homomorphism'(ZZdFactorizationMap) -- get the element of Hom from a map of factorizations

Function: homomorphism'
Usage:

f = homomorphism g
Inputs:
- g, a map of ZZ/d-graded factorizations, from $C$ to $D$
Optional inputs:
- DegreeLimit => ..., default value null,
- MinimalGenerators => ..., default value true,
- Strategy => ..., default value null,
Outputs:
- f, a map of ZZ/d-graded factorizations, a map of the form $f : R^1 \to Hom(C, D)$, where $R$ is the underlying ring of these factorizations

Description

As a first example, consider two matrix factorizations $C$ and $D$. From a random map $f : R^1 \to Hom(C, D)$, we construct the corresponding map of ZZ/d-graded factorizations $g : C \to D$.

i1 : S = ZZ/101[a,b];

i2 : R = S/(a^3+b^3);

i3 : m = ideal vars R

o3 = ideal (a, b)

o3 : Ideal of R

i4 : C = tailMF m

      2      2      2
o4 = S  <-- S  <-- S
                    
     0      1      0

o4 : ZZdFactorization

i5 : D = tailMF (m^2)

      3      3      3
o5 = S  <-- S  <-- S
                    
     0      1      0

o5 : ZZdFactorization

i6 : g = randomFactorizationMap(D, C, InternalDegree => 2)

          3                                        2
o6 = 0 : S  <------------------------------------ S  : 0
               {3} | 24a-36b  -10a2-29ab-8b2  |
               {3} | -30a-29b -22a2-29ab-24b2 |
               {3} | 19a+19b  -38a2-16ab+39b2 |

          3                                 2
     1 : S  <----------------------------- S  : 1
               {5} | 21a+34b  -13a-43b |
               {5} | 19a-47b  -15a-28b |
               {5} | -39a-18b -47a+38b |

o6 : ZZdFactorizationMap

i7 : isWellDefined g

o7 = true

i8 : f = homomorphism' g

          12                               1
o8 = 0 : S   <--------------------------- S  : 0
                {1} | 24a-36b         |
                {1} | -30a-29b        |
                {1} | 19a+19b         |
                {0} | -10a2-29ab-8b2  |
                {0} | -22a2-29ab-24b2 |
                {0} | -38a2-16ab+39b2 |
                {1} | 21a+34b         |
                {1} | 19a-47b         |
                {1} | -39a-18b        |
                {1} | -13a-43b        |
                {1} | -15a-28b        |
                {1} | -47a+38b        |

          12
     1 : S   <----- 0 : 1
                0

o8 : ZZdFactorizationMap

i9 : isWellDefined f

o9 = true

The map $g : C \to D$ corresponding to a random map into $Hom(C,D)$ does not generally commute with the differentials. However, if the element of $Hom(C,D)$ is a cycle, then the corresponding map does commute.

i10 : g = randomFactorizationMap(D, C, Cycle => true, InternalDegree => 3)

           3                                                      2
o10 = 0 : S  <-------------------------------------------------- S  : 0
                {3} | 40a2+41ab+48b2  2a3-31a2b-48ab2+26b3   |
                {3} | 32a2-24ab-19b2  -45a3+36a2b+49ab2+45b3 |
                {3} | -37a2-27ab+22b2 -47a3-9a2b+47ab2+10b3  |

           3                                               2
      1 : S  <------------------------------------------- S  : 1
                {5} | 40a2-21ab-10b2  2a2+16ab+22b2   |
                {5} | -32a2+8ab+26b2  45a2-34ab-48b2  |
                {5} | -37a2-42ab+45b2 -47a2+47ab+19b2 |

o10 : ZZdFactorizationMap

i11 : isWellDefined g

o11 = true

i12 : f = homomorphism' g

           12                                      1
o12 = 0 : S   <---------------------------------- S  : 0
                 {1} | 40a2+41ab+48b2         |
                 {1} | 32a2-24ab-19b2         |
                 {1} | -37a2-27ab+22b2        |
                 {0} | 2a3-31a2b-48ab2+26b3   |
                 {0} | -45a3+36a2b+49ab2+45b3 |
                 {0} | -47a3-9a2b+47ab2+10b3  |
                 {1} | 40a2-21ab-10b2         |
                 {1} | -32a2+8ab+26b2         |
                 {1} | -37a2-42ab+45b2        |
                 {1} | 2a2+16ab+22b2          |
                 {1} | 45a2-34ab-48b2         |
                 {1} | -47a2+47ab+19b2        |

           12
      1 : S   <----- 0 : 1
                 0

o12 : ZZdFactorizationMap

i13 : isWellDefined f

o13 = true

i14 : assert isCommutative g

i15 : assert(degree f === 0)

i16 : assert(source f == ZZdfactorization(S^{-3}, 2))

i17 : assert(target g === D)

i18 : assert(homomorphism f == g)

Ways to use this method:

homomorphism'(ZZdFactorizationMap) -- get the element of Hom from a map of factorizations

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MatrixFactorizations/MatrixFactorizationsDOC.m2:3625:0.

homomorphism'(ZZdFactorizationMap) -- get the element of Hom from a map of factorizations

Description

See also

Ways to use this method: