Structure of finite free resolutions

Abstract: Codimension $2$ Cohen Macaulay varieties (respectively codimension $3$ Gorenstein varieties) come from rank condition on $n\times (n+1)$ matrices (respectively a skew symmetric matrix). Weyman related problem of codimension $3$ varieties with the classification of semi-simple Lie algebra. In the first part of the talk, for Dynkin type $E_6$, we define an interesting family of perfect ideals of codimension $3$, with $5$ generators, of Cohen Macaulay type $2$ with the trivial multiplication on Tor algebra. In the second part of the talk, we explore spinor structure on free resolutions of codimension $4$ Gorenstein ideals. For such ideals with $4$, $6$, $7$, $8$ and $9$ generators, we present the minimal number of generators of ideals among spinor coordinates.