# makeS2 -- compute the S2ification of a reduced ring

## Synopsis

• Usage:
(F,G) = makeS2 R
• Inputs:
• R, a ring, an equidimensional reduced (or just unmixed and genericaly Gorenstein) affine ring
• Optional inputs:
• Verbosity => an integer, default value 0, larger values give more information.
• Variable => ..., default value "w", Sets the name of the indexed variables introduced in computing the S2-ification.
• Outputs:
• F, , $R \rightarrow S$, where $S$ is the so-called S2-ification of $R$
• G, , $frac S \rightarrow frac R$, giving the corresponding fractions

## Description

A ring $S$ satisfies Serre's S2 condition if every codimension 1 ideal contains a nonzerodivisor and every principal ideal generated by a nonzerodivisor is equidimensional of codimension one. If $R$ is an affine reduced ring, then there is a unique smallest extension $R\subset S\subset {\rm frac}(R)$ satisfying S2, and $S$ is finite as an $R$-module.

Uses the method of Vasconcelos, "Computational Methods..." p. 161, taking the idealizer of a canonical ideal.

There are other methods to compute $S$, not currently implemented in this package. See for example the function (S2,Module) in the package "CompleteIntersectionResolutions".

We compute the S2-ification of the rational quartic curve in $P^3$

 i1 : A = ZZ/101[a..d]; i2 : I = monomialCurveIdeal(A,{1,3,4}) 3 2 2 2 3 2 o2 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c) o2 : Ideal of A i3 : R = A/I; i4 : (F,G) = makeS2 R ZZ ---[w , a..d] 101 0,0 o4 = (map (------------------------------------------------------------------ 2 2 2 (b*c - a*d, w d - c , w c - b*d, w b - a*c, w a - b , w 0,0 0,0 0,0 0,0 0,0 ------------------------------------------------------------------------ -------, R, {a, b, c, d}), map (frac R, - a*d) ------------------------------------------------------------------------ / ZZ | ---[w , a..d] | 101 0,0 frac|------------------------------------------------------------------- | 2 2 2 |(b*c - a*d, w d - c , w c - b*d, w b - a*c, w a - b , w \ 0,0 0,0 0,0 0,0 0,0 ------------------------------------------------------------------------ \ | | b*d ------|, {---, a, b, c, d})) | c - a*d)| / o4 : Sequence

## Caveat

Assumes that first element of canonicalIdeal R is a nonzerodivisor; else returns error. The return value of this function is likely to change in the future