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Complexes :: yonedaProduct(Module,Module)

yonedaProduct(Module,Module) -- make the product map between Ext modules

Synopsis

Description

For any triple $(L, M, N)$ of $R$-modules, the Yoneda product is a pairing between $\operatorname{Ext}$-modules

$\phantom{WWWW} \operatorname{Ext}_R^d(L,M) \otimes \operatorname{Ext}_R^e(M,N) \to \operatorname{Ext}_R^{d+e}(L,N). $

Given $D = \operatorname{Ext}_R^d(L,M)$ and $E = \operatorname{Ext}_R^e(M,N)$, this method returns this pairing. To compute the product of a pair of elements, see yonedaProduct(Matrix,Matrix).

Specifically, for an element of $\operatorname{Ext}_R^{e}(M,N)$, thought of as an extension

$\phantom{WWWW} 0 \leftarrow M \leftarrow F_{0} \leftarrow F_{1} \leftarrow \dotsb \leftarrow F_{e-2} \leftarrow P \leftarrow N \leftarrow 0, $

and for an element of $\operatorname{Ext}_R^{d}(L,M)$, thought of as an extension

$\phantom{WWWW} 0 \leftarrow L \leftarrow G_{0} \leftarrow G_1 \leftarrow \dotsb \leftarrow G_{d-2} \leftarrow Q \leftarrow M \leftarrow 0, $

the Yoneda product corresponds to

$\phantom{WWWW} 0 \leftarrow L \leftarrow G_{0} \leftarrow G_{1} \leftarrow \dotsb \leftarrow Q \leftarrow F_{0} \leftarrow F_{1} \leftarrow \dotsb \leftarrow P \leftarrow N \leftarrow 0, $

where the map from $F_0$ to $Q$ factors through $M$. For more information about extensions, see yonedaExtension.

Alternatively, the module $\operatorname{Ext}^d_R(L,M)$ is constructed from a free resolution $G$ of $L$,

$\phantom{WWWW} 0 \leftarrow L \leftarrow G_0 \leftarrow G_1 \leftarrow \dotsb \leftarrow G_d \leftarrow \dotsb, $

by taking the homology of the complex $\operatorname{Hom}_R(G, M)$. An element of $\operatorname{Ext}^d_R(L,M)$ is represented by an element of $\operatorname{Hom}_R(G_d, M)$. This map extends to a complex map having degree $-d$ from $G$ to the free resolution $F$ of $M$. The Yoneda product is the composition of the map of chain complexes from $G$ to $F$ with the map of chain complexes having degree $-e$ from $F$ to a free resolution of $N$. For more information about these maps, see yonedaMap.

In our first example, the image of the tensor product of two $\operatorname{Ext}^1$-modules happens to generate the the $\operatorname{Ext}^2$-module.

i1 : S = ZZ/101[x_0..x_3];
i2 : I = borel monomialIdeal(x_1*x_2)

                     2         2
o2 = monomialIdeal (x , x x , x , x x , x x )
                     0   0 1   1   0 2   1 2

o2 : MonomialIdeal of S
i3 : E1 = Ext^1(S^1/I, S^1/I)

o3 = subquotient ({-2} | x_1 x_0 0   0   0   0   0   0   0   0   0   0   |, {-2} | 0 x_1x_2 x_0x_2 x_1^2 x_0x_1 x_0^2 0      0      0     0      0     0      0      0     0      0     0      0      0     0      0     0      0      0     0      0     |)
                  {-2} | 0   0   x_1 x_0 0   0   0   0   0   0   0   0   |  {-2} | 0 0      0      0     0      0     x_1x_2 x_0x_2 x_1^2 x_0x_1 x_0^2 0      0      0     0      0     0      0      0     0      0     0      0      0     0      0     |
                  {-2} | 0   0   0   0   x_1 x_0 0   0   0   0   0   0   |  {-2} | 0 0      0      0     0      0     0      0      0     0      0     x_1x_2 x_0x_2 x_1^2 x_0x_1 x_0^2 0      0      0     0      0     0      0      0     0      0     |
                  {-2} | 0   0   0   0   0   0   x_2 x_1 x_0 0   0   0   |  {-2} | 0 0      0      0     0      0     0      0      0     0      0     0      0      0     0      0     x_1x_2 x_0x_2 x_1^2 x_0x_1 x_0^2 0      0      0     0      0     |
                  {-2} | 0   0   0   0   0   0   0   0   0   x_2 x_1 x_0 |  {-2} | 0 0      0      0     0      0     0      0      0     0      0     0      0      0     0      0     0      0      0     0      0     x_1x_2 x_0x_2 x_1^2 x_0x_1 x_0^2 |

