If R is a Gorenstein local ring and M is an R-module, then the essential MCM approximation is a map phi: M'-->M, where M' is an MCM R-module, obtained as the k-th cosyzygy of the k-th syzygy of M, where k >= the co-depth of M. The Auslander invariant is the number of generators of coker phi. Thus if R is regular the Auslander invariant is just the minimal number of generators of M, and if M is already an MCM module with no free summands then the Auslander invariant is 0.
Ding showed that if R is a hypersurface ring, then auslanderInvariant (R^1)/((ideal vars R)^i) is zero precisely for i<multiplicity R.
Experimentally, it looks as if for a complete intersection the power is the a-invariant plus 1, but NOT for the codim 3 Pfaffian example.
i1 : R = ZZ/101[a..d]/ideal"a3" o1 = R o1 : QuotientRing |
i2 : apply(5, i -> auslanderInvariant ((R^1)/(ideal(vars R))^(i+1)))
o2 = {0, 0, 1, 1, 1}
o2 : List
|
i3 : R = ZZ/101[a..d]/ideal"a3,b4" o3 = R o3 : QuotientRing |
i4 : apply(6, i -> auslanderInvariant ((R^1)/(ideal(vars R))^(i+1)))
o4 = {0, 0, 0, 0, 0, 1}
o4 : List
|
i5 : S = ZZ/101[a,b,c] o5 = S o5 : PolynomialRing |
i6 : N = matrix{{0,a,0,0,c},
{0,0,b,c,0},
{0,0,0,a,0},
{0,0,0,0,b},
{0,0,0,0,0}}
o6 = | 0 a 0 0 c |
| 0 0 b c 0 |
| 0 0 0 a 0 |
| 0 0 0 0 b |
| 0 0 0 0 0 |
5 5
o6 : Matrix S <--- S
|
i7 : M = N-transpose N
o7 = | 0 a 0 0 c |
| -a 0 b c 0 |
| 0 -b 0 a 0 |
| 0 -c -a 0 b |
| -c 0 0 -b 0 |
5 5
o7 : Matrix S <--- S
|
i8 : J = pfaffians(4,M)
2 2 2
o8 = ideal (a , b*c, a*b + c , a*c, b )
o8 : Ideal of S
|
i9 : R = S/J o9 = R o9 : QuotientRing |
i10 : I = ideal vars R o10 = ideal (a, b, c) o10 : Ideal of R |
i11 : scan(5, i->print auslanderInvariant ((R^1)/(I^i))) 0 0 0 1 1 |
The object auslanderInvariant is a method function with options.