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A concept of Cohen–Macaulay in codimension t is defined and characterized for arbitrary finitely generated modules and coherent sheaves by Miller, Novik, and Swartz in 2011. Soon after, Haghighi, Yassemi, and Zaare-Nahandi defined and studied CMt simplicial complexes, which is the pure version of the abovementioned concept and naturally generalizes both Cohen–Macaulay and Buchsbaum properties. The purpose of this paper is to survey briefly recent results of CMt simplicial complexes.

Let R be a commutative ring, and let C be a semidualizing R-module. We introduce the notion of finitely presented C-injective modules, finitely presented C-flat modules, weak C-injective modules and weak C-flat modules. Some properties of these modules are investigated. It is proved that over weak C-injective ring R, if for a super finitely presented R-module M, $${\text {Hom}}_{R}(M,R)$$ is super finitely presented and $${\text {Hom}}_{R}(M,R) \in \mathcal {A}_C(R)$$ , then the following statements hold

We provide necessary and sufficient conditions for the amalgamation of rings A⋈fJ to be a divided ring, locally divided ring, going-down ring, and Gaussian ring.

A misuse of terminology has occurred in the statement and proof of Theorem 4.1 in our paper [Proc. Amer. Math. Soc. 143 (2015), pp. 1981–1989] which caused some justifiable misinterpretation of this result. To recover this result we provide a new definition and give the statement of our result in terms of this definition. The proof of the new version is an improvement of the old proof. The effect of the new definition on further relevant results in our paper has been adopted in a remark.

Local dimension is an ordinal valued invariant that is in some sense a measure of how far a ring is from being local and denoted [Formula: see text]. The purpose of this paper is to study the local dimension of ring extensions such as homomorphic image, trivial ring extension and the amalgamation of rings.

We give a generalization of Eagon-Reiner’s theorem relating Betti numbers of the Stanley-Reisner ideal of a simplicial complex and the CMt property of its Alexander dual. Then we characterize bi-CMt bipartite graphs and bi-CMt chordal graphs. These are generalizations of recent results due to Herzog and Rahimi.

Let G be a graph with n vertices, let $$S={\mathbb {K}}[x_1,\dots ,x_n]$$ be the polynomial ring in n variables over a field $${\mathbb {K}}$$ and let I(G) denote the edge ideal of G. For every collection $${\mathcal {H}}$$ of connected graphs with $$K_2\in {\mathcal {H}}$$ , we introduce the notions of $${{\mathrm{ind-match}}}_{{\mathcal {H}}}(G)$$ and $${{\mathrm{min-match}}}_{{\mathcal {H}}}(G)$$ . It will be proved that the inequalities $${{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\le \mathrm{reg}(S/I(G))\le {{\mathrm{min-match}}}_{\{K_2, C_5\}}(G)$$ are true. Moreover, we show that if G is a Cohen–Macaulay graph with girth at least five, then $$\mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)$$ . Furthermore, we prove that if G is a paw-free and doubly Cohen–Macaulay graph, then $$\mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)$$ if and only if every connected component of G is either a complete graph or a 5-cycle graph. Among other results, we show that for every doubly Cohen–Macaulay simplicial complex, the equality $$\mathrm{reg}({\mathbb {K}}[\Delta ])=\mathrm{dim}({\mathbb {K}}[\Delta ])$$ holds.

A very well-covered graph is an unmixed graph whose covering number is half of the number of vertices. We construct an explicit minimal free resolution of the cover ideal of a Cohen–Macaulay very well-covered graph. Using this resolution, we characterize the projective dimension of the edge ideal of a very well-covered graph in terms of a pairwise$3$-disjoint set of complete bipartite subgraphs of the graph. We also show nondecreasing property of the projective dimension of symbolic powers of the edge ideal of a very well-covered graph with respect to the exponents.

We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs which have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes and Toledo.

Let R be a commutative ring with Z(R) the set of zero-divisors and U(R) the set of unit elements of R. The total graph of R, denoted by T(Γ(R)), is the (undirected) graph with all elements of R as vertices, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). We study the domination number of T(Γ(R)). It is shown that if R = Z(R) ∪ U(R), then the domination number of T(∪(R)) is finite provided R has a maximal ideal of finite index. Moreover, if R = ∏ i = 1 n F i $R = \prod\limits_{i = 1}^n {{F_i}} $ , where Fi is a field for each 1 ≤ i ≤ n and t = |F 1| ≤ |F 2| ≤ ··· ≤ |Fn |, then the domination number of T(Γ(R)) is equal to t - 1 provided t = |Fi | for every 1 ≤ i ≤ n, and is equal to t otherwise. Finally, for an R-module M it is shown that the total domination number of the total graph of the idealization (Nagata extension) R(+)M is equal to the domination number of the total graph of R provided M is a torsion free R-module or R = Z(R) ∪ U(R).

Let R0 be a Noetherian local ring and R a standard graded R0-algebra with maximal ideal 𝔪 and residue class field 𝕂 = R/𝔪. For a graded ideal I in R we show that for k ≫ 0: (1) the Artin-Rees number of the syzygy modules of Ik as submodules of the free modules from a free resolution is constant, and thereby the Artin-Rees number can be presented as a proper replacement of regularity in the local situation; and (2) R is a polynomial ring over the regular R0, the ring R/Ik is Golod, its Poincaré-Betti series is rational and the Betti numbers of the free resolution of 𝕂 over R/Ik are polynomials in k of a specific degree. The first result is an extension of the work by Swanson on the regularity of Ik for k ≫ 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of the work by Kodiyalam on the behaviour of Betti numbers of the minimal free resolution of R/Ik over R.

In this paper we study some algebraic and combinatorial behaviors of expansion functor. We show that on monomial ideals some properties like polymatroidalness, weakly polymatroidalness and having linear quotients are preserved under taking the expansion functor. The main part of the paper is devoted to study of toric ideals associated to the expansion of subsets of monomials which are minimal with respect to divisibility. It is shown that, for a given discrete polymatroid $P$, if toric ideal of $P$ is generated by double swaps then toric ideal of any expansion of $P$ has such a property. This result, in a special case, says that White's conjecture is preserved under taking the expansion functor. Finally, the construction of Gr\"{o}bner bases and some homological properties of toric ideals associated to expansions of subsets of monomials is investigated., Comment: 15 pages

Let G be a simple graph on the vertex set V (G) and S = {x11,...,xn1} a subset of V (G). Let m1,...,mn ? 2 be integers and G1,...,Gn connected simple graphs on the vertex sets V (Gi) = {xi1,..., ximi} for i = 1,..., n. The graph G(G1,...,Gn) is obtained from G by attaching Gi to G at the vertex xi1 for i = 1,...,n. We give a characterization of G(G1,...,Gn) for being vertex decomposable. This generalizes a result due to Mousivand, Seyed Fakhari, and Yassemi.

In this paper we introduce a class of monomial ideals, called k-decomposable ideals. It is shown that the class of k-decomposable ideals is contained in the class of monomial ideals with linear quotients, and when k is large enough, the class of k-decomposable ideals is equal to the class of ideals with linear quotients. In addition, it is shown that a d-dimensional simplicial complex is k-decomposable if and only if the Stanley-Reisner ideal of its Alexander dual is a k-decomposable ideal, where k ≤ d. Moreover, it is shown that every k-decomposable ideal is componentwise k-decomposable.

Let \({S=\mathbb{K}[x_1,\dots,x_n]}\) be the polynomial ring in n variables over the field \({\mathbb{K}}\). Suppose that \({J\subsetneq I}\) are polymatroidal ideals of S. We provide a lower bound for the Stanley depth of I/J. Using this lower bound, we prove that \({{\rm sdepth}(I^k/I^{k+1})\geq {\rm depth}(I^k/I^{k+1})}\) for every integer \({k\gg0}\). We also prove that if I is the edge ideal of a forest graph with p connected components, then \({{\rm sdepth}(I^k/I^{k+1})\geq p}\) and conclude that \({{\rm sdepth}(I^k/I^{k+1})\geq {\rm depth}(I^k/I^{k+1})}\) for every integer \({k\gg0}\).

Let G be a finite simple graph on a vertex set V(G) = {x 11,…, x n1}. Also let m 1,…, m n ≥ 2 be integers and G 1,…, G n be connected simple graphs on the vertex sets V(G i ) = {x i1,…, x im i }. In this article, we provide necessary and sufficient conditions on G 1,…, G n for which the graph obtained by attaching the G i to G is unmixed or vertex decomposable. Then we characterize Cohen–Macaulay and sequentially Cohen–Macaulay graphs obtained by attaching the cycle graphs or connected chordal graphs to arbitrary graphs.

