Projective Resolutions of Generic Order Ideals

Kim, Saeja Oh

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https://hdl.handle.net/2142/71265 Copy

Description

Title

Projective Resolutions of Generic Order Ideals

Author(s)

Kim, Saeja Oh

Issue Date

1988

Doctoral Committee Chair(s)

Grayson, Daniel,

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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$\sb{\rm 2n}$).

Graduation Semester

1988

Type of Resource

text

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http://hdl.handle.net/2142/71265

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