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

Synopsis

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

See also

Ways to use auslanderInvariant :

For the programmer

The object auslanderInvariant is a method function with options.