Since Ext is a bifunctor, there are two different connecting maps. The first comes from applying the covariant functor $\operatorname{Hom}_S(M, -)$ to a short exact sequence of modules. The second comes from applying the contravariant functor $\operatorname{Hom}_S(-, M)$ to a short exact sequence of modules. More explicitly, given the short exact sequence
$\phantom{WWWW} 0 \leftarrow C \xleftarrow{g} B \xleftarrow{f} A \leftarrow 0, $
the $(-i)$-th connecting homomorphism is, in the first case, a map $\operatorname{Ext}_S^i(M, C) \to \operatorname{Ext}_S^{i+1}(M, A)$ and, in the second case, a map $\operatorname{Ext}_S^i(A, M) \to \operatorname{Ext}_S^{i+1}(C, M)$. Observe that the connecting homomorphism is indexed homologically, whereas Ext modules are indexed cohomologically, explaining the different signs for the index $i$.
As a first example, applying the functor $\operatorname{Hom}(S/I, -)$ 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 the long exact sequence in Ext modules having the form
$\phantom{WWWW} \dotsb \leftarrow \operatorname{Ext}^{i+1}(S/I, S(-\deg h)) \xleftarrow{\delta_{-i}} \operatorname{Ext}^i(S/I, S/h) \leftarrow \operatorname{Ext}^i(S/I, S) \leftarrow \operatorname{Ext}^i(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 : g = map(S^1/h, S^1, 1) o4 = | 1 | o4 : Matrix |
i5 : f = map(S^1, S^{-degree h}, {{h}})
o5 = | ac2+acd+bd2 |
1 1
o5 : Matrix S <--- S
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i6 : assert isShortExactSequence(g,f) |
i7 : delta = connectingExtMap(S^1/I, g, f)
o7 = -2 : 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 |) : -2
{-3, -2} | 0 -d c 0 0 0 0 -a 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 |
{-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 0 0 0 0 -d2 c2 c2+cd b2 0 | {-2, -4} | 0 -d2 0 c2 0 0 0 ac2+acd+bd2 |
-1 : 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 |) : -1
{-3, 0} | d2 0 b2 0 a2 0 | {-3, 0} | 0 0 -b2 a2 | {-3, 2} | 0 0 0 0 0 0 0 -ab -ac2-acd+bd2 | {-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 |
{-1, -2} | 0 d2 -c2 0 0 a2 | {-1, -2} | -d2 0 c2 0 | {-1, 0} | 0 0 0 0 0 0 0 0 0 | {-2, -2} | 0 0 ac2+acd+bd2 0 a2d2 -a3d a3c+a3d -ad4 d6 | {-2, -2} | a2d2 0 0 ac2+acd+bd2 0 |
{-1, -2} | 0 0 0 -d2 c2 b2 | {-1, -2} | 0 -d2 0 c2 | {-1, 0} | 0 0 0 0 0 0 0 0 0 | {-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 |
{-1, 0} | 0 0 0 0 0 0 0 0 0 |
{-1, 0} | 0 0 0 0 0 0 0 0 0 |
{1, -2} | 0 0 0 0 0 d -c 0 0 |
o7 : ComplexMap
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i8 : assert isWellDefined delta |
i9 : assert(degree delta == 0) |
i10 : assert(source delta_(-1) == Ext^1(comodule I, S^1/h)) |
i11 : assert(target delta_(-1) == Ext^2(comodule I, S^{{-1,-2}}))
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As a second example, applying the functor $\operatorname{Hom}(-, S)$ to the short exact sequence of modules
$\phantom{WWWW} 0 \leftarrow S/(I+J) \leftarrow S/I \oplus S/J \leftarrow S/I \cap J \leftarrow 0 $
gives rise to the long exact sequence of Ext modules having the form
$\phantom{WWWW} \dotsb \leftarrow \operatorname{Ext}^{i+1}(S/(I+J), S) \xleftarrow{\delta_{-i}} \operatorname{Ext}^i(S/I \cap J, S) \leftarrow \operatorname{Ext}^i(S/I \oplus S/J, S) \leftarrow \operatorname{Ext}^i(S/(I+J), S) \leftarrow \dotsb. $
i12 : S = ZZ/101[a..d]; |
i13 : I = ideal(c^3-b*d^2, b*c-a*d)
3 2
o13 = ideal (c - b*d , b*c - a*d)
o13 : Ideal of S
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i14 : J = ideal(a*c^2-b^2*d, b^3-a^2*c)
2 2 3 2
o14 = ideal (a*c - b d, b - a c)
o14 : Ideal of S
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i15 : g = map(S^1/(I+J), S^1/I ++ S^1/J, {{1,1}})
o15 = | 1 1 |
o15 : Matrix
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i16 : f = map(S^1/I ++ S^1/J, S^1/intersect(I,J), {{1},{-1}})
o16 = | 1 |
| -1 |
o16 : Matrix
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i17 : assert isShortExactSequence(g,f) |
i18 : delta = connectingExtMap(g, f, S^1)
o18 = -2 : cokernel {-5} | d -c -b a | <------------------ cokernel {-5} | d -c b -a | : -2
{-5} | 1 0 0 | {-5} | 0 0 -d c |
{-6} | -ac b2 0 0 |
o18 : ComplexMap
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i19 : assert isWellDefined delta |
i20 : assert(source delta_-2 == Ext^2(comodule intersect(I,J), S)) |
i21 : assert(target delta_-2 == Ext^3(comodule (I+J), S)) |