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For $k,n\ge 2$, a $(k,n)$-firecracker graph, denoted by $F_{k,n}$, is obtained by concatenation of $k$$n$-star graphs by connecting one leaf from each $n$-star graph. Given a finite simple graph $G$, one can associate a simplicial complex $\mathrm{\Delta }(G)$. In this paper, we compute all the graded Betti numbers of the edge ideal $I(F_{2,n})$ of the firecracker graph $F_{2,n}$ by using the combinatorial data associated with the simplicial complex $\mathrm{\Delta }(F_{2,n}).$ We also find the regularity of the ideals $I(F_{k,n})$. Further, using the domination parameters of the graphs, we explicitly compute the projective dimension of $I(F_{k,n})$.

For integers $k,n\ge 2$, the graph obtained by joining one leaf of each of $k$-copies of $n$-star graph with a single vertex distinct from all the vertices of $k$-copies of $n$-star graph is known as Banana tree graph, written as $B_{k,n}$. In this paper, we shall investigate certain homological invariants of edge rings of Banana tree graphs $B_{k,n}$ in terms of associated combinatorial data. More precisely, we shall show that the regularity and projective dimension of edge ring of $B_{k,n}$ are $k$ and $nk-k$, respectively, and deduce combinatorial formulae for computing all the nonzero graded Betti numbers of edge ring of $B_{2,n}$.

We show that every finite group G has a set of cohomological elements satisfying ceratin algebraic property which can be regarded as a generalized notion of an algebraic counterpart to the topological phenomenon of free actions on finite dimensional homotopy spheres. We extend this result to a certain class of groups which contains groups of finite virtual cohomological dimension.

(Dual) hypergraphs have been used by Kimura, Rinaldo and Terai to characterize squarefree monomial ideals $J$ with $\mathrm{pd}(R/J)\le \mu (J)-1$, i.e. whose projective dimension equals the minimal number of generators of $J$ minus 1. In this paper, we prove sufficient and necessary combinatorial conditions for $\mathrm{pd}(R/J)\le \mu (J)-2$. The second main result is an effective explicit procedure to compute the projective dimension of a large class of 1-dimensional hypergraphs ${\mathscr{H}}$ (the ones in which every connected component contains at most one cycle). An algorithm to compute the projective dimension is also provided. Applications of these results are given; they include, for instance, computing the projective dimension of monomial ideals whose associated hypergraph has a spanning Ferrers graph.

This work is devoted to interpretation of concepts of Zariski dimension of an algebraic variety over a field and of Krull dimension of a coordinate ring in algebraic geometry over algebraic structures of an arbitrary signature. Proposed dimensions are ordinal numbers (ordinals).

Let G be a finite simple graph on n non-isolated vertices, and let $J_G$ be its binomial edge ideal. We determine almost all pairs $(\mathrm{\text{proj dim}}(J_G),\mathrm{\text{reg}}(J_G))$, where G ranges over all finite simple graphs on n non-isolated vertices, for any n.

Benson and Goodearl [Periodic flat modules, and flat modules for finite groups, Pacific J. Math.196(1) (2000) 45–67] proved that if M is a flat module over a ring R such that there exists an exact sequence of R-modules 0 → M → P → M → 0 with P a projective module, then M is projective. The main purpose of this paper is to generalize this theorem to any exact sequence of the form 0 → M → G → M → 0, where G is an arbitrary module over R. Moreover, we seek counterpart entities in the Gorenstein homological algebra of pure projective and pure injective modules.

We show that for the edge ideals of the graphs consisting of one cycle or two cycles of any length connected through a vertex, the arithmetical rank equals the projective dimension of the corresponding quotient ring.

For a simplicial complex Δ, we introduce a simplicial complex attached to Δ, called the expansion of Δ, which is a natural generalization of the notion of expansion in graph theory. We are interested in knowing how the properties of a simplicial complex and its Stanley–Reisner ring relate to those of its expansions. It is shown that taking expansion preserves vertex decomposable and shellable properties and in some cases Cohen–Macaulayness. Also it is proved that some homological invariants of Stanley–Reisner ring of a simplicial complex relate to those invariants in the Stanley–Reisner ring of its expansions.

The purpose of this paper is to examine the freeness of nonzero reflexive modules M and N over a regular local ring R, under the condition is zero.

Let $\mathrm{\Lambda }$ be a finite dimensional Auslander algebra. For a $\mathrm{\Lambda }$-module $N$, we prove that the projective dimension of $N$ is at most one if and only if the projective dimension of its socle soc$N$ is at most one. As an application, we give a new characterization of Auslander algebras $\mathrm{\Lambda }$ and prove that a finite dimensional algebra $\mathrm{\Lambda }$ is an Auslander algebra provided its global dimension gl.d$\mathrm{\Lambda }\le 2$ and an injective $\mathrm{\Lambda }$-module is projective if and only if the projective dimension of its socle is at most one.

By generalizing the notion of the path ideal of a graph, we study some algebraic properties of some path ideals associated to a line graph. We show that the quotient ring of these ideals are always sequentially Cohen–Macaulay and also provide some exact formulas for the projective dimension and the regularity of these ideals. As some consequences, we give some exact formulas for the depth of these ideals.

In this paper, we study the generalized path ideals, which is a new class of path ideals of cycle graphs. These ideals naturally generalize the standard path ideals of cycles, as studied by Alilooee and Faridi [On the resolution of path ideals of cycles, Comm. Algebra43 (2015) 5413–5433]. We give some formulas to compute all the top degree graded Betti numbers of these path ideals of cycle graphs. As a consequence, we can give some formulas to compute their projective dimension and regularity.

Let $(R,𝔪)$ be a Noetherian local ring and $M$, $N$ be two finitely generated $R$-modules. In this paper, it is shown that

In this paper, we provide some precise formulas for regularity of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs. For the projective dimension of such an edge ideal, we give its exact formula. Meanwhile, we also give the upper and lower bounds of projective dimension of higher powers of such an edge ideal. As an application, we present regularity and projective dimension of powers of edge ideal of some gap-free bipartite undirected graphs. Some examples show that these formulas are related to direction selection.

In this paper, we obtain a combinatorial formula for computing the Betti numbers in the linear strand of edge ideals of bipartite Kneser graphs. We deduce lower and upper bounds for regularity of powers of edge ideals of these graphs in terms of associated combinatorial data and show that the lower bound is attained in some cases. Also, we obtain bounds on the projective dimension of edge ideals of these graphs in terms of combinatorial data.

Given a graph $G$ whose edges are labeled by ideals of a commutative ring $R$ with identity, a generalized spline is a vertex labeling of $G$ by the elements of $R$ so that the difference of labels on adjacent vertices is an element of the corresponding edge ideal. The set of all generalized splines on a graph $G$ with base ring $R$ has a ring and an $R$-module structure. In this paper, we focus on the freeness of generalized spline modules over certain graphs with the base ring $R=k[x_1,\dots ,x_d]$ where $k$ is a field. We first show the freeness of generalized spline modules on graphs with no interior edges over $k[x,y]$ such as cycles or a disjoint union of cycles with free edges. Later, we consider graphs that can be decomposed into disjoint cycles without changing the isomorphism class of the syzygy modules. Then we use this decomposition to show that generalized spline modules are free over $k[x,y]$ and later we extend this result to the base ring $R=k[x_1,\dots ,x_d]$ under some restrictions.

Let denote an ideal of a d-dimensional Gorenstein local ring R, and M and N two finitely generated R-modules with pd M < ∞. It is shown that if and only if for all .

The study of the cohomological dimension of algebraic varieties has produced some interesting results and problems in local algebra. Let 𝔞 be an ideal of a commutative Noetherian ring R. For finitely generated R-modules M and N, the concept of cohomological dimension cd_𝔞(M, N) of M and N with respect to 𝔞 is introduced. If 0 → N' → N → N'' → 0 is an exact sequence of finitely generated R-modules, then it is shown that cd_𝔞(M, N) = max{cd_𝔞(M, N'), cd_𝔞(M, N'')} whenever proj dim M < ∞. Also, if L is a finitely generated R-module with Supp(N/Γ_𝔞(N)) ⊆ Supp(L/Γ_𝔞(L)), then it is proved that cd_𝔞(M, N) ≤ max{cd_𝔞(M,L), proj dim M}. Finally, as a consequence, a result of Brodmann is improved.

The groups of local cohomology with supports in the non-free locus of a module are used in order to obtain three classifications and one characterization of four classes of modules.