Projective Resolutions of Generic Order Ideals

Let Y$\sp{\rm (n)}$ be the 1 x n matrix containing indeterminate entries $\{$Y$\sb1,\dots$,Y$\sb{\rm n}\}$ and X$\sp{\rm (n)}$ be the n x n alternating matrix containing indeterminate entries $\{$X$\sb{\rm ij}\vert$1 $\leq$ i $<$ j $\leq$ n$\}$, where we adopt the convention that X$\sb{\rm ii}$ = 0 and X$\sb{\rm ji}$ = $-$ X$\sb{\rm ij}$. Let $\{$g$\sbsp{1}{\rm (n)},\dots$,g$\sbsp{\rm n}{\rm (n)}\}$ be the entries of the product Y$\sp{\rm (n)}$X$\sp{\rm (n)}$. Let R$\sb{\rm n}$ be the ring $\doubz$ (X$\sb{\rm ij}$,Y$\sb1,\dots$,Y$\sb{\rm n}\rbrack\sb{\rm 1 \leq i < j \leq n}$. Then $\{$g$\sbsp{1}{\rm (n)},\dots$,g$\sbsp{\rm n-1}{\rm (n)}\}$ is a regular sequence and $\sum\sbsp{\rm i=1}{\rm n}$ Y$\sb{\rm i}$g$\sbsp{\rm i}{\rm (n)}$ = 0. Let I$\sb{\rm n}$ be the ideal of R$\sb{\rm n}$ generated by $\{$g$\sbsp{1}{\rm (n)},\dots$,g$\sbsp{\rm n}{\rm (n)}\}$ and J$\sb{\rm n}$ be the ideal of R$\sb{\rm n}$ generated by $\{$g$\sbsp{1}{\rm (n)},\dots$,g$\sbsp{\rm n}{\rm (n)},(-1)\sp{\rm n+1}$pf(X$\sp{\rm (n)})\}$. Since the pfaffian of X$\sp{\rm (2m+1)}$ is zero, J$\sb{\rm 2m+1}$ = I$\sb{\rm 2m+1}$. I$\sb{\rm n}$ is the generic order ideal of the second syzygy M of the Koszul complex resolution of $\doubz$ (Y$\sb1,\dots$,Y$\sb{\rm n}$) /(Y$\sb1,\dots$,Y$\sb{\rm n}$) for n $\geq$ 3, and is the ideal of relations of Sym$\sb{\doubz\lbrack\rm Y\sb1,\dots,Y\sb{n}\rbrack}(\Lambda\sp{\rm n-2}(\doubz$ (Y$\sb1,\dots$,Y$\sb{\rm n}\rbrack)\sp{\rm n}$/M*). Since I$\sb{\rm n}$ has grade n $-$ 1 and is generated by n elements, it is an almost complete intersection. Huneke and Ulrich showed that J$\sb{\rm n}$ is a perfect prime ideal of grade n $-$ 1 and the ideals J$\sb{\rm n+1}$ and (J$\sb{\rm n}$,Y$\sb{\rm n+1}$) are linked by the regular sequence $\{$g$\sbsp{1}{\rm (n+1)},\dots$,g$\sbsp{\rm n}{\rm (n+1)}\}$.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Kim, Saeja Oh

Contributors dc:contributor

• Grayson, Daniel,

• Mathematics

Identifier

(UMI)AAI8823170

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/71265