# Ext(Module,Module) -- total Ext module

## Synopsis

• Scripted functor: Ext
• Usage:
Ext(M,N)
• Inputs:
• Outputs:
• , the $Ext$ module of $M$ and $N$, as a multigraded module, with the modules $Ext^i(M,N)$ for all values of $i$ appearing simultaneously.

## Description

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 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); 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] 1 2 o4 : PolynomialRing i5 : S = ring H; 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 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 i10 : rank source basis( { -4,-5 }, H) o10 = 4 i11 : rank source basis( { -5 }, Ext^4(N,N) ) o11 = 4 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 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

The result of the computation is cached for future reference.