise, we have a more direct description of when it has the Ulrich property. Consequently, when H has all nonzero elements with both entries positive and (1, 1) is in the relative interior of the cone C then we obtain a simple arithmetic criterion for the almost Gorenstein property of R in terms of the entries of a 1 and a 2 alone.
Given a numerical semigroup ring R = k[[S]], an ideal E of S and an odd element b ∈ S, the numeri... more Given a numerical semigroup ring R = k[[S]], an ideal E of S and an odd element b ∈ S, the numerical duplication S ✶ b E is a numerical semigroup, whose associated ring k[[S ✶ b E]] shares many properties with the Nagata's idealization and the amalgamated duplication of R along the monomial ideal I = (t e | e ∈ E). In this paper we study the associated graded ring of the numerical duplication characterizing when it is Cohen-Macaulay, Gorenstein or complete intersection. We also study when it is a homogeneous numerical semigroup, a property that is related to the fact that a ring has the same Betti numbers of its associated graded ring. On the way we also characterize when gr m (I) is Cohen-Macaulay and when gr m (ωR) is a canonical module of gr m (R) in terms of numerical semigroup's properties, where ωR is a canonical module of R.
We introduce the concept of homogeneous numerical semigroups and show that all homogeneous numeri... more We introduce the concept of homogeneous numerical semigroups and show that all homogeneous numerical semigroups with Cohen-Macaulay tangent cones are of homogeneous type. In embedding dimension three, we classify all numerical semigroups of homogeneous type into numerical semigroups with complete intersection tangent cones and the homogeneous ones which are not symmetric with Cohen-Macaulay tangent cones. We also study the behavior of the homogeneous property by gluing and shiftings to construct large families of homogeneous numerical semigroups with Cohen-Macaulay tangent cones. In particular we show that these properties fulfill asymptotically in the shifting classes. Several explicit examples are provided along the paper to illustrate the property. 1. introduction Let S be a numerical semigroup minimally generated by a sequence of positive integers n : 0 < n 1 < • • • < n d. For any nonnegative integer j one may consider the shifted sequence n + j : 0 < n 1 + j < • • • < n d + j. Let k be a field and k[S] := k[t n 1 ,. .. , t n d ] ⊂ k[t], where t is a free variable, be the numerical semigroup ring defined by S. This is the homogeneous coordinate ring of the affine monomial curve in A d k defined parametrically by x 1 = t n 1 ,. .. , x d = t n d. Denote by I(n) ⊂ k[x 1 ,. .. , x d ] the defining ideal of k[S] obtained from the natural presentation k[x 1 ,. .. , x d ] → k[S] → 0. J. Herzog and H. Srinivasan conjectured that for j ≫ 0 the Betti numbers of the ideals I(n + j) become periodic on j with period n d − n 1. In 2013, the conjecture was proven to be true by A. V. Jayanthan and H. Srinivasan for d = 3 [17], by A. Marzullo for some particular cases if d = 4 [20], and by P. Gimenez, I. Sengupta and H. Srinivasan in the case of arithmetic sequences [10]. Finally, in 2014, T. Vu gave a completely general positive answer in [35]. One of the main ingredients of Vu's proof is that there exists a positive integer N such that, for all j > N, any minimal binomial non-homogeneous generator of I(n+j) is of the form x a 1 1 u−vx a d d , where a 1 , a d are positive integers, u, v are monomials in the variables x 2 ,. .. , x d−1 , and deg x a 1 1 u > deg vx a d d. It is noteworthy that the bound N depends on the Castelnuovo-Mumford regularity of J(n), the ideal generated by the homogeneous elements in I(n). Let I(n) * be the homogeneous ideal generated by the initial forms of the elements in I(n). Then, k[x 1 ,. .. , x d ]/I(n) * ≃ G(S), the tangent cone of k[S]. By using this main ingredient in Vu's proof of the conjecture, J. Herzog and D. I. Stamate proved in [15] that for any j > N, the Betti numbers
Let A be a K-subalgebra of the polynomial ring S = K [x 1 ,. .. , x d ] of dimension d, generated... more Let A be a K-subalgebra of the polynomial ring S = K [x 1 ,. .. , x d ] of dimension d, generated by finitely many monomials of degree r. Then, the Gauss algebra G(A) of A is generated by monomials of degree (r − 1)d in S. We describe the generators and the structure of G(A), when A is a Borel fixed algebra, a squarefree Veronese algebra, generated in degree 2, or the edge ring of a bipartite graph with at least one loop. For a bipartite graph G with one loop, the embedding dimension of G(A) is bounded by the complexity of the graph G.
