ABSTRACT
Liaison relates the cohomology of the ideal sheaf of a scheme to the cohomology of the canonical module of its link. We here refer to Gorenstein liaison in a projective space over a field: each ideal is the residual of the other in one Gorenstein homogeneous ideal of a polynomial ring. Assuming that the linked schemes (or equivalently one of them) are Cohen-Macaulay, Serre duality expresses the cohomology of the canonical module in terms of the cohomology of the ideal sheaf. Therefore, in the case of Cohen-Macaulay linked schemes, the cohomology of ideal sheaves can be computed one from another: up to shifts in ordinary and homological degrees, they are exchanged and dualized. In terms of free resolutions this means that, up to a degree shift, they may be obtained one from another by dualizing the corresponding complexes (for instance, the generators of one cohomology module corresponds to the last syzygies of another cohomology module
of the link). If the linked schemes are not Cohen-Macaulay, this property fails. Neverthe-
less, experience on a computer shows that these modules are closely related. We here investigate this relation for surfaces and three-dimensionnal schemes. To describe our results, notice that the graded duals of the local cohomology modules are Ext modules (into the polynomial ring or the Gorenstein quotient that provides the linkage), let us set — for the graded dual and DiM : Hi
M, the i-th deficiency module when i dimM, while ωM DdimMM.
