On Stably free modules over Laurent polynomial rings

We prove constructively that for any finite-dimensional commu- tative ring R, every stably free module over R[X;X^{1}] of rank > dim R is free, i.e., R[X;X^{-1}] is (dimR)-Hermite.Comment:

On the Eisenbud-Green-Harris Conjecture

It has been conjectured by Eisenbud, Green and Harris that if $I$ is a homogeneous ideal in $k[x_1,...,x_n]$ containing a regular sequence $f_1,...,f_n$ of degrees $\deg(f_i)=a_i$, where $2\leq a_1\leq ... \leq a_n$, then there is a homogeneous ideal $J$ containing $x_1^{a_1},...,x_n^{a_n}$ with the same Hilbert function. In this paper we prove the Eisenbud-Green-Harris conjecture when $f_i$ splits into linear factors for all $i$

Macaulay–Lex rings

AbstractWe present simple proofs of Macaulayʼs theorem and Clements–Lindströmʼs theorem. We generalize Shakinʼs theorem by proving that a stable ideal I of S is Macaulay–Lex if and only if I is a piecewise lexsegment ideal. We also study Macaulay–Lex ideals of the form 〈x1e1,x1t1x2e2,x1t1x2t2x3e3,…,x1t1⋯xn−1tn−1xnen〉, where 2⩽e1⩽⋯⩽en⩽∞ and ti<ei for all i, and generalize Clements–Lindströmʼs theorem

On the subadditivity condition of edge ideal

Let $S=K[x_1,\ldots,x_n]$, where $K$ is a field, and $t_i(S/I)$ denotes the maximal shift in the minimal graded free $S$-resolution of the graded algebra $S/I$ at degree $i$, where $I$ is an edge ideal. In this paper, we prove that if $t_b(S/I)\geq \lceil \frac{3b}{2} \rceil$ for some $b\geq 0$, then the subadditivity condition $t_{a+b}(S/I)\leq t_a(S/I)+t_b(S/I)$ holds for all $a\geq 0$. In addition, we prove that $t_{a+4}(S/I)\leq t_a(S/I)+t_4(S/I)$ for all $a\geq 0$ (the case $b=0,1,2,3$ is known). We conclude that if the projective dimension of $S/I$ is at most $9$, then $I$ satisfies the subadditivity condition.Comment: 8 page