Let Δ be a simplicial complex and χ be an s-coloring of Δ. Biermann and Van Tuyl have introduced the simplicial complex Δχ. As a corollary of Theorems 5 and 7 in their 2013 article, we obtain that the Stanley–Reisner ring of Δχ over a field is Cohen–Macaulay. In this note, we generalize this corollary by proving that the Stanley–Reisner ideal of Δχ over a field is set-theoretic complete intersection. This also generalizes a result of Macchia.

Let R be a commutative Cohen–Macaulay ring, and let C be a semidualizing module of R. In this paper, we show that C is generically dualizing if and only if the tensor products of injective and C-injective R-modules are injective. This leads to a characterization of dualizing modules as well as generalizes a result of Enochs and Jenda.

Let C be a semidualizing module for a commutative ring R. It is shown that the [Formula: see text]-injective dimension has the ability to detect the regularity of R as well as the [Formula: see text]-projective dimension. It is proved that if D is dualizing for a Noetherian ring R such that id R(D) = n < ∞, then [Formula: see text] for every flat R-module F. This extends the result due to Enochs and Jenda. Finally, over a Noetherian ring R, it is shown that if M is a pure submodule of an R-module N, then [Formula: see text]. This generalizes the result of Enochs and Holm.

The theory of finitely generated relative (co)tilting modules has been established in the 1980s by Auslander and Solberg, and infinitely generated relative tilting modules have recently been studied by many authors in the context of Gorenstein homological algebra. In this work, we build on the theory of infinitely generated Gorenstein tilting modules by developing "Gorenstein tilting approximations" and employing these approximations to study Gorenstein tilting classes and their associated relative cotorsion pairs. As applications of our results, we discuss the problem of existence of complements to partial Gorenstein tilting modules as well as some connections between Gorenstein tilting modules and finitistic dimension conjectures., Comment: Comments welcome; a gap in the proof of Thm.(3.11) was fixed

Let $R$ be a commutative noetherian local ring. We define a new invariant for $R$-modules which we call the little dimension. Using it, we extend the improved new intersection theorem., Comment: 7 pages, to appear in Math. Scand

We relate the homological behavior of an associative ring R and those of the rings R / x R and R x when x is a regular central element in R . For left weak global dimensions we prove wgldim ( R ) ≤ max ⁡ { 1 + wgldim ( R / x R ) , wgldim ( R x ) } with equality if wgldim ( R / x R ) is finite. The key point is a formula for flat dimensions of R -modules: fd R M = max ⁡ { fd R / x R ( ( R / x R ) ⊗ R L M ) , fd R x M x } . For left noetherian R we recover results of Li, Van den Bergh and Van Oystaeyen [3] on global and projective dimensions. Similar formulae hold for injective dimensions.

Let R be a commutative Noetherian ring and let C be a semidualizing R-module. It is shown that a finitely generated R-module M with finite G C -dimension is C-reflexive if and only if $M_{\mathfrak {p}}$ is $C_{\mathfrak {p}}$ -reflexive for $\mathfrak {p} \in \text {Spec}\,(R) $ with $\text {depth}\,(R_{\mathfrak {p}}) \leq 1$ , and $G_{C_{\mathfrak {p}}}-\dim _{R_{\mathfrak {p}}} (M_{\mathfrak {p}}) \leq \text {depth}\,(R_{\mathfrak {p}})-2 $ for $\mathfrak {p} \in \text {Spec}\, (R) $ with $\text {depth}\,(R_{\mathfrak {p}})\geq 2 $ . As the ring R itself is a semidualizing module, this result gives a generalization of a natural setting for extension of results due to Serre and Samuel (see Czech. Math. J. 62(3) (9) 663–672 and Beitrage Algebra Geom. 50(2) (3) 353–362). In addition, it is shown that over ring R with $\dim R \leq n$ , where n≥2 is an integer, $G_{D}-\dim _{R} (\text {Hom}\,_{R} (M,D)) \leq n-2$ for every finitely generated R-module M and a dualizing R-module D.

Let G G be an unmixed bipartite graph of dimension d − 1 d-1 . Assume that K n , n K_{n,n} , with n ≥ 2 n\ge 2 , is a maximal complete bipartite subgraph of G G of minimum dimension. Then G G is Cohen-Macaulay in codimension t t if and only if t ≥ d − n + 1 t\ge d-n+1 . This is derived from a characterization of Cohen-Macaulay bipartite graphs by Herzog and Hibi and generalizes a recent result of Cook and Nagel on unmixed Buchsbaum graphs. Furthermore, we show that any unmixed bipartite graph G G which is Cohen-Macaulay in codimension t t , is obtained from a Cohen-Macaulay graph by replacing certain edges of G G with complete bipartite graphs. Thus, in light of combinatorial characterization of Cohen-Macaulay bipartite graphs, our result may be considered purely combinatorial.

