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Complexes :: longExactSequence(ComplexMap,ComplexMap)

longExactSequence(ComplexMap,ComplexMap) -- make the long exact sequence in homology

Synopsis

Description

Every short exact sequence of complexes

$\phantom{WWWW} 0 \leftarrow C \xleftarrow{g} B \xleftarrow{f} A \leftarrow 0 $

gives rise to a long exact sequence $L$ in homology having the form

$\phantom{WWWW} \dotsb \leftarrow H_{i-1}(C) \xleftarrow{H_{i-1}(g)} H_{i-1}(B) \xleftarrow{H_{i-1}(f)} H_{i-1}(A) \xleftarrow{\delta_i} H_{i}(C) \xleftarrow{H_{i}(g)} H_{i}(B) \xleftarrow{H_{i}(f)} H_{i}(A) \leftarrow \dotsb. $

This method returns the complex $L$ such that, for all integers $i$, we have $L_{3i} = H_i(C)$, $L_{3i+1} = H_i(B)$, and $L_{3i+2} = H_i(A)$. The differentials $\operatorname{dd}^L_{3i}$ are the connecting homomorphisms $\delta_i \colon H_i(C) \to H_{i-1}(A)$. Moreover, we have $\operatorname{dd}^L_{3i+1} = H_{i}(g)$ and $\operatorname{dd}^L_{3i+2} = H_{i}(f)$.

For example, consider a free resolution $F$ of $S/I$. Applying the Hom functor $\operatorname{Hom}(F, -)$ to a short exact sequence of modules

$\phantom{WWWW} 0 \leftarrow S/h \leftarrow S \xleftarrow{h} S(- \deg h) \leftarrow 0 $

gives rise to a short exact sequence of complexes. The corresponding long exact sequence $L$ in homology has the form

$\phantom{WWWW} \dotsb \leftarrow \operatorname{Ext}^{d+1}(S/I, S(-\deg h)) \xleftarrow{\delta} \operatorname{Ext}^d(S/I, S/h) \leftarrow \operatorname{Ext}^d(S/I, S) \leftarrow \operatorname{Ext}^d(S/I, S(-\deg h)) \leftarrow \dotsb. $

i1 : S = ZZ/101[a..d, Degrees=>{2:{1,0},2:{0,1}}];
i2 : h = a*c^2 + a*c*d + b*d^2;
i3 : I = (ideal(a,b) * ideal(c,d))^[2]

             2 2   2 2   2 2   2 2
o3 = ideal (a c , a d , b c , b d )

o3 : Ideal of S
i4 : F = freeResolution comodule I;
i5 : g = Hom(F, map(S^1/h, S^1, 1))

                                                                 1
o5 = -3 : cokernel {-4, -4} | ac2+acd+bd2 | <------------------ S  : -3
                                               {-4, -4} | 1 |

                                                                                                           4
     -2 : cokernel {-4, -2} | ac2+acd+bd2 0           0           0           | <------------------------ S  : -2
                   {-4, -2} | 0           ac2+acd+bd2 0           0           |    {-4, -2} | 1 0 0 0 |
                   {-2, -4} | 0           0           ac2+acd+bd2 0           |    {-4, -2} | 0 1 0 0 |
                   {-2, -4} | 0           0           0           ac2+acd+bd2 |    {-2, -4} | 0 0 1 0 |
                                                                                   {-2, -4} | 0 0 0 1 |

                                                                                                           4
     -1 : cokernel {-2, -2} | ac2+acd+bd2 0           0           0           | <------------------------ S  : -1
                   {-2, -2} | 0           ac2+acd+bd2 0           0           |    {-2, -2} | 1 0 0 0 |
                   {-2, -2} | 0           0           ac2+acd+bd2 0           |    {-2, -2} | 0 1 0 0 |
                   {-2, -2} | 0           0           0           ac2+acd+bd2 |    {-2, -2} | 0 0 1 0 |
                                                                                   {-2, -2} | 0 0 0 1 |

                                              1
     0 : cokernel | ac2+acd+bd2 | <--------- S  : 0
                                     | 1 |

o5 : ComplexMap
i6 : f = Hom(F, map(S^1, S^{-degree h}, {{h}}))

           1                                1
o6 = -3 : S  <---------------------------- S  : -3
                {-4, -4} | ac2+acd+bd2 |

           4                                                                    4
     -2 : S  <---------------------------------------------------------------- S  : -2
                {-4, -2} | ac2+acd+bd2 0           0           0           |
                {-4, -2} | 0           ac2+acd+bd2 0           0           |
                {-2, -4} | 0           0           ac2+acd+bd2 0           |
                {-2, -4} | 0           0           0           ac2+acd+bd2 |

           4                                                                    4
     -1 : S  <---------------------------------------------------------------- S  : -1
                {-2, -2} | ac2+acd+bd2 0           0           0           |
                {-2, -2} | 0           ac2+acd+bd2 0           0           |
                {-2, -2} | 0           0           ac2+acd+bd2 0           |
                {-2, -2} | 0           0           0           ac2+acd+bd2 |

          1                       1
     0 : S  <------------------- S  : 0
               | ac2+acd+bd2 |

o6 : ComplexMap
i7 : assert isWellDefined g
i8 : assert isWellDefined f
i9 : assert isShortExactSequence(g, f)
i10 : L = longExactSequence(g,f)

o10 = cokernel {-4, -4} | d2 -c2 -b2 a2 ac2+acd+bd2 | <-- cokernel {-4, -4} | d2 -c2 -b2 a2 | <-- cokernel {-3, -2} | d2 -c2 -b2 a2 | <-- subquotient ({-4, -2} | c2 ac+bd -bc   b2 0   a2  0  ab    0  0           |, {-4, -2} | -b2 a2  0   0  ac2+acd+bd2 0           0           0           |) <-- subquotient ({-4, -2} | c2 b2 0   a2  0  0  |, {-4, -2} | -b2 a2  0   0  |) <-- subquotient ({-3, 0}  | c2 b2 0   a2  0  0  |, {-3, 0}  | -b2 a2  0   0  |) <-- subquotient ({-2, -2} | ac2+acd+bd2 0           0           0           -a2cd-abd2 a3c+a2bd -a2bc     acd3+bd4   c2d4            |, {-2, -2} | -a2cd-abd2 ac2+acd+bd2 0           0           0           |) <-- subquotient ({-2, -2} | a2c2 |, {-2, -2} | a2c2 |) <-- subquotient ({-1, 0} | a2c2 |, {-1, 0} | a2c2 |) <-- subquotient (| ac2+acd+bd2 |, | ac2+acd+bd2 |) <-- image 0 <-- image 0 <-- 0
                                                                                                                                                       {-4, -2} | d2 -ad   ac+ad 0  b2  0   a2 0     0  0           |  {-4, -2} | 0   0   -b2 a2 0           ac2+acd+bd2 0           0           |                   {-4, -2} | d2 0  b2  0   a2 0  |  {-4, -2} | 0   0   -b2 a2 |                   {-3, 0}  | d2 0  b2  0   a2 0  |  {-3, 0}  | 0   0   -b2 a2 |                   {-2, -2} | 0           ac2+acd+bd2 0           0           b2c2       ab2c+b3d -b3c      bc4+bc3d   c6+2c5d+c4d2    |  {-2, -2} | b2c2       0           ac2+acd+bd2 0           0           |                   {-2, -2} | b2c2 |  {-2, -2} | b2c2 |                   {-1, 0} | b2c2 |  {-1, 0} | b2c2 |                                                                                  
      -9                                                  -8                                      -7                                                   {-2, -4} | 0  0     0     d2 -c2 0   0  0     a2 ac2+acd+bd2 |  {-2, -4} | -d2 0   c2  0  0           0           ac2+acd+bd2 0           |                   {-2, -4} | 0  d2 -c2 0   0  a2 |  {-2, -4} | -d2 0   c2  0  |                   {-1, -2} | 0  d2 -c2 0   0  a2 |  {-1, -2} | -d2 0   c2  0  |                   {-2, -2} | 0           0           ac2+acd+bd2 0           a2d2       -a3d     a3c+a3d   -ad4       d6              |  {-2, -2} | a2d2       0           0           ac2+acd+bd2 0           |                   {-2, -2} | a2d2 |  {-2, -2} | a2d2 |                   {-1, 0} | a2d2 |  {-1, 0} | a2d2 |      0                                                  1           2           3
                                                                                                                                                       {-2, -4} | 0  0     0     0  0   -d2 c2 c2+cd b2 0           |  {-2, -4} | 0   -d2 0   c2 0           0           0           ac2+acd+bd2 |                   {-2, -4} | 0  0  0   -d2 c2 b2 |  {-2, -4} | 0   -d2 0   c2 |                   {-1, -2} | 0  0  0   -d2 c2 b2 |  {-1, -2} | 0   -d2 0   c2 |                   {-2, -2} | 0           0           0           ac2+acd+bd2 b2d2       -ab2d    ab2c+ab2d bc2d2+bcd3 c4d2+2c3d3+c2d4 |  {-2, -2} | b2d2       0           0           0           ac2+acd+bd2 |                   {-2, -2} | b2d2 |  {-2, -2} | b2d2 |                   {-1, 0} | b2d2 |  {-1, 0} | b2d2 |
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                          -6                                                                                                                                                            -5                                                                              -4                                                                              -3                                                                                                                                                                                                               -2                                                     -1

o10 : Complex
i11 : assert isWellDefined L
i12 : assert(HH L == 0)

We verify that the indexing on $L$ in this example matches the description above.

i13 : delta = connectingMap(g, f);
i14 : assert(dd^L_-9 === delta_-3)
i15 : assert(dd^L_-8 === HH_-3 g)
i16 : assert(dd^L_-7 === HH_-3 f)
i17 : assert(dd^L_-6 === delta_-2)
i18 : assert(dd^L_-5 === HH_-2 g)
i19 : assert(dd^L_-4 === HH_-2 f)
i20 : assert(dd^L_-3 === delta_-1)

See also