Abed Abedelfatah
Among other results, we prove that if $I$ is a monomial ideal of $S=K[x_1,\ldots,x_n]$, where $K$ is a field, and $a\geq b-1\geq0$ are integers such that $a+b\leq\mathrm{proj~dim}(S/I)$, then $$t_{a+b}\leq t_a+t_1+t_2+\cdots+t_b-\frac{b(b-1)}{2},$$ where $t_1,t_2,\dots$ are the maximal shifts in the minimal graded free $S$-resolution of $S/I$.
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We prove the subadditivity property for the maximal degrees of the syzygies of facet ideals simplicial forests. For such an ideal $I$, if the $i$-th Betti number is nonzero and $i=a+b$, we show that there are monomials in the lcm lattice of $I$ that are complements in part of the lattice, each supporting a nonvanishing $a$-th and $b$-th Betti numbers. The subadditivity formula follows from this observation.
We consider the following question: if a simplicial complex $Δ$ has $d$-homology, then does the corresponding $d$-cycle always induce cycles of smaller dimension that are not boundaries in $Δ$? We provide an answer to this question in a fixed dimension. We use the breaking of homology to show the subadditivity property for the maximal degrees of syzygies of monomial ideals in a fixed homological degree.
Let $R=S/I$ be a graded algebra with $t_i$ and $T_i$ being the minimal and maximal shifts in the minimal $S$ resolution of $R$ at degree $i$. In this paper we prove that $t_n\leq t_1+T_{n-1}$, for all $n$ and as a consequence, we show that for Gorenstein algebras of codimension $h$, the subadditivity of maximal shifts $T_i$ in the minimal resolution holds for $i \geq h-1$, i.e, we show that $T_i \leq T_a+T_{i-a}$ for $i\geq h-1$.
Estimates are obtained for the degrees of minimal syzygies of quotient algebras of polynomial rings. For a class that includes Koszul algebra in almost all characteristics, these degrees are shown to increase by at most 2 from one syzygy module to the next one. Even slower growth is proved if, in addition, the algebra satisfies Green and Lazarsfeld's condition N_q with q > 1.
Well ordered covers of square-free monomial ideals are subsets of the minimal generating set ordered in a certain way that give rise to a Lyubeznik resolution for the ideal, and have guaranteed nonvanishing Betti numbers in certain degrees. This paper is about square-free monomial ideals which have a well ordered cover. We consider the question of subadditivity of syzygies of square-free monomial ideals via complements in the lcm lattice of the ideal, and examine how lattice complementation breaks well ordered covers of the ideal into (well ordered) covers of subideals. We also introduce a fam…
On the subadditivity condition of edge ideal
本文研究边理想商代数分次自由消解中最大移位的次可加性，证明在移位阈值条件及 $b=4$ 情形下的不等式，并推出投射维数不超过 $9$ 的边理想均满足该条件。
We study families of cellular resolutions by looking at them as a category and applying tools from representation stability. We obtain sufficient conditions on the structure of the family to have a noetherian representation category and apply this to concrete examples of families. In the study of syzygies we make use of defining the syzygy module as a representation and find conditions for the finite generation of this representation. We then show that many families of cellular resolutions coming from powers of ideals satisfy these conditions and have finitely generated syzygies, including the…
Let $R = \mathbb{K}[x_1, \ldots, x_n]$ and $I \subset R$ be a homogeneous ideal. In this article, we first obtain certain sufficient conditions for the subadditivity of $R/I$. As a consequence, we prove that if $I$ is generated by homogeneous complete intersection, then subadditivity holds for $R/I$. We then study a conjecture of Avramov, Conca and Iyengar on subadditivity, when $I$ is a monomial ideal with $R/I$ Koszul. We identify several classes of edge ideals of graphs $G$ such that the subadditivity holds for $R/I(G)$. We then study the strand connectivity of edge ideals and obtain severa…
Linear syzygies of curves in weighted projective space
本文将Green的$N_p$条件推广到加权射影空间，证明曲线高次嵌入满足这些条件，并在广泛情形下建立了正性与低次虚拟合冲之间的联系。
Vector space summands of lower syzygies
本文研究局部环剩余域作为合冲模直和项的条件，证明了第二、第三次合冲模直和项与极大理想直和项的等价性，并利用 Eliahou-Kervaire 预解式构造了一类使内射包络对偶为向量空间的阿廷局部环。
Under certain effective positivity conditions, we show that the secant variety to a smooth variety satisfies $N_{3,p}$. For smooth curves, we provide the best possible effective bound on the degree $d$ of the embedding, $d\geq 2g+3+p$.
Syzygies, constant rank, and beyond
本文研究了常系数线性PDE组中常秩条件与primary decomposition概念间的联系，并在PDE无常秩时，推进了PDE约束场序列上积分泛函弱下半连续性的研究。
Let $F$ be a finite set of monomials of the same degree $d\geq 2$ in a polynomial ring $R=k[x_1,...,x_n]$ over an arbitrary field $k$. We give some necessary and/or sufficient conditions for the birationality of the ring extension $k[F]\subset R^{(d)}$, where $R^{(d)}$ is the $d${\it th} Veronese subring of $R$. One of our results extends to arbitrary characteristic, in the case of rational monomial maps, a previous syzygy-theoretic birationality criterion in characteristic zero
On the first non-trivial strand of syzygies of projective schemes and Condition ${\mathrm ND}(l)$
本文研究射影概型合冲第一非平凡链，引入推广非退化的条件 ND(ℓ)，证明分次 Betti 数尖锐上界，并刻画 d-线性算术 Cohen-Macaulay 分解，推广合冲刚性定理至一般 d-正则性。
Weighted Syzygies of Pointed Curves
针对亏格g光滑射影曲线上的点P，研究非次数1生成的截面环R_d的合冲，界定其生成元次数、Betti表支撑界，证明满足加权N_p条件的d阈值，给出Betti数确定的充分条件并得普通点R_{g+1}分解为纯的。
Previously, Ohsugi and Hibi gave a combinatorial description of bipartite graphs $G$ whose toric edge ideal $I_G$ is generated by quadrics, showing that every cycle of $G$ of length at least $6$ must have a chord. This corresponds to the Green-Lazarsfeld condition $\mathbf{N}_1$. In this paper, we investigate the higher syzygies of $I_G$ and give combinatorial descriptions of the Green-Lazarsfeld conditions $\mathbf{N}_p$ of toric edge ideals of bipartite graphs for all $p \ge 1$. In particular, we show that $I_G$ is linearly presented (i.e. satisfies condition $\mathbf{N}_2$) if and only if t…
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