Let G be a Cohen–Macaulay very well-covered graph. We prove that the h-vector of the independence complex of G is precisely the f-vector of a flag complex. Moreover, we show that the h-vector of clique-whiskered graphs is exactly the h-vector of Cohen–Macaulay very well-covered graphs. In particular, we show that Kalai’s conjecture holds if G is a very well covered Cohen–Macaulay graph.

Let R be a commutative ring with identity. Let M be an R-module and T (M)* be the set of nonzero torsion elements. The set T(M)* makes up the vertices of the corresponding torsion graph, Γ R (M), with two distinct vertices x, y ∈ T(M)* forming an edge if Ann(x) ∩ Ann(y) ≠ 0. In this paper we study the case where the graph Γ R (M) is connected with diam(Γ R (M)) ≤ 3 and we investigate the relationship between the diameters of Γ R (M) and Γ R (R). Also we study girth of Γ R (M), it is shown that if Γ R (M) contains a cycle, then gr(Γ R (M)) = 3.

Let 𝕂 be a field, and let R = 𝕂[x 1,…, x n ] be the polynomial ring over 𝕂 in n indeterminates x 1,…, x n . Let G be a graph with vertex-set {x 1,…, x n }, and let J be the cover ideal of G in R. For a given positive integer k, we denote the kth symbolic power and the kth bracket power of J by J (k) and J [k], respectively. In this paper, we give necessary and sufficient conditions for R/J k , R/J (k), and R/J [k] to be Cohen–Macaulay. We also study the limit behavior of the depths of these rings.

The unit graph of a ring R with nonzero identity is the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x C y is a unit in R. In this paper, we derive several necessary conditions for the nonplanarity of the unit graphs of finite commutative rings with nonzero identity, and determine, up to isomorphism, all finite commutative rings with nonzero identity whose unit graphs are toroidal. Algebraic combinatorics is an area of mathematics which employs methods of abstract algebra in various combinatorial contexts and vice versa. Associating a graph to an algebraic structure is a research subject in this area and has attracted considerable attention. The research in this subject aims at exposing the relationship between algebra and graph theory and at advancing the application of one to the other. In fact, there are three major problems in this area: (1) characterization of the resulting graphs, (2) characterization of the algebraic structures with isomorphic graphs, and (3) realization of the connections between the algebraic structures and the corresponding graphs. Beck (1988) introduced the idea of a zero-divisor graph of a commutative ring R with nonzero identity. He defined00.R/ to be the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x yD 0. He was mostly concerned with coloring of 00.R/. Beck conjectured that . R/D!.R/, where . R/ and !.R/ denote, respectively, the chromatic number and the clique number of00.R/. Such graphs are called weakly perfect graphs. This investigation of coloring of a commutative ring was then continued by Anderson and Naseer (1993). They gave a counterexample for the above conjecture of Beck. Anderson and Livingston (1999) proposed a different

Let G(R) be the unitary Cayley graph corresponding to a finite commutative ring R with nonzero identity. Let ΔG(R) be the simplicial complex associated to G(R), whose faces correspond to the independent sets of G(R). We study well-coverednees of G(R) and Cohen–Macaulayness of ΔG(R), i.e., its Stanley–Reisner ring k [ΔG(R)] is a Cohen–Macaulay ring. Furthermore, we show that a unitary Cayley graph is shellable and Gorenstein if it is Cohen–Macaulay.

A classical question of Yoneda asks when the tensor product of two injective modules is injective. A complete answer to this question was given by Enochs and Jenda in 1991. In this paper the analogue question for pure-injective modules is studied.

Let C be a semidualizing module for a commutative ring R. In this paper, we study the resulting modules of finite G C -projective dimension in Bass class, showing that they admit G C -projective precover. Over local ring, we prove that dim R (M) ≤ 𝒢ℐ C − id R (M) for any nonzero finitely generated R-module M, which generalizes a result due to Bass.

A theorem due to Froberg states that a simple graph is chordal if and only if the edge ideal of its complementary graph has a linear resolution. Actually, this is a characterization of the edge ideals of 2-uniform hypergraphs which have a linear resolution. In this article, edge ideals of a special kind of d-uniform hypergraphs is investigated. It is shown that if a d-uniform hypergraph is chordal then the edge ideal of its complementary hypergraph is weakly polymatroidal. This is an improvement of a theorem due to Emtander, Mohammadi and Moradi. Also, it is proved that all powers of the edge ideal of a Ferrers hypergraph are weakly polymatroidal. Finally, we show that the edge ideals of a complete admissible uniform hypergraph ℋ and are weakly polymatroidal, where is an uniform hypergraph with the edges e c for all edges e of ℋ.

