# minimize(Complex) -- a quasi-isomorphic complex whose terms have minimal rank

## Synopsis

• Function: minimize
• Usage:
D = minimize C
• Inputs:
• C, , graded, whose terms are all free modules
• Outputs:
• D, , graded, whose terms are all free modules of minimal rank
• Consequences:
• The projection morphism $g : C \to D$ is available as g = D.cache.minimizingMap.

## Description

This method essentially removes all scalar units from the matrices in the differential of $C$.

We illustrate this in a simple example.

 i1 : S = ZZ/32003[a,b]; i2 : I = ideal(a^2-b^2, a*b) 2 2 o2 = ideal (a - b , a*b) o2 : Ideal of S i3 : C = freeResolution(I, FastNonminimal=>true) 1 3 2 o3 = S <-- S <-- S 0 1 2 o3 : Complex i4 : betti C 0 1 2 o4 = total: 1 3 2 0: 1 . . 1: . 2 1 2: . 1 1 o4 : BettiTally i5 : D = minimize C 1 2 1 o5 = S <-- S <-- S 0 1 2 o5 : Complex i6 : assert(isWellDefined D and isHomogeneous D) i7 : betti D 0 1 2 o7 = total: 1 2 1 0: 1 . . 1: . 2 . 2: . . 1 o7 : BettiTally i8 : g = D.cache.minimizingMap 1 1 o8 = 0 : S <--------- S : 0 | 1 | 2 3 1 : S <------------------ S : 1 {2} | 1 0 -b | {2} | 0 1 a | 1 2 2 : S <--------------- S : 2 {4} | 0 1 | o8 : ComplexMap i9 : assert isWellDefined g i10 : assert(isComplexMorphism g and isQuasiIsomorphism g) i11 : assert(source g == C) i12 : assert(target g == D) i13 : assert(coker g == 0)

The minimal complex $D$ is a direct summand of the original complex $C$. The natural inclusion of $D$ into $C$ can be constructed as follows.

 i14 : f = liftMapAlongQuasiIsomorphism(id_D, g) 1 1 o14 = 0 : S <--------- S : 0 | 1 | 3 2 1 : S <--------------- S : 1 {2} | 1 0 | {2} | 0 1 | {3} | 0 0 | 2 1 2 : S <------------- S : 2 {3} | a | {4} | 1 | o14 : ComplexMap i15 : g*f == id_D o15 = true i16 : assert(source f == D) i17 : assert(target f == C) i18 : assert(ker f == 0) i19 : f*g 1 1 o19 = 0 : S <--------- S : 0 | 1 | 3 3 1 : S <------------------ S : 1 {2} | 1 0 -b | {2} | 0 1 a | {3} | 0 0 0 | 2 2 2 : S <--------------- S : 2 {3} | 0 a | {4} | 0 1 | o19 : ComplexMap

The chain complex $D$ is a direct summand of $C$, giving rise to a split short exact sequence of chain complexes.

 i20 : h = prune canonicalMap(C, ker g) 3 1 o20 = 1 : S <-------------- S : 1 {2} | -b | {2} | a | {3} | -1 | 2 1 2 : S <------------- S : 2 {3} | 1 | {4} | 0 | o20 : ComplexMap i21 : assert isShortExactSequence(g, h)

Warning: If the input complex is not homogeneous, then the output is probably not what one would expect.

 i22 : S = ZZ/32003[a..d] o22 = S o22 : PolynomialRing i23 : J = ideal(a*b*c-b*c, a*d-c, a^3-d^2*c) 3 2 o23 = ideal (a*b*c - b*c, a*d - c, a - c*d ) o23 : Ideal of S i24 : CJ = freeResolution J 1 4 4 1 o24 = S <-- S <-- S <-- S 0 1 2 3 o24 : Complex i25 : assert not isHomogeneous CJ i26 : D = minimize CJ 1 3 1 o26 = S <-- cokernel | 0 | <-- S <-- S | 0 | 0 | -c+d | 2 3 | a-1 | 1 o26 : Complex i27 : isWellDefined D o27 = true i28 : prune HH D == prune HH CJ o28 = true