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Complexes :: yonedaExtension(Matrix)

yonedaExtension(Matrix) -- creates a chain complex representing an extension of modules

Synopsis

Description

The module $\operatorname{Ext}^d(M,N)$ corresponds to equivalence classes of extensions of $N$ by $M$. In particular, an element of this module is represented by an exact sequence of the form \[ 0 \leftarrow M \leftarrow F_0 \leftarrow F_1 \leftarrow \dots \leftarrow F_{d-2} \leftarrow P \leftarrow N \leftarrow 0 \] where $F_0 \leftarrow F_1 \leftarrow \dots$ is a free resolution of $M$, and $P$ is the pushout of the maps $g : F_d \rightarrow N$ and $F_d \rightarrow F_{d-1}$. The element corresponding to $f$ in $\operatorname{Ext}^d(M,N)$ lifts to the map $g$.

In our first example, the module $\operatorname{Ext}^1(M,R^1)$ has one generator, in degree 0. The middle term in the corresponding short exact sequence determines an irreducible rank 2 vector bundle on the elliptic curve, which can be verified by computing Fitting ideals.

i1 : R = ZZ/101[x,y,z]/(y^2*z-x*(x-z)*(x-2*z));
i2 : M = truncate(1,R^1)

o2 = image | z y x |

                             1
o2 : R-module, submodule of R
i3 : f = basis(0, Ext^1(M, R^1))

o3 = {-1} | 0 |
     {-1} | 0 |
     {-1} | 0 |
     {0}  | 1 |

o3 : Matrix
i4 : C = yonedaExtension f

                                                                           1
o4 = image | z y x | <-- cokernel {1} | y          x  0  0          | <-- R
                                  {1} | -z         0  x  -yz        |      
     0                            {1} | 0          -z -y x2-3xz+2z2 |     2
                                  {0} | x2-3xz+2z2 yz 0  0          |
                          
                         1

o4 : Complex
i5 : assert isWellDefined C
i6 : assert isShortExactSequence(dd^C_1, dd^C_2)
i7 : E = C_1

o7 = cokernel {1} | y          x  0  0          |
              {1} | -z         0  x  -yz        |
              {1} | 0          -z -y x2-3xz+2z2 |
              {0} | x2-3xz+2z2 yz 0  0          |

                            4
o7 : R-module, quotient of R
i8 : fittingIdeal(1, E)

o8 = ideal ()

o8 : Ideal of R
i9 : saturate fittingIdeal(2, E)

o9 = ideal 1

o9 : Ideal of R

For higher Ext modules, we get longer exact sequences. When the map $f$ has degree 0, the corresponding exact sequence is homogeneous.

i10 : x = symbol x;
i11 : S = ZZ/101[x_0..x_5];
i12 : I = borel monomialIdeal(x_2*x_3)

                      2         2               2
o12 = monomialIdeal (x , x x , x , x x , x x , x , x x , x x , x x )
                      0   0 1   1   0 2   1 2   2   0 3   1 3   2 3

o12 : MonomialIdeal of S
i13 : E = Ext^4(S^1/I, S^{-5})

o13 = cokernel | -x_3 0    0    x_2 0   -x_1 x_0  0    0   0   0    0   |
               | 0    -x_3 0    0   x_2 0    0    -x_1 0   x_0 0    0   |
               | 0    0    -x_3 0   0   0    -x_2 x_2  x_2 0   -x_1 x_0 |

                             3
o13 : S-module, quotient of S
i14 : f = E_{0}

o14 = | 1 |
      | 0 |
      | 0 |

o14 : Matrix
i15 : assert(isHomogeneous f and degree f === {0})
i16 : C = yonedaExtension f

                                                                                    1      9      17                                          1
o16 = cokernel | x_0^2 x_0x_1 x_1^2 x_0x_2 x_1x_2 x_2^2 x_0x_3 x_1x_3 x_2x_3 | <-- S  <-- S  <-- S   <-- cokernel {4} | -x_3 0    0    | <-- S
                                                                                                                  {4} | 0    -x_3 0    |      
      0                                                                            1      2      3                {4} | 0    0    -x_3 |     5
                                                                                                                  {4} | x_2  0    0    |
                                                                                                                  {4} | 0    x_2  0    |
                                                                                                                  {4} | -x_1 0    0    |
                                                                                                                  {4} | x_0  0    -x_2 |
                                                                                                                  {4} | 0    -x_1 x_2  |
                                                                                                                  {4} | 0    0    x_2  |
                                                                                                                  {4} | 0    x_0  0    |
                                                                                                                  {4} | 0    0    -x_1 |
                                                                                                                  {4} | 0    0    x_0  |
                                                                                                                  {5} | 1    0    0    |
                                                                                                          
                                                                                                         4

o16 : Complex
i17 : assert isWellDefined C
i18 : assert isHomogeneous C
i19 : assert(HH C == 0)

The inverse operation is given by yonedaExtension'.

i20 : f' = yonedaExtension' C

o20 = | 1 |
      | 0 |
      | 0 |

o20 : Matrix
i21 : assert(f' == f)

See also