For a given (d−1)-dimensional simplicial complex Γ, we denote its h-vector by h(Γ)=(h0(Γ),h1(Γ),. . .,hd(Γ)) and set h−1(Γ)=0. The known Swartz equality implies that if Δ is a (d−1)-dimensional Buchsbaum simplicial complex over a field, then for every 0 ≤ i ≤ d, the inequality ihi(Δ)+(d−i+1)hi−1(Δ) ≥ 0 holds true. In this paper, by using these inequalities, we give a simple proof for a result of Terai (N. Terai, On h-vectors of Buchsbaum Stanley–Reisner rings, Hokkaido Math. J. 25(1) (1996), 137–148) on the h-vectors of Buchsbaum simplicial complexes. We then generalize the Swartz equality (E. Swartz, Lower bounds for h-vectors of k-CM, independence, and broken circuit complexes, SIAM J. Discrete Math. 18(3) (2004/05), 647–661), which in turn leads to a generalization of the above-mentioned inequalities for Cohen–Macaulay simplicial complexes in co-dimension t.

This paper aims at generalizing the notions of “Auslander dual”, “k-torsionless” and “k-th syzygy” modules to the relative setting with respect to a semidualizing module. It is shown that the “relative” Auslander dual shares many nice properties with the Auslander dual originally introduced by Auslander and Bridger. It is also shown the interplay between the “relative” version of “k-torsionless” and “k-th syzygy” modules parallels that of k-torsionless and k-th syzygy modules.

Let K \mathbb {K} be a field and S = K [ x 1 , … , x n ] S=\mathbb {K}[x_1,\dots ,x_n] be the polynomial ring in n n variables over the field K \mathbb {K} . Let G G be a forest with p p connected components G 1 , … , G p G_1,\ldots ,G_p and let I = I ( G ) I=I(G) be its edge ideal in S S . Suppose that d i d_i is the diameter of G i G_i , 1 ≤ i ≤ p 1\leq i\leq p , and consider d = max { d i ∣ 1 ≤ i ≤ p } d =\max \hspace {0.04cm}\{d_i\mid 1\leq i\leq p\} . Morey has shown that for every t ≥ 1 t\geq 1 , the quantity max { ⌈ d − t + 2 3 ⌉ + p − 1 , p } \max \{\lceil \frac {d-t+2}{3}\rceil +p-1,p\} is a lower bound for depth ( S / I t ) \textrm {depth}(S/I^t) . In this paper, we show that for every t ≥ 1 t\geq 1 , the mentioned quantity is also a lower bound for sdepth ( S / I t ) \textrm {sdepth}(S/I^t) . By combining this inequality with Burch’s inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem.

Recently, Haghighi, Terai, Yassemi, and Zaare-Nahandi introduced the notion of a sequentially (S r ) simplicial complex. This notion gives a generalization of two properties for simplicial complexes: being sequentially Cohen–Macaulay and satisfying Serre’s condition (S r ). Let Δ be a (d−1)-dimensional simplicial complex with Γ(Δ) as its algebraic shifting. Also let (h i,j (Δ))0≤j≤i≤d be the h-triangle of Δ and (h i,j (Γ(Δ)))0≤j≤i≤d be the h-triangle of Γ(Δ). In this paper, it is shown that for a Δ being sequentially (S r ) and for every i and j with 0≤j≤i≤r−1, the equality h i,j (Δ)=h i,j (Γ(Δ)) holds true.

Given a non-unit, non-zero-divisor, central element $x$ of a ring $\Lambda$, it is well known that many properties or invariants of $\Lambda$ determine, and are determined by, those of $\Lambda / x \Lambda$ and $\Lambda_x$. In the present paper, we investigate how the property of "being tilting" behaves in this situation. It turns out that any tilting module over $\Lambda$ gives rise to tilting modules over $\Lambda_x$ and $\Lambda / x \Lambda$ after localization and passing to quotient respectively. On the other hand, it is proved that under some mild conditions, a module over $\Lambda$ is tilting if its corresponding localization and quotient are tilting over $\Lambda_x$ and $\Lambda / x \Lambda$ respectively., Comment: A gap in the statement of Proposition 2.5 and in the proof of Theorem 2.10 has been fixed. Minor editorial changes have been made. To appear in "Journal of Pure and Applied Algebra"

In this paper we compute the weak Gorenstein global dimension of a coherent power series ring. It is shown that the weak Gorenstein global dimension of R[[x]] is equal to the weak Gorenstein global dimension of R plus one, provided R[[x]] is coherent.

