On the projective dimension and the unmixed part of three cubics

Regularity, Depth and Arithmetic Rank of Bipartite Edge Ideals

We study minimal free resolutions of edge ideals of bipartite graphs. We associate a directed graph to a bipartite graph whose edge ideal is unmixed, and give expressions for the regularity and the depth of the edge ideal in terms of invariants of the directed graph. For some classes of unmixed edge ideals, we show that the arithmetic rank of the ideal equals projective dimension of its quotient.

Bounds on the regularity and projective dimension of ideals associated to graphs

In this paper we give new upper bounds on the regularity of edge ideals whose resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free monomial ideals of codimension two. We also discuss and connect these results to more classical topics in commutative algebra.

The projective dimension of three cubics is at most 5

Let $R$ be a polynomial ring over a field and $I$ an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension $\mathrm{pd}(R/I)$ of $R/I$ is at most 36, although the example with largest projective dimension he constructed has $\mathrm{pd}(R/I)=5$. Based on computational evidence, it had been conjectured that $\mathrm{pd}(R/I)\leq 5$. In the present paper we prove this conjectured sharp bound.

Ext的 annihilator

Annihilator of $Ext^g_R(R/I,R)$

本文研究了Ext的 annihilator 的结构和性质，探讨了其在不同理想类中的包含关系，并给出了相关应用。

Bound on the projective dimension of three cubics

We show that given any polynomial ring R over a field, and any ideal J in R which is generated by three cubic forms, the projective dimension of R/J is at most 36. We also settle the question whether ideals generated by three cubic forms can have projective dimension greater than 4, by constructing one with projective dimension equal to 5.

三次循环图边理想的代数不变量

Algebraic invariants of edge ideals of cubic circulant graphs

本文研究所有三次循环图对应边理想的商环，给出其深度与投射维数的精确值，并建立 Stanley 深度的下界。