We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The comp... more We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The complete intersection property of numerical semigroup algebras is investigated using factorizations of monomials into minimal ones. The goal is to study whether a flat rectangular algebra is a complete intersection. Along this direction, special types of algebras generated by few monomials are worked out in detail.
We introduce the concept of monomial ideals with stable projective dimension, as a generalization... more We introduce the concept of monomial ideals with stable projective dimension, as a generalization of the Cohen–Macaulay property. Indeed, we study the class of monomial ideals [Formula: see text], whose projective dimension is stable under monomial localizations at monomial prime ideals [Formula: see text], with [Formula: see text]. We study the relations between this property and other sorts of Cohen–Macaulayness. Finally, we characterize some classes of polymatroidal ideals with stable projective dimension.
Algebraic and combinatorial properties of a monomial ideal are studied in terms of its associated... more Algebraic and combinatorial properties of a monomial ideal are studied in terms of its associated radical ideals. In particular, we present some applications to the symbolic powers of square-free monomial ideals.
The class of equidimensional polymatroidal ideals is studied. In particular, we show that an unmi... more The class of equidimensional polymatroidal ideals is studied. In particular, we show that an unmixed polymatroidal ideal is connected in codimension one if and only if it is Cohen-Macaulay. Especially a matroidal ideal is connected in codimension one precisely when it is a squarefree Veronese ideal. As a consequence we indicate that for polymatroidal ideals, Serre's condition (S n) for some n ≥ 2 is equivalent to Cohen-Macaulay property. We also give a classification of generalized Cohen-Macaulay polymatroidal ideals.
In this thesis, the class of modules whose Cousin complexes have finitely generated cohomologies ... more In this thesis, the class of modules whose Cousin complexes have finitely generated cohomologies are studied as a subclass of modules which have uniform local cohomological annihilators and it is shown that these two classes coincide over local rings with Cohen-Macaulay formal fibres. This point of view enables us to obtain some properties of modules with finite Cousin complexes and find some characterizations of them. In this connection we discuss attached prime ideals of certain local cohomology modules in terms of cohomologies of Cousin complexes. In continuation, we study the top local cohomology modules with specified set of attached primes. Our approach to study Cousin complexes leads us to characterization of generalized Cohen-Macaulay modules in terms of uniform annihilators of local cohomology. We use these results to study the Cohen-Macaulay loci of modules and find two classes of rings over which the Cohen-Macaulay locus of any finitely generated module is a Zariski--open...
In this paper, we use the Apéry table of the numerical semigroup associated to an affine monomial... more In this paper, we use the Apéry table of the numerical semigroup associated to an affine monomial curve in order to characterize arithmetic properties and invariants of its tangent cone. In particular, we precise the shape of the Apéry table of a numerical semigroup of embedding dimension 3, when the tangent cone of its monomial curve is Buchsbaum or 2-Buchsbaum, and give new proofs for two conjectures raised by V.
We study arithmetic properties of tangent cones associated to affine monomial curves, using the c... more We study arithmetic properties of tangent cones associated to affine monomial curves, using the concept of gluing. In particular we characterize the Cohen-Macaulay and Gorenstein properties of tangent cones of some families of monomial curves obtained by gluing. Moreover, we provide new families of monomial curves with non-decreasing Hilbert functions.
Let A be a Noetherian ring. It is shown that any finite A-module M of finite Krull dimension with... more Let A be a Noetherian ring. It is shown that any finite A-module M of finite Krull dimension with finite Cousin complex cohomologies has a uniform local cohomological annihilator. The converse is also true for a finite module M satisfying (S 2) which is over a local ring with Cohen-Macaulay formal fibres.
The Cohen-Macaulay locus of any finite module over a noetherian local ring A is studied and it is... more The Cohen-Macaulay locus of any finite module over a noetherian local ring A is studied and it is shown that it is a Zariski-open subset of Spec A in certain cases. In this connection, the rings whose formal fibres over certain prime ideals are Cohen-Macaulay are studied.
