# auslanderInvariant -- measures failure of surjectivity of the essential MCM approximation

## Synopsis

• Usage:
a = auslanderInvariant M
• Inputs:
• M, ,
• Optional inputs:
• CoDepth => ..., default value -1
• Outputs:

## Description

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