Let \({\mathbb{K}}\) be a field and \({S = \mathbb{K}[x_1,\dots,x_n]}\) be the polynomial ring in n variables over the field \({\mathbb{K}}\). In this paper, it is shown that Stanley’s conjecture holds for I and S/I if I is a product of monomial prime ideals or I is a high enough power of a polymatroidal or a stable ideal generated in a single degree.

We consider relative Tor functors built from resolutions described by a semidualizing module C over a commutative noetherian ring R. We show that the bifunctors Tor^{F_CM}_i (-,-) and Tor^{P_CM}_i (-,-), defined using flat-like and projective-like resolutions, are isomorphic. We show how the vanishing of these functors characterizes the finiteness of the homological dimension F_C-pd, and we use this to give a relation between the F_C-pd of a given module and that of a pure submodule. On the other hand, we show that other relations that one may expect to hold similarly, fail in general. In fact, such relations force the semidualizing modules under consideration to be trivial., 29 pages

Any finitely generated module M over a local ring R is endowed with a complete intersection dimension CI-dim RM and a Gorenstein dimension G-dim RM. The Gorenstein dimension can be extended to all modules over the ring R. This paper presents a similar extension for the complete intersection dimension, and mentions the relation between this dimension and the Gorenstein flat dimension. In addition, we show that in the intersection theorem, the flat dimension can be replaced by the complete intersection flat dimension.

Let S be a polynomial ring over a field K and let R be the Stanley-Reisner ring of a matroid complex. In this paper, as a comparison of multiplicity and final Betti number of R over S, the inequality $\beta_{p}(R)\leq \frac{p}{M_p}\, e(R)$ is obtained.

For a positive integer k and a non-negative integer t, a class of simplicial complexes, to be denoted by k-CMt, is introduced. This class generalizes two notions for simplicial complexes: being k-Cohen–Macaulay and k-Buchsbaum. In analogy with the Cohen–Macaulay and Buchsbaum complexes, we give some characterizations of CMt (=1−CMt) complexes, in terms of vanishing of some homologies of its links, and in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We give a result on the behavior of the CMt property under the operation of join of two simplicial complexes. We show that a complex is k-CMt if and only if the links of its non-empty faces are k-CMt−1. We prove that for an integer s≤d, the (d−s−1)-skeleton of a (d−1)-dimensional k-CMt complex is (k+s)-CMt. This result generalizes Hibi’s result for Cohen–Macaulay complexes and Miyazaki’s result for Buchsbaum complexes.

In this paper, we deal with zero-divisor graphs of posets. We prove that the diameter of such a graph is either 1, 2 or 3 while its girth is either 3, 4 or ~. We also characterize zero-divisor graphs of posets in terms of their diameter and girth.

Let \({(R, \mathfrak{m})}\) be a commutative Noetherian local ring of Krull dimension d, and let C be a semidualizing R-module. In this paper, it is shown that if R is complete, then C is a dualizing module if and only if the top local cohomology module of \({R, H _{\mathfrak{m}} ^{d} (R)}\), has finite GC-injective dimension. This generalizes a recent result due to Yoshizawa, where the ring is assumed to be complete Cohen-Macaulay.

Let Δ be a ( d − 1 ) -dimensional simplicial complex and let h ( Δ ) = ( h 0 , h 1 , … , h d ) be its h -vector. A recent result of Murai and Terai guarantees that if Δ satisfies Serre’s condition ( S r ), then ( h 0 , h 1 , … , h r ) is an M -vector and h r + h r + 1 + ⋯ + h d is nonnegative. In this article, we extend the result of Murai and Terai by giving r extra necessary conditions. More precisely, we prove that if Δ satisfies Serre’s condition ( S r ), then i i h r + i + 1 i h r + 1 + ⋯ + i + d − r i h d , 0 ≤ i ≤ r ≤ d , are all nonnegative.

Let 𝜑 : R → S be a ring homomorphism of Noetherian rings and let 𝔞 be an ideal of R. Let M be a finitely generated S-module with dim R(M)=d. In this paper, we prove that the R-module [Formula: see text] has a secondary representation and we exhibit its set of attached primes (under some technical conditions).