Bulletin of the Australian Mathematical Society, 2013
Let $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that... more Let $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that $I$ is MHC (that is, $I$ satisfies the maximal height condition for the associated primes of $I$) if there exists a prime ideal $\mathfrak{p}\in {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I$ for which $\mathrm{ht} (\mathfrak{p})$ equals the number of indeterminates that appear in the minimal set of monomials generating $I$. Let $I= { \mathop{\bigcap }\nolimits}_{i= 1}^{k} {Q}_{i} $ be the irreducible decomposition of $I$ and let $m(I)= \max \{ \vert Q_{i}\vert - \mathrm{ht} ({Q}_{i} ): 1\leq i\leq k\} $, where $\vert {Q}_{i} \vert $ denotes the total degree of ${Q}_{i} $. Then it can be seen that when $I$ is primary, $\mathrm{reg} (S/ I)= m(I)$. In this paper we improve this result and show that whenever $I$ is MHC, then $\mathrm{reg} (S/ I)= m(I)$ provided $\vert {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I\vert \leq 2$. We also prove that $m({I}^{n} )\leq n\max \{ \vert Q_{i}\vert : 1\leq i...
Let (R, 𝔪) be a complete Noetherian local ring and let M be a finite R-module of positive Krull d... more Let (R, 𝔪) be a complete Noetherian local ring and let M be a finite R-module of positive Krull dimension n. It is shown that any subset T of Assh R(M) can be expressed as the set of attached primes of the top local cohomology module [Formula: see text] for some ideal 𝔞 of R. Moreover, if 𝔞 is an ideal of R such that the set of attached primes of [Formula: see text] is a non-empty proper subset of Assh R(M), then [Formula: see text] for some ideal 𝔟 of R with dim R(R/𝔟) = 1.
The shedding vertices of simplicial complexes are studied from an algebraic point of view. Based ... more The shedding vertices of simplicial complexes are studied from an algebraic point of view. Based on this perspective, we introduce the class of ass-decomposable monomial ideals which is a generalization of the class of Stanley-Reisner ideals of vertex decomposable simplicial complexes. The recursive structure of ass-decomposable monomial ideals allows us to find a simple formula for the depth, and in squarefree case, an upper bound for the regularity of such ideals. Introduction A simplicial complex ∆ on the vertex set V is a collection of subsetes of V , such that ∪F∈∆F = V and ∆ is closed under the operation of taking subsets. The elements of ∆ are called faces. The maximal faces of ∆, under inclusion, are called the facets of ∆ and a simplicial complex with facets F1, . . . , Fm is often denoted by 〈F1, . . . , Fm〉. A simplicial complex with only one facet is called a simplex. For any face F ∈ ∆ the link and the deletion of F in ∆ are defined as link∆(F ) = {G ∈ ∆: G ∩ F = ∅ and ...
We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has a monom... more We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has a monomial minimal reduction. When this is the case, we study the Cohen-Macaulay and Gorenstein properties of the associated graded ring and provide several bounds for the reduction number with respect to the monomial minimal reduction.
We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of... more We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of the simplicial affine semigroups. In this way, we characterize the Cohen-Macaulay type of the simplicial affine semigroup ring K[S]. We define the type of S, type(S), in terms of some Apéry sets of S and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when K[S] is Cohen-Macaulay. If K[S] is a d-dimensional Cohen-Macaulay ring of embedding dimension at most d + 2, then type(S) ≤ 2. Otherwise, type(S) might be arbitrary large and it has no upper bound in terms of the embedding dimension. Finally, we present a generating set for the conductor of S as an ideal of its normalization.
Numerical semigroup rings are investigated from the relative viewpoint. It is known that algebrai... more Numerical semigroup rings are investigated from the relative viewpoint. It is known that algebraic properties such as singularities of a numerical semigroup ring are properties of a flat numerical semigroup algebra. In this paper, we show that arithmetic and set-theoretic properties of a numerical semigroup ring are properties of an equi-gcd numerical semigroup algebra.
