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ZZdFactorization == ZZdFactorization -- whether two ZZ/d-graded factorizations are equal

Description

Two factorizations are equal if the corresponding objects and corresponding maps at each index are equal.

i1 : S = ZZ/101[a..c]/(a^2+b^2+c^2)

o1 = S

o1 : QuotientRing
 
i2 : C = tailMF coker vars S

       ZZ       4       ZZ       4       ZZ       4
o2 = (---[a..c])  <-- (---[a..c])  <-- (---[a..c])
      101              101              101
                                        
     0                1                0

o2 : ZZdFactorization
 
i3 : C.dd

           ZZ       4                           ZZ       4
o3 = 1 : (---[a..c])  <---------------------- (---[a..c])  : 0
          101            {3} | a  -b c  0 |    101
                         {3} | b  a  0  c |
                         {3} | -c 0  a  b |
                         {3} | 0  -c -b a |

           ZZ       4                           ZZ       4
     0 : (---[a..c])  <---------------------- (---[a..c])  : 1
          101            {2} | a  b -c 0  |    101
                         {2} | -b a 0  -c |
                         {2} | c  0 a  -b |
                         {2} | 0  c b  a  |

o3 : ZZdFactorizationMap
 
i4 : D = C[3][-3]

       ZZ       4       ZZ       4       ZZ       4
o4 = (---[a..c])  <-- (---[a..c])  <-- (---[a..c])
      101              101              101
                                        
     0                1                0

o4 : ZZdFactorization
 
i5 : C === D

o5 = false
 
i6 : C == D

o6 = true

Both the maps and the objects must be equal.

i7 : E = ZZdfactorization {0*id_(C_0), 0*id_(C_1)}

       ZZ       4       ZZ       4       ZZ       4
o7 = (---[a..c])  <-- (---[a..c])  <-- (---[a..c])
      101              101              101
                                        
     0                1                0

o7 : ZZdFactorization
 
i8 : dd^E

           ZZ       4          ZZ       4
o8 = 1 : (---[a..c])  <----- (---[a..c])  : 0
          101            0    101

           ZZ       4          ZZ       4
     0 : (---[a..c])  <----- (---[a..c])  : 1
          101            0    101

o8 : ZZdFactorizationMap
 
i9 : C == E

o9 = false
 
i10 : E == 0

o10 = false
 
i11 : dd^E == 0

o11 = true

A complex is equal to zero if all the objects and maps are zero. This could require computation to determine if something that is superficially not zero is in fact zero.

i12 : f = id_C

            ZZ       4                        ZZ       4
o12 = 0 : (---[a..c])  <------------------- (---[a..c])  : 0
           101            {2} | 1 0 0 0 |    101
                          {2} | 0 1 0 0 |
                          {2} | 0 0 1 0 |
                          {2} | 0 0 0 1 |

            ZZ       4                        ZZ       4
      1 : (---[a..c])  <------------------- (---[a..c])  : 1
           101            {3} | 1 0 0 0 |    101
                          {3} | 0 1 0 0 |
                          {3} | 0 0 1 0 |
                          {3} | 0 0 0 1 |

o12 : ZZdFactorizationMap
 
i13 : D = coker f

o13 = cokernel {2} | 1 0 0 0 | <-- cokernel {3} | 1 0 0 0 | <-- cokernel {2} | 1 0 0 0 |
               {2} | 0 1 0 0 |              {3} | 0 1 0 0 |              {2} | 0 1 0 0 |
               {2} | 0 0 1 0 |              {3} | 0 0 1 0 |              {2} | 0 0 1 0 |
               {2} | 0 0 0 1 |              {3} | 0 0 0 1 |              {2} | 0 0 0 1 |
                                                                 
      0                            1                            0

o13 : ZZdFactorization
 
i14 : D == 0

o14 = true

Testing for equality is not the same testing for isomorphism. In particular, different presentations of a factorization need not be equal.

i15 : use ambient S

       ZZ
o15 = ---[a..c]
      101

o15 : PolynomialRing
 
i16 : K = koszulMF({a,b,c}, a^2+b^2+c^2)

        ZZ       4       ZZ       4       ZZ       4
o16 = (---[a..c])  <-- (---[a..c])  <-- (---[a..c])
       101              101              101
                                         
      0                1                0

o16 : ZZdFactorization
 
i17 : C == K

o17 = false

The matrix factorizations $C$ and $K$ above are isomorphic, however.

See also

Ways to use this method:

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:876:0.