Even Linkage Classes

In this paper we generalize the $\mathcal {E}$ and $\mathcal {N}$-type resolutions used by Martin-Deschamps and Perrin for curves in $\mathbb {P}^{3}$ to subschemes of pure codimension in projective space, and shows that these resolutions are interchanged by the mapping cone procedure under a simple linkage. Via these resolutions, Rao’s correspondence is extended to give a bijection between even linkage classes of subschemes of pure codimension two and stable equivalence classes of reflexive sheaves $\mathcal {E}$ satisfying $H^{1}_{*}( \mathcal {E})=0$ and $\mathcal {E}xt^{1}( \mathcal {E}^{\vee }, \mathcal {O})=0$. Further, these resolutions are used to extend the work of Martin-Deschamps and Perrin for Cohen-Macaulay curves in $\mathbb {P}^{3}$ to subschemes of pure codimension two in $\mathbb {P}^{n}$. In particular, even linkage classes of such subschemes satisfy the Lazarsfeld-Rao property and any minimal subscheme for an even linkage class links directly to a minimal subscheme for the dual class.