Let H ⊂ N d be a normal simplicial affine semigroup, R = K[H] its semigroup ring over the field K... more Let H ⊂ N d be a normal simplicial affine semigroup, R = K[H] its semigroup ring over the field K and ω R its canonical module, which is identified with an ideal in R. The Ulrich elements for H are those h in H such that for the multiplication map by x h from R into ω R , the cokernel is an Ulrich module. We say that the ring R is almost Gorenstein if Ulrich elements exist in H. We provide algebraic and combinatorial criteria to test the Ulrich property for arbitrary elements in H. In particular, unless R is a regular ring, we show that Ulrich elements are located among the exponents of the minimal monomial generators of ω R. Assume d = 2 and a 1 , a 2 are the vectors with coprime integer entries on the extremal rays of the cone C over H. Then any b in the Hilbert basis B H , different from a 1 , a 2 , is an Ulrich element in H if and only if c 1 + c 2 ∈ (a 1 + H) ∪ (a 2 + H) ∪ (b + H) for all c 1 , c 2 ∈ B H. For the bottom element of H, which is defined as the smallest among the exponents of the monomials in ω R ordered componentwise, we have a more direct description of when it has the Ulrich property. Consequently, when H has all nonzero elements with both entries positive and (1, 1) is in the relative interior of the cone C then we obtain a simple arithmetic criterion for the almost Gorenstein property of R in terms of the entries of a 1 and a 2 alone.
Given a numerical semigroup ring R = k[[S]], an ideal E of S and an odd element b ∈ S, the numeri... more Given a numerical semigroup ring R = k[[S]], an ideal E of S and an odd element b ∈ S, the numerical duplication S ✶ b E is a numerical semigroup, whose associated ring k[[S ✶ b E]] shares many properties with the Nagata's idealization and the amalgamated duplication of R along the monomial ideal I = (t e | e ∈ E). In this paper we study the associated graded ring of the numerical duplication characterizing when it is Cohen-Macaulay, Gorenstein or complete intersection. We also study when it is a homogeneous numerical semigroup, a property that is related to the fact that a ring has the same Betti numbers of its associated graded ring. On the way we also characterize when gr m (I) is Cohen-Macaulay and when gr m (ωR) is a canonical module of gr m (R) in terms of numerical semigroup's properties, where ωR is a canonical module of R.
We introduce the concept of homogeneous numerical semigroups and show that all homogeneous numeri... more We introduce the concept of homogeneous numerical semigroups and show that all homogeneous numerical semigroups with Cohen-Macaulay tangent cones are of homogeneous type. In embedding dimension three, we classify all numerical semigroups of homogeneous type into numerical semigroups with complete intersection tangent cones and the homogeneous ones which are not symmetric with Cohen-Macaulay tangent cones. We also study the behavior of the homogeneous property by gluing and shiftings to construct large families of homogeneous numerical semigroups with Cohen-Macaulay tangent cones. In particular we show that these properties fulfill asymptotically in the shifting classes. Several explicit examples are provided along the paper to illustrate the property. 1. introduction Let S be a numerical semigroup minimally generated by a sequence of positive integers n : 0 < n 1 < • • • < n d. For any nonnegative integer j one may consider the shifted sequence n + j : 0 < n 1 + j < • • • < n d + j. Let k be a field and k[S] := k[t n 1 ,. .. , t n d ] ⊂ k[t], where t is a free variable, be the numerical semigroup ring defined by S. This is the homogeneous coordinate ring of the affine monomial curve in A d k defined parametrically by x 1 = t n 1 ,. .. , x d = t n d. Denote by I(n) ⊂ k[x 1 ,. .. , x d ] the defining ideal of k[S] obtained from the natural presentation k[x 1 ,. .. , x d ] → k[S] → 0. J. Herzog and H. Srinivasan conjectured that for j ≫ 0 the Betti numbers of the ideals I(n + j) become periodic on j with period n d − n 1. In 2013, the conjecture was proven to be true by A. V. Jayanthan and H. Srinivasan for d = 3 [17], by A. Marzullo for some particular cases if d = 4 [20], and by P. Gimenez, I. Sengupta and H. Srinivasan in the case of arithmetic sequences [10]. Finally, in 2014, T. Vu gave a completely general positive answer in [35]. One of the main ingredients of Vu's proof is that there exists a positive integer N such that, for all j > N, any minimal binomial non-homogeneous generator of I(n+j) is of the form x a 1 1 u−vx a d d , where a 1 , a d are positive integers, u, v are monomials in the variables x 2 ,. .. , x d−1 , and deg x a 1 1 u > deg vx a d d. It is noteworthy that the bound N depends on the Castelnuovo-Mumford regularity of J(n), the ideal generated by the homogeneous elements in I(n). Let I(n) * be the homogeneous ideal generated by the initial forms of the elements in I(n). Then, k[x 1 ,. .. , x d ]/I(n) * ≃ G(S), the tangent cone of k[S]. By using this main ingredient in Vu's proof of the conjecture, J. Herzog and D. I. Stamate proved in [15] that for any j > N, the Betti numbers
