# evenExtModule -- even part of Ext^*(M,k) over a complete intersection as module over CI operator ring

## Synopsis

• Usage:
E = evenExtModule M
• Inputs:
• M, , over a complete intersection ring
• Optional inputs:
• OutRing => ..., default value 0
• Outputs:
• E, , over a polynomial ring with gens in degree 1

## Description

Extracts the even degree part from ExtModule M If the optional argument OutRing => T is given, and class T === PolynomialRing, then the output will be a module over T.

