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Title:Projective Resolutions of Generic Order Ideals
Author(s):Kim, Saeja Oh
Doctoral Committee Chair(s):Grayson, Daniel,
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Mathematics
Abstract: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$\sbsp{\rm n}{\rm (n+1)}\}$.
In this thesis, we produce a minimal free resolution of R$\sb{\rm 2n}$/I$\sb{\rm 2n}$. From this resolution, we read that I$\sb{\rm 2n}$ is an almost perfect ideal (i.e. pd(R$\sb{\rm 2n}$/I$\sb{\rm 2n}$) = grade(I$\sb{\rm 2n}$) + 1), Ext$\sbsp{\rm R\sb{2n}}{\rm 2n}$(R$\sb{\rm 2n}$/I$\sb{\rm 2n}$,R$\sb{\rm 2n}$) = R$\sb{\rm 2n}$/(Y$\sb1,\dots$,Y$\sb{\rm 2n}$), (Y$\sb1,\dots$,Y$\sb{\rm 2n}$) $\in$ Ass(R$\sb{\rm 2n}$/I$\sb{\rm 2n}$) and J$\sb{\rm 2n}$ $\in$ Ass(R$\sb{\rm 2n}$/I$\sb{\rm 2n}$).
Issue Date:1988
Type:Text
Description:107 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988.
URI:http://hdl.handle.net/2142/71265
Other Identifier(s):(UMI)AAI8823170
Date Available in IDEALS:2014-12-16
Date Deposited:1988


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