Let A be a K-subalgebra of the polynomial ring S = K [x 1 ,. .. , x d ] of dimension d, generated... more Let A be a K-subalgebra of the polynomial ring S = K [x 1 ,. .. , x d ] of dimension d, generated by finitely many monomials of degree r. Then, the Gauss algebra G(A) of A is generated by monomials of degree (r − 1)d in S. We describe the generators and the structure of G(A), when A is a Borel fixed algebra, a squarefree Veronese algebra, generated in degree 2, or the edge ring of a bipartite graph with at least one loop. For a bipartite graph G with one loop, the embedding dimension of G(A) is bounded by the complexity of the graph G.
We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The comp... more We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The complete intersection property of numerical semigroup algebras is investigated using factorizations of monomials into minimal ones. The goal is to study whether a flat rectangular algebra is a complete intersection. Along this direction, special types of algebras generated by few monomials are worked out in detail.
We introduce the concept of monomial ideals with stable projective dimension, as a generalization... more We introduce the concept of monomial ideals with stable projective dimension, as a generalization of the Cohen–Macaulay property. Indeed, we study the class of monomial ideals [Formula: see text], whose projective dimension is stable under monomial localizations at monomial prime ideals [Formula: see text], with [Formula: see text]. We study the relations between this property and other sorts of Cohen–Macaulayness. Finally, we characterize some classes of polymatroidal ideals with stable projective dimension.
Algebraic and combinatorial properties of a monomial ideal are studied in terms of its associated... more Algebraic and combinatorial properties of a monomial ideal are studied in terms of its associated radical ideals. In particular, we present some applications to the symbolic powers of square-free monomial ideals.
The class of equidimensional polymatroidal ideals is studied. In particular, we show that an unmi... more The class of equidimensional polymatroidal ideals is studied. In particular, we show that an unmixed polymatroidal ideal is connected in codimension one if and only if it is Cohen-Macaulay. Especially a matroidal ideal is connected in codimension one precisely when it is a squarefree Veronese ideal. As a consequence we indicate that for polymatroidal ideals, Serre's condition (S n) for some n ≥ 2 is equivalent to Cohen-Macaulay property. We also give a classification of generalized Cohen-Macaulay polymatroidal ideals.
In this thesis, the class of modules whose Cousin complexes have finitely generated cohomologies ... more In this thesis, the class of modules whose Cousin complexes have finitely generated cohomologies are studied as a subclass of modules which have uniform local cohomological annihilators and it is shown that these two classes coincide over local rings with Cohen-Macaulay formal fibres. This point of view enables us to obtain some properties of modules with finite Cousin complexes and find some characterizations of them. In this connection we discuss attached prime ideals of certain local cohomology modules in terms of cohomologies of Cousin complexes. In continuation, we study the top local cohomology modules with specified set of attached primes. Our approach to study Cousin complexes leads us to characterization of generalized Cohen-Macaulay modules in terms of uniform annihilators of local cohomology. We use these results to study the Cohen-Macaulay loci of modules and find two classes of rings over which the Cohen-Macaulay locus of any finitely generated module is a Zariski--open...
In this paper, we use the Apéry table of the numerical semigroup associated to an affine monomial... more In this paper, we use the Apéry table of the numerical semigroup associated to an affine monomial curve in order to characterize arithmetic properties and invariants of its tangent cone. In particular, we precise the shape of the Apéry table of a numerical semigroup of embedding dimension 3, when the tangent cone of its monomial curve is Buchsbaum or 2-Buchsbaum, and give new proofs for two conjectures raised by V.
