The modules
M and
N should be graded (homogeneous) modules over the same ring.
If
M or
N is an ideal or ring, it is regarded as a module in the evident way.
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 2, i.e., is bigraded, with the degree
i part of
Ext^n(M,N) appearing as the degree
{-n,i} part of
Ext(M,N). We illustrate this in the following example.
i1 : R = QQ[x,y]/(x^3,y^2);
|
i2 : N = cokernel matrix {{x^2, x*y}}
o2 = cokernel | x2 xy |
1
o2 : R-module, quotient of R
|
i3 : H = Ext(N,N);
|
i4 : ring H
o4 = QQ [X , X , x, y, Degrees => {{-2, -2}, {-2, -3}, {0, 1}, {0, 1}}]
1 2
o4 : PolynomialRing
|
i5 : S = ring H;
|
i6 : H
o6 = cokernel {0, 0} | 0 0 0 0 0 0 0 0 y2 xy x2 0 0 0 0 X_1y X_1x |
{-1, -1} | y 0 0 0 x 0 0 0 0 0 0 0 0 X_1 0 0 0 |
{-1, -1} | 0 y 0 0 0 x 0 0 0 0 0 0 0 0 X_1 0 0 |
{-1, -1} | 0 0 y 0 0 0 x 0 0 0 0 0 0 0 0 0 0 |
{-1, -1} | 0 0 0 y 0 0 0 x 0 0 0 0 0 0 0 0 0 |
{-2, -2} | 0 0 0 0 0 0 0 0 0 0 0 y x 0 0 0 0 |
6
o6 : S-module, quotient of S
|
i7 : isHomogeneous H
o7 = true
|
i8 : rank source basis( {-2,-3}, H)
o8 = 1
|
i9 : rank source basis( {-3}, Ext^2(N,N) )
o9 = 1
|
The result of the computation is cached for future reference.