If M is a graded module over a ring S, then the S2ification of M is \sum_{d \in ZZ} H^0((sheaf M)(d)), which may be computed as lim_{d>\infty} Hom(I_d,M), where I_d is any sequence of ideals contained in higher and higher powers of S_+. There is a natural restriction map f: M = Hom(S,M) \to Hom(I_d,M). We compute all this using the ideals I_d generated by the dth powers of the variables in S.
Since the result may not be finitely generated (this happens if and only if M has an associated prime of dimension 1), we compute only up to a specified degree bound b. For the result to be correct down to degree b, it is sufficient to compute Hom(I,M) where I \subset (S_+)^{rb}.








At one time DE hoped that, if M were a module over the complete intersection R with residue field k, then the natural map from "complete" Ext module "(widehat Ext)_R(M,k)" to the S2ification of Ext_R(M,k) would be surjective; equivalently, if N were a sufficiently negative syzygy of M, then the first local cohomology module of Ext_R(M,k) would be zero. This is false, as shown by the following example:
















Text S2ification is related to computing cohomology and to computing integral closure; there are scripts in those packages that produce an S2ification, but one takes a ring as argument and the other doesn't produce the comparison map.
The object S2 is a method function.