We study arithmetic properties of tangent cones associated to affine monomial curves, using the c... more We study arithmetic properties of tangent cones associated to affine monomial curves, using the concept of gluing. In particular we characterize the Cohen-Macaulay and Gorenstein properties of tangent cones of some families of monomial curves obtained by gluing. Moreover, we provide new families of monomial curves with non-decreasing Hilbert functions.
Let A be a Noetherian ring. It is shown that any finite A-module M of finite Krull dimension with... more Let A be a Noetherian ring. It is shown that any finite A-module M of finite Krull dimension with finite Cousin complex cohomologies has a uniform local cohomological annihilator. The converse is also true for a finite module M satisfying (S 2) which is over a local ring with Cohen-Macaulay formal fibres.
The Cohen-Macaulay locus of any finite module over a noetherian local ring A is studied and it is... more The Cohen-Macaulay locus of any finite module over a noetherian local ring A is studied and it is shown that it is a Zariski-open subset of Spec A in certain cases. In this connection, the rings whose formal fibres over certain prime ideals are Cohen-Macaulay are studied.
Bulletin of the Australian Mathematical Society, 2013
Let $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that... more Let $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that $I$ is MHC (that is, $I$ satisfies the maximal height condition for the associated primes of $I$) if there exists a prime ideal $\mathfrak{p}\in {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I$ for which $\mathrm{ht} (\mathfrak{p})$ equals the number of indeterminates that appear in the minimal set of monomials generating $I$. Let $I= { \mathop{\bigcap }\nolimits}_{i= 1}^{k} {Q}_{i} $ be the irreducible decomposition of $I$ and let $m(I)= \max \{ \vert Q_{i}\vert - \mathrm{ht} ({Q}_{i} ): 1\leq i\leq k\} $, where $\vert {Q}_{i} \vert $ denotes the total degree of ${Q}_{i} $. Then it can be seen that when $I$ is primary, $\mathrm{reg} (S/ I)= m(I)$. In this paper we improve this result and show that whenever $I$ is MHC, then $\mathrm{reg} (S/ I)= m(I)$ provided $\vert {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I\vert \leq 2$. We also prove that $m({I}^{n} )\leq n\max \{ \vert Q_{i}\vert : 1\leq i...
Let (R, 𝔪) be a complete Noetherian local ring and let M be a finite R-module of positive Krull d... more Let (R, 𝔪) be a complete Noetherian local ring and let M be a finite R-module of positive Krull dimension n. It is shown that any subset T of Assh R(M) can be expressed as the set of attached primes of the top local cohomology module [Formula: see text] for some ideal 𝔞 of R. Moreover, if 𝔞 is an ideal of R such that the set of attached primes of [Formula: see text] is a non-empty proper subset of Assh R(M), then [Formula: see text] for some ideal 𝔟 of R with dim R(R/𝔟) = 1.
The shedding vertices of simplicial complexes are studied from an algebraic point of view. Based ... more The shedding vertices of simplicial complexes are studied from an algebraic point of view. Based on this perspective, we introduce the class of ass-decomposable monomial ideals which is a generalization of the class of Stanley-Reisner ideals of vertex decomposable simplicial complexes. The recursive structure of ass-decomposable monomial ideals allows us to find a simple formula for the depth, and in squarefree case, an upper bound for the regularity of such ideals. Introduction A simplicial complex ∆ on the vertex set V is a collection of subsetes of V , such that ∪F∈∆F = V and ∆ is closed under the operation of taking subsets. The elements of ∆ are called faces. The maximal faces of ∆, under inclusion, are called the facets of ∆ and a simplicial complex with facets F1, . . . , Fm is often denoted by 〈F1, . . . , Fm〉. A simplicial complex with only one facet is called a simplex. For any face F ∈ ∆ the link and the deletion of F in ∆ are defined as link∆(F ) = {G ∈ ∆: G ∩ F = ∅ and ...
We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has a monom... more We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has a monomial minimal reduction. When this is the case, we study the Cohen-Macaulay and Gorenstein properties of the associated graded ring and provide several bounds for the reduction number with respect to the monomial minimal reduction.
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Papers by Raheleh Jafari