minimize -- minimal quotient complex of a free ChainComplex

Synopsis

Usage:

m = minimize F
Inputs:
- F, a chain complex, chain complex of free modules
Outputs:
- m, a chain complex map, quasi-isomorphism F -> F', where F' is a minimal free complex

Description

For the quasi-isomorphism from a minimal subcomplex use

dual minimize dual F

To simplify the notation consider the complex C = E[min E] that is shifted so that the first module is C_0. The algorithm: Set dbar = the reduction of the differential d mod the maximal ideal. a complement of ker dbar, and compute the idempotent rho: E -> E. the map rho is not a chain complex map, but the image of (rho | d*rho): C ++ C[1] --> C is a subcomplex and the minimization of C is the complex C/image(rho|d*rho). The script returns the ChainComplexMap from the minimization to C.

To illustrate we first make a nonminimal complex by adding trivial complexes to a minimal complex and then mixing things up by conjugating with general isomorphisms:

i1 : S = ZZ/32003[a,b,c]

o1 = S

o1 : PolynomialRing

i2 : red = map(S,S,toList(numgens S:0_S))

o2 = map (S, S, {0, 0, 0})

o2 : RingMap S <-- S

i3 : C = koszul gens (ideal vars S)^2

      1      6      15      20      15      6      1
o3 = S  <-- S  <-- S   <-- S   <-- S   <-- S  <-- S
                                                   
     0      1      2       3       4       5      6

o3 : ChainComplex

i4 : G = S^{0,-1,-2,-3,-4,-5,-6}

      7
o4 = S

o4 : S-module, free, degrees {0..6}

i5 : D = apply(length C+1, i-> C_i++G++G)

       15   20   29   34   29   20   15
o5 = {S  , S  , S  , S  , S  , S  , S  }

o5 : List

i6 : zG = map(G,G,0)

o6 = 0

             7      7
o6 : Matrix S  <-- S

i7 : difs0 = apply(length C, i-> (map(D_i, D_(i+1), matrix{{C.dd_(i+1), map(C_i,G,0), map(C_i,G,0)},{map(G,C_(i+1),0), zG, zG},{map(G,C_(i+1),0), id_G, zG}})));

i8 : len = #difs0

o8 = 6

i9 : Q = apply(len, i-> random(target difs0_i, target difs0_i))|
       {random(source difs0_(len-1), source difs0_(len-1))};

i10 : difs1 = apply(len, i-> Q_i*difs0_i*Q_(i+1)^(-1));

i11 : E = chainComplex difs1

       15      20      29      34      29      20      15
o11 = S   <-- S   <-- S   <-- S   <-- S   <-- S   <-- S
                                                       
      0       1       2       3       4       5       6

o11 : ChainComplex

i12 : isMinimalChainComplex E

o12 = false

Now we minimize the result. The free summand we added to the end maps to zero, and thus is part of the minimization.

i13 : time m = minimize (E[1]);
     -- used 0.322958 seconds

i14 : isQuasiIsomorphism m

o14 = true

i15 : E[1] == source m

o15 = true

i16 : E' = target m

       8      6      15      20      15      6      8
o16 = S  <-- S  <-- S   <-- S   <-- S   <-- S  <-- S
                                                    
      -1     0      1       2       3       4      5

o16 : ChainComplex

i17 : isChainComplex E'

o17 = true

i18 : isMinimalChainComplex E'

o18 = true

Ways to use minimize :

minimize(ChainComplex)

For the programmer

The object minimize is a method function.