The computation of the total Ext module is possible for modules over the ring $R$ of a complete intersection, according the algorithm of Shamash-Eisenbud-Avramov-Buchweitz. The result is provided as a finitely presented module over a new ring with one additional variable of degree {-2,-d} for each equation of degree d defining $R$. The variables in this new ring have degree length 1 more than the degree length of the original ring, i.e., is multigraded, with the degree d part of $Ext^n(M,N)$ appearing as the degree prepend(-n,d) part of Ext(M,N). We illustrate this in the following example.
i1 : R = QQ[x,y]/(x^3,y^2);
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i2 : N = cokernel matrix {{x^2, x*y}}
o2 = cokernel | x2 xy |
1
o2 : R-module, quotient of R
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i3 : H = Ext(N,N);
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i4 : ring H
o4 = QQ[X ..X , x..y]
1 2
o4 : PolynomialRing
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i5 : S = ring H;
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i6 : H
o6 = cokernel {0, 0} | y2 xy x2 0 0 0 0 0 0 0 0 X_1y X_1x 0 0 0 0 |
{-1, -1} | 0 0 0 y x 0 0 0 0 0 0 0 0 X_1 0 0 0 |
{-1, -1} | 0 0 0 0 0 y x 0 0 0 0 0 0 0 X_1 0 0 |
{-1, -1} | 0 0 0 0 0 0 0 y x 0 0 0 0 0 X_1 0 0 |
{-1, -1} | 0 0 0 0 0 0 0 0 0 y x 0 0 0 0 0 0 |
{-2, -2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y x |
6
o6 : S-module, quotient of S
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i7 : isHomogeneous H
o7 = true
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i8 : rank source basis( { -2,-3 }, H)
o8 = 1
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i9 : rank source basis( { -3 }, Ext^2(N,N) )
o9 = 1
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i10 : rank source basis( { -4,-5 }, H)
o10 = 4
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i11 : rank source basis( { -5 }, Ext^4(N,N) )
o11 = 4
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i12 : hilbertSeries H
-1 -1 2 -1 -2 -2 3 -1 -2 -1 -3 -3 -2 -3 -2 -2 -3 -1
1 + 4T T - 3T - 8T + T T + 2T + 4T T - 4T T - 2T T + 5T + 4T T - 2T T - 2T T
0 1 1 0 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1
o12 = --------------------------------------------------------------------------------------------------------
-2 -2 -2 -3 2
(1 - T T )(1 - T T )(1 - T )
0 1 0 1 1
o12 : Expression of class Divide
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i13 : hilbertSeries(H,Order=>11)
-1 -1 -2 -3 -2 -2 -3 -4 -4 -6 -3 -3
o13 = 1 + 2T + 4T T + T T + 4T T + 4T T + T T + 2T T +
1 0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-4 -5 -5 -7 -6 -9 -4 -4 -5 -6 -6 -8 -7 -10
4T T + 4T T + T T + 2T T + 2T T + 4T T + 4T T +
0 1 0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-8 -12 -5 -5 -6 -7 -7 -9 -8 -11 -9 -13 -10 -15
T T + 2T T + 2T T + 2T T + 4T T + 4T T + T T
0 1 0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-6 -6 -7 -8 -8 -10 -9 -12 -10 -14 -11 -16
+ 2T T + 2T T + 2T T + 2T T + 4T T + 4T T +
0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-12 -18 -7 -7 -8 -9 -9 -11 -10 -13 -11 -15
T T + 2T T + 2T T + 2T T + 2T T + 2T T +
0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-12 -17 -13 -19 -14 -21 -8 -8 -9 -10 -10 -12
4T T + 4T T + T T + 2T T + 2T T + 2T T +
0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-11 -14 -12 -16 -13 -18 -14 -20 -15 -22 -16 -24
2T T + 2T T + 2T T + 4T T + 4T T + T T +
0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-9 -9 -10 -11 -11 -13 -12 -15 -13 -17 -14 -19
2T T + 2T T + 2T T + 2T T + 2T T + 2T T +
0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-15 -21 -16 -23 -17 -25 -18 -27 -10 -10 -11 -12
2T T + 4T T + 4T T + T T + 2T T + 2T T +
0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-12 -14 -13 -16 -14 -18 -15 -20 -16 -22 -17 -24
2T T + 2T T + 2T T + 2T T + 2T T + 2T T +
0 1 0 1 0 1 0 1 0 1 0 1
-----------------------------------------------------------------------
-18 -26 -19 -28 -20 -30
4T T + 4T T + T T
0 1 0 1 0 1
o13 : ZZ[T ..T ]
0 1
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The result of the computation is cached for future reference.