# complexity -- complexity of a module over a complete intersection

## Synopsis

• Usage:
c = complexity M
c = complexity mf
• Inputs:
• M, , module over a complete intersection
• mf, a list, output of a matrix factorization computation
• Outputs:
• c, an integer, 1+dimension of Ext(M,k) over the ring of CI operators

## Description

The minimal resolution of a module over a complete intersection has betti numbers that grow as a polynomial of degree at most equal to the codimension-1. The complexity is one more than the degree of this polynomial.

 i1 : setRandomSeed 0 o1 = 0 i2 : S = ZZ/101[a,b,c,d]; i3 : ff1 = matrix"a3,b3,c3,d3"; 1 4 o3 : Matrix S <--- S i4 : ff =ff1*random(source ff1, source ff1); 1 4 o4 : Matrix S <--- S i5 : R = S/ideal ff; i6 : M = highSyzygy (R^1/ideal"a2b2"); i7 : complexity M o7 = 2 i8 : mf = matrixFactorization (ff, M) o8 = {{7} | -a -36b 0 a |, {8} | 35a2 48b 0 -33b 0 |, {6} | 0 36 {6} | b2 a2 0 0 | {8} | -35b2 -35a 0 0 0 | {7} | -36 0 {7} | 0 0 b a | {8} | 0 0 33b2 33a -33b2 | {7} | 1 0 {8} | 0 0 -43a2 -33b 0 | ------------------------------------------------------------------------ 0 |} 36 | 0 | o8 : List i9 : complexity mf o9 = 2 i10 : betti res (R^1/ideal"a2b2", LengthLimit=>10) 0 1 2 3 4 5 6 7 8 9 10 o10 = total: 1 1 2 3 4 5 6 7 8 9 10 0: 1 . . . . . . . . . . 1: . . . . . . . . . . . 2: . . . . . . . . . . . 3: . 1 2 1 . . . . . . . 4: . . . 2 4 2 . . . . . 5: . . . . . 3 6 3 . . . 6: . . . . . . . 4 8 4 . 7: . . . . . . . . . 5 10 o10 : BettiTally