                               5
o3 : S-module, subquotient of S
i4 : h = yonedaProduct(E1, E1)

o4 = {-2} | 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 -1 0  0 0 0
     {-2} | 0 0 0 0 0 1 0 0 0 0  0 0 0 0 0 1 0 0 0  0 0 0 0 0 0 0 0  -1 0 0 0
     {-2} | 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0  1 0 0
     {-2} | 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 1 0
     {-2} | 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0
     {-2} | 0 0 0 0 0 0 0 0 0 0  1 0 0 0 0 0 0 0 0  1 0 0 0 0 0 0 0  0  0 0 0
     {-2} | 0 0 0 0 0 0 0 0 0 0  0 1 0 0 0 0 0 0 0  0 1 0 0 0 0 0 0  0  0 0 0
     {-2} | 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0
     {-2} | 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0
     {-2} | 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0
     {-2} | 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0
     {-2} | 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0
     {-2} | 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0
     {-2} | 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0
     {-2} | 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0
     {-2} | 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0
     ------------------------------------------------------------------------
     0 0 0  0 0 -1 0  0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0 0  0  0
     0 0 1  0 0 0  -1 0 0 0 0 1  0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0 0  0  0
     0 0 -1 0 0 0  0  1 0 0 0 -1 0 0 0 0 0 0 0 -1 0  0 0 0 0 0 0 0  0 -1 0  0
     0 0 0  0 0 0  0  0 1 0 0 0  0 0 0 0 0 0 0 0  -1 0 0 0 0 0 1 0  0 0  -1 0
     0 0 0  0 0 0  0  0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0 0  0  0
     0 0 0  0 0 0  0  0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0 0  0  0
     0 0 0  0 0 0  0  0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0 0  0  0
     0 0 0  0 0 0  0  0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0 0  0  0
     0 0 0  1 0 0  0  0 0 0 0 0  1 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0 0  0  0
     0 0 0  0 1 0  0  0 0 0 0 0  0 1 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0 0  0  0
     0 0 0  0 0 0  0  0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0 0  0  0
     0 0 0  0 0 0  0  0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0 0  0  0
     0 0 0  0 0 0  0  0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 -1 0 0  0  0
     0 0 0  0 0 0  0  0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0 0  0  0
     0 0 0  0 0 0  0  0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0 0  0  0
     0 0 0  0 0 0  0  0 0 0 0 0  0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  1 0  0  0
     ------------------------------------------------------------------------
     0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 1 0  0  0  0 0  0 0 0  0  0 0  0 0 0 0 0
     0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0  0  -1 0 0  0 0 0  0  0 0  0 0 0 0 0
     0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0  0  1  0 0  0 0 0  0  0 0  0 0 0 0 0
     0 0 0 1 0  0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0  0 0 0  0  0 0  0 0 0 0 0
     0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0  0 0 0  0  0 0  0 0 0 0 0
     0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 -1 0  0  0 0  0 0 -1 0  0 0  0 0 0 0 0
     0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0  -1 0  1 0  0 0 0  -1 1 0  0 0 0 0 0
     0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0  0 0 0  0  0 0  0 0 0 0 0
     0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0  0 0 0  0  0 0  0 0 0 0 0
     0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 -1 0 0 0  0  0 -1 0 0 0 0 0
     0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0  0  1  0 0  0 0 0  0  0 0  0 0 0 0 0
     0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0  0 0 0  0  0 0  0 0 0 0 0
     0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 1  0 0 0  0  0 1  0 0 0 0 0
     0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0  0 0 0  0  0 0  0 0 0 0 0
     0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0  0 0 0  0  0 0  0 0 0 0 0
     0 0 0 0 0  1 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0  0 0 0  0  0 0  0 0 0 0 0
     ------------------------------------------------------------------------
     0 0  0  0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0 0  0 0 0  0  0 0 
     0 0  0  0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0 0  0 0 0  0  0 0 
     0 0  0  0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0 0  0 0 0  0  0 0 
     0 0  0  0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  -1 0 0  0 0 0  0  0 0 
     0 0  0  0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0 0  0 0 0  0  0 0 
     0 -1 0  0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0 0  0 0 0  0  0 0 
     0 0  -1 1 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  -1 0  0  0 0  0 0 -1 0  0 0 
     0 0  0  0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0 0  0 0 0  0  0 0 
     0 0  0  0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  -1 0  0  0 0  0 0 -1 0  0 0 
     0 0  0  0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0 0  0 0 0  0  0 0 
     0 0  0  0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0  0  0  0 0  0 0 0  0  0 0 
     0 0  0  0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0 0  0 0 0  0  0 0 
     0 0  0  0 1  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  -1 0  1 0  0 0 0  -1 1 0 
     0 0  0  0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0 0  0 0 0  0  0 0 
     0 0  0  0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0 0  0 0 0  0  0 0 
     0 0  0  0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0  0 -1 0 0 0  0  0 -1
     ------------------------------------------------------------------------
     0 0 0 0 0 0 0  0  0 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0  0  0 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0  0  0 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0  0  0 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0  0  0 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0  0  0 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 -1 0  0 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0  0  0 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 -1 0  0 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0  0  0 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0  0  0 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0  0  0 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0  -1 1 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0  0  0 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0  0  0 0  0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0  0  0 -1 0 0 0 0 0 0 0 0 |

