Macaulay2 » Documentation
Packages » MatrixFactorizations :: suspension
next | previous | forward | backward | up | index | toc

# suspension -- Constructs the suspension of a d-fold factorization

- Usage:
    `suspension(ZZdFactorization)`
    `suspension(ZZ, ZZdFactorization)`
    `suspension(ZZdFactorizationMap)`
- Inputs:
  - ZZdFactorization, a ZZ/d graded factorization, A ZZdFactorization object representing a d-fold factorization
  - ZZ, an integer, An integer representing the dimension
  - ZZdFactorizationMap, a map of ZZ/d-graded factorizations, A ZZdFactorizationMap object representing a map of factorizations
- Outputs:
  - a ZZ/d graded factorization,

## Description

The suspension of a d-fold factorization may be canonically defined by taking the cokernel of the natural embedding of a factorization into its injective cover. When $d=2$, this recovers the shift functor of a factorization, but when $d > 2$ this object is an indecomposable yet nonminimal ZZ/d-graded factorization.

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

    i2 : D = koszulMF(a^3+b^3+c^3)

          4      4      4
    o2 = Q  <-- Q  <-- Q

         0      1      0

    o2 : ZZdFactorization

    i3 : D.dd

              4                        4
    o3 = 1 : Q  <-------------------- Q  : 0
                   | c2 0  b  a   |
                   | 0  c2 a2 -b2 |
                   | b2 a  -c 0   |
                   | a2 -b 0  -c  |

              4                         4
         0 : Q  <--------------------- Q  : 1
                   | c  0  b   a   |
                   | 0  c  a2  -b2 |
                   | b2 a  -c2 0   |
                   | a2 -b 0   -c2 |

    o3 : ZZdFactorizationMap

    i4 : suspension D

          4      4      4
    o4 = Q  <-- Q  <-- Q

         0      1      0

    o4 : ZZdFactorization

    i5 : oo == D[1]

    o5 = true

    i6 : suspension(2, D) == D

    o6 = true

    i7 : C = linearMF(a^3+b^3,t)

         /   Q[t]   \3     /   Q[t]   \3     /   Q[t]   \3     /   Q[t]   \3
    o7 = |----------|  <-- |----------|  <-- |----------|  <-- |----------|
         | 2        |      | 2        |      | 2        |      | 2        |
         \t  + t + 1/      \t  + t + 1/      \t  + t + 1/      \t  + t + 1/

         0                 1                 2                 0

    o7 : ZZdFactorization

    i8 : sC = suspension(C)

         /   Q[t]   \6     /   Q[t]   \6     /   Q[t]   \6     /   Q[t]   \6
    o8 = |----------|  <-- |----------|  <-- |----------|  <-- |----------|
         | 2        |      | 2        |      | 2        |      | 2        |
         \t  + t + 1/      \t  + t + 1/      \t  + t + 1/      \t  + t + 1/

         0                 1                 2                 0

    o8 : ZZdFactorization

    i9 : sC.dd

             /   Q[t]   \6                                 /   Q[t]   \6
    o9 = 2 : |----------|  <------------------------------ |----------|  : 0
             | 2        |     | 0 0 0 -b2 -abt-ab -a2  |   | 2        |
             \t  + t + 1/     | 0 0 0 -a2 b2t+b2  ab   |   \t  + t + 1/
                              | 0 0 0 abt -a2     -b2t |
                              | 1 0 0 -b  -a      0    |
                              | 0 1 0 0   -bt     -a   |
                              | 0 0 1 -a  0       bt+b |

             /   Q[t]   \6                                 /   Q[t]   \6
         0 : |----------|  <------------------------------ |----------|  : 1
             | 2        |     | -b2 -abt-ab -a2  0 0 0 |   | 2        |
             \t  + t + 1/     | -a2 b2t+b2  ab   0 0 0 |   \t  + t + 1/
                              | abt -a2     -b2t 0 0 0 |
                              | -b  -a      0    1 0 0 |
                              | 0   -bt     -a   0 1 0 |
                              | -a  0       bt+b 0 0 1 |

             /   Q[t]   \6                                 /   Q[t]   \6
         1 : |----------|  <------------------------------ |----------|  : 2
             | 2        |     | 1 0 0 -b  -a      0    |   | 2        |
             \t  + t + 1/     | 0 1 0 0   -bt     -a   |   \t  + t + 1/
                              | 0 0 1 -a  0       bt+b |
                              | 0 0 0 -b2 -abt-ab -a2  |
                              | 0 0 0 -a2 b2t+b2  ab   |
                              | 0 0 0 abt -a2     -b2t |

    o9 : ZZdFactorizationMap

    i10 : sC.dd^3

              /   Q[t]   \6                                              /   Q[t]   \6
    o10 = 0 : |----------|  <------------------------------------------- |----------|  : 0
              | 2        |     | a3+b3 0     0     0     0     0     |   | 2        |
              \t  + t + 1/     | 0     a3+b3 0     0     0     0     |   \t  + t + 1/
                               | 0     0     a3+b3 0     0     0     |
                               | 0     0     0     a3+b3 0     0     |
                               | 0     0     0     0     a3+b3 0     |
                               | 0     0     0     0     0     a3+b3 |

              /   Q[t]   \6                                              /   Q[t]   \6
          1 : |----------|  <------------------------------------------- |----------|  : 1
              | 2        |     | a3+b3 0     0     0     0     0     |   | 2        |
              \t  + t + 1/     | 0     a3+b3 0     0     0     0     |   \t  + t + 1/
                               | 0     0     a3+b3 0     0     0     |
                               | 0     0     0     a3+b3 0     0     |
                               | 0     0     0     0     a3+b3 0     |
                               | 0     0     0     0     0     a3+b3 |

              /   Q[t]   \6                                              /   Q[t]   \6
          2 : |----------|  <------------------------------------------- |----------|  : 2
              | 2        |     | a3+b3 0     0     0     0     0     |   | 2        |
              \t  + t + 1/     | 0     a3+b3 0     0     0     0     |   \t  + t + 1/
                               | 0     0     a3+b3 0     0     0     |
                               | 0     0     0     a3+b3 0     0     |
                               | 0     0     0     0     a3+b3 0     |
                               | 0     0     0     0     0     a3+b3 |

    o10 : ZZdFactorizationMap

    i11 : suspension(3,C)

          /   Q[t]   \24     /   Q[t]   \24     /   Q[t]   \24     /   Q[t]   \24
    o11 = |----------|   <-- |----------|   <-- |----------|   <-- |----------|
          | 2        |       | 2        |       | 2        |       | 2        |
          \t  + t + 1/       \t  + t + 1/       \t  + t + 1/       \t  + t + 1/

          0                  1                  2                  0

    o11 : ZZdFactorization

    i12 : isdFactorization oo

                  3    3
    o12 = (true, a  + b )

    o12 : Sequence

## See also

- injectiveCover -- Constructs the injective hull of a d-fold matrix factorization
- projectiveCover -- Constructs the projective cover of a d-fold matrix factorization
- kernel -- kernel of a map
- prune -- prune, e.g., compute a minimal presentation

## Ways to use suspension:

- `suspension(ZZ,ZZdFactorization)`
- `suspension(ZZdFactorization)`
- `suspension(ZZdFactorizationMap)`

## For the programmer

The object suspension is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MatrixFactorizations/MatrixFactorizationsDOC.m2:5944:0.