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directSum(ZZdFactorization) -- direct sum of ZZ/d-graded factorizations

Description

The direct sum of two factorizations is another factorization, assuming the inputs factor the same ring element. The differentials are simply the direct sum of the constituent differentials.

i1 : S = ZZ/101[x,y];
 
i2 : C1 = randomLinearMF(2,S)

      4      4      4
o2 = S  <-- S  <-- S
                    
     0      1      0

o2 : ZZdFactorization
 
i3 : C2 = randomLinearMF(2,S)

      4      4      4
o3 = S  <-- S  <-- S
                    
     0      1      0

o3 : ZZdFactorization
 
i4 : C12 = C1 ++ C2

      8      8      8
o4 = S  <-- S  <-- S
                    
     0      1      0

o4 : ZZdFactorization
 
i5 : potential(C1) == potential(C2)

o5 = false
 
i6 : isdFactorization(C12)

o6 = (false, no potential)

o6 : Sequence
 
i7 : C12.dd^2

          8                                                                                                                                           8
o7 = 0 : S  <--------------------------------------------------------------------------------------------------------------------------------------- S  : 0
               {3} | 24x2-36xy-30y2 0              0              0              0               0               0               0               |
               {3} | 0              24x2-36xy-30y2 0              0              0               0               0               0               |
               {3} | 0              0              24x2-36xy-30y2 0              0               0               0               0               |
               {3} | 0              0              0              24x2-36xy-30y2 0               0               0               0               |
               {3} | 0              0              0              0              -29x2+19xy+19y2 0               0               0               |
               {3} | 0              0              0              0              0               -29x2+19xy+19y2 0               0               |
               {3} | 0              0              0              0              0               0               -29x2+19xy+19y2 0               |
               {3} | 0              0              0              0              0               0               0               -29x2+19xy+19y2 |

          8                                                                                                                                           8
     1 : S  <--------------------------------------------------------------------------------------------------------------------------------------- S  : 1
               {3} | 24x2-36xy-30y2 0              0              0              0               0               0               0               |
               {3} | 0              24x2-36xy-30y2 0              0              0               0               0               0               |
               {3} | 0              0              24x2-36xy-30y2 0              0               0               0               0               |
               {3} | 0              0              0              24x2-36xy-30y2 0               0               0               0               |
               {3} | 0              0              0              0              -29x2+19xy+19y2 0               0               0               |
               {3} | 0              0              0              0              0               -29x2+19xy+19y2 0               0               |
               {3} | 0              0              0              0              0               0               -29x2+19xy+19y2 0               |
               {3} | 0              0              0              0              0               0               0               -29x2+19xy+19y2 |

o7 : ZZdFactorizationMap

As we can see in the above example, the problem stems from the fact that the direct sum of matrix factorizations with different potentials is no longer a matrix factorization. Constructing factorizations with the same potential will solve this:

i8 : Q = ZZ/101[x,y]

o8 = Q

o8 : PolynomialRing
 
i9 : f = x^3+y^3;
 
i10 : R = Q/(f)

o10 = R

o10 : QuotientRing
 
i11 : m = ideal vars R

o11 = ideal (x, y)

o11 : Ideal of R
 
i12 : F1 = tailMF m

       2      2      2
o12 = Q  <-- Q  <-- Q
                     
      0      1      0

o12 : ZZdFactorization
 
i13 : F2 = tailMF (m^2)

       3      3      3
o13 = Q  <-- Q  <-- Q
                     
      0      1      0

o13 : ZZdFactorization
 
i14 : potential(F1) == potential(F2)

o14 = true
 
i15 : F12 = F1++ F2

       5      5      5
o15 = Q  <-- Q  <-- Q
                     
      0      1      0

o15 : ZZdFactorization
 
i16 : isdFactorization F12

              3    3
o16 = (true, x  + y )

o16 : Sequence

The direct sum is an n-ary operator with projection and inclusion maps from each component satisfying appropriate identities.

i17 : C4 = directSum(first => F1, second => F2)

       5      5      5
o17 = Q  <-- Q  <-- Q
                     
      0      1      0

o17 : ZZdFactorization
 
i18 : C4_[first] -- inclusion map C1 --> C4

           5                   2
o18 = 0 : Q  <--------------- Q  : 0
                {2} | 1 0 |
                {3} | 0 1 |
                {3} | 0 0 |
                {3} | 0 0 |
                {3} | 0 0 |

           5                   2
      1 : Q  <--------------- Q  : 1
                {4} | 1 0 |
                {4} | 0 1 |
                {5} | 0 0 |
                {5} | 0 0 |
                {5} | 0 0 |

o18 : ZZdFactorizationMap
 
i19 : C4^[first] -- projection map C4 --> C1

           2                         5
o19 = 0 : Q  <--------------------- Q  : 0
                {2} | 1 0 0 0 0 |
                {3} | 0 1 0 0 0 |

           2                         5
      1 : Q  <--------------------- Q  : 1
                {4} | 1 0 0 0 0 |
                {4} | 0 1 0 0 0 |

o19 : ZZdFactorizationMap
 
i20 : C4^[first] * C4_[first] == 1

o20 = true
 
i21 : C4^[second] * C4_[second] == 1

o21 = true
 
i22 : C4^[first] * C4_[second] == 0

o22 = true
 
i23 : C4^[second] * C4_[first] == 0

o23 = true
 
i24 : C4_[first] * C4^[first] + C4_[second] * C4^[second] == 1

o24 = true

There are two short exact sequences associated to a direct sum.

i25 : isShortExactSequence(C4^[first], C4_[second])

o25 = true
 
i26 : isShortExactSequence(C4^[second], C4_[first])

o26 = true

Given a factorization which is a direct sum, we obtain the component complexes and their names (indices) as follows (even if the resulting factorization is not well-defined, we can still obtain the components).

i27 : components C12

        4      4      4   4      4      4
o27 = {S  <-- S  <-- S , S  <-- S  <-- S }
                                        
       0      1      0   0      1      0

o27 : List
 
i28 : indices C12

o28 = {0, 1}

o28 : List
 
i29 : components C4

        2      2      2   3      3      3
o29 = {Q  <-- Q  <-- Q , Q  <-- Q  <-- Q }
                                        
       0      1      0   0      1      0

o29 : List
 
i30 : indices C4

o30 = {first, second}

o30 : List

See also

Ways to use this method:

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:1195:0.