o4 : Matrix
i5 : assert isWellDefined h
i6 : assert(target h == Ext^2(S^1/I, S^1/I))
i7 : coker h == 0

o7 = true

In our second example, all three modules in the triple are distinct and the image of the Yoneda product is not surjective.

i8 : R = S/(x_0*x_1, x_2*x_3);
i9 : E1 = Ext^1(R^1/(x_0, x_2), R^1/(x_0, x_2, x_3))

o9 = subquotient ({-1} | 0 x_3 x_2 x_0 |, {-1} | 0 x_3 x_2 x_0 0   0   0   |)
                  {-1} | 1 0   0   0   |  {-1} | 0 0   0   0   x_3 x_2 x_0 |

                               2
o9 : R-module, subquotient of R
i10 : E2 = Ext^2(R^1/(x_0, x_2, x_3), R^1/(x_0, x_1, x_2, x_3))

o10 = cokernel {-2} | 0 0 0 x_3 x_2 x_1 x_0 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   |
               {-2} | 0 0 0 0   0   0   0   x_3 x_2 x_1 x_0 0   0   0   0   0   0   0   0   0   0   0   0   |
               {-2} | 0 0 0 0   0   0   0   0   0   0   0   x_3 x_2 x_1 x_0 0   0   0   0   0   0   0   0   |
               {-2} | 0 0 0 0   0   0   0   0   0   0   0   0   0   0   0   x_3 x_2 x_1 x_0 0   0   0   0   |
               {-2} | 0 0 0 0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   x_3 x_2 x_1 x_0 |

                             5
o10 : R-module, quotient of R
i11 : E3 = Ext^3(R^1/(x_0, x_2), R^1/(x_0, x_1, x_2, x_3))

o11 = cokernel {-3} | 0 0 0 x_3 x_2 x_1 x_0 0   0   0   0   0   0   0   0   0   0   0   0   |
               {-3} | 0 0 0 0   0   0   0   x_3 x_2 x_1 x_0 0   0   0   0   0   0   0   0   |
               {-3} | 0 0 0 0   0   0   0   0   0   0   0   x_3 x_2 x_1 x_0 0   0   0   0   |
               {-3} | 0 0 0 0   0   0   0   0   0   0   0   0   0   0   0   x_3 x_2 x_1 x_0 |

                             4
o11 : R-module, quotient of R
i12 : h = yonedaProduct(E1, E2)

o12 = {-3} | 0 1 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
      {-3} | 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
      {-3} | 0 0 0  0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
      {-3} | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |

o12 : Matrix
i13 : assert isWellDefined h
i14 : assert(target h == E3)
i15 : prune coker h

o15 = cokernel {-3} | x_3 x_2 x_1 x_0 |

                             1
o15 : R-module, quotient of R

See also