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
|
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
The object makeS2 is a method function with options.