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}}];
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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
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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
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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
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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
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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) |