Seyedamin Seyedfakhari

Suppose that M is a nonzero finitely generated graded S-module with minimal free resolutionThe Castelnuovo-Mumford regularity (or simply, regularity) of M, denoted by reg(M), is defined asThe regularity of M is an important invariant in commutative algebra and algebraic geometry. Several researchers have studied the regularity of squarefree monomialideals (see e.g., [3], [8], [13] and [26]).Cutkosky, Herzog, Trung, [7], and independently Kodiyalam [19], proved that for a homogenous ideal I in the polynomial ring, the function is linear, for s ≫ 0, i.e., there exist integers a(I), b(I) and such that It is known that a(I) is bounded above by the maximum degree of elements in a minimal generating set of I. But a general combinatorial bound for b(I) and is not known, even if I is monomial ideal. The first general aim of this project is to extend the information about b(I) and when I is the edge ideal of a graph. The next aim of this project is to study the regularity of symbolic powers of edge ideals. In particular, we investigate a conjecture due to N. C. Minh, which states that for each positive integer s and for each graph G, the regularity of the s-th power of I(G) is equal to the regularity of the s-th symbolic power of I(G).We also study the depth of (symbolic) powers of edge ideals. We know from the celebrated Auslander-Buchsbaum formula that computing the depth of a homogenous ideal I ⊆ S is equivalent to computing its projective dimension. Several researchers have studied the depth of edge ideals and their powers (see for instance, [8], [11], [18], [28]). Trung [28] proves that for every graph G and for every integer k ≫ 0, the depth of is equal to the number of bipartite connected components of G. On the other hand, we know from [14] that for every graph G, the depth of is constant for large k. We are going to study this constant number. We also, try to study the depth of small (symbolic) powers of edge ideals and its relation to the combinatorial invariants of the underlying graph.

Suppose that M is a nonzero finitely generated graded S-module with minimal free resolutionThe Castelnuovo-Mumford regularity (or simply, regularity) of M, denoted by reg(M), is defined asThe regularity of M is an important invariant in commutative algebra and algebraic geometry. Several researchers have studied the regularity of squarefree monomialideals (see e.g., [3], [8], [13] and [26]).Cutkosky, Herzog, Trung, [7], and independently Kodiyalam [19], proved that for a homogenous ideal I in the polynomial ring, the function is linear, for s ≫ 0, i.e., there exist integers a(I), b(I) and such that It is known that a(I) is bounded above by the maximum degree of elements in a minimal generating set of I. But a general combinatorial bound for b(I) and is not known, even if I is monomial ideal. The first general aim of this project is to extend the information about b(I) and when I is the edge ideal of a graph. The next aim of this project is to study the regularity of symbolic powers of edge ideals. In particular, we investigate a conjecture due to N. C. Minh, which states that for each positive integer s and for each graph G, the regularity of the s-th power of I(G) is equal to the regularity of the s-th symbolic power of I(G).We also study the depth of (symbolic) powers of edge ideals. We know from the celebrated Auslander-Buchsbaum formula that computing the depth of a homogenous ideal I ⊆ S is equivalent to computing its projective dimension. Several researchers have studied the depth of edge ideals and their powers (see for instance, [8], [11], [18], [28]). Trung [28] proves that for every graph G and for every integer k ≫ 0, the depth of is equal to the number of bipartite connected components of G. On the other hand, we know from [14] that for every graph G, the depth of is constant for large k. We are going to study this constant number. We also, try to study the depth of small (symbolic) powers of edge ideals and its relation to the combinatorial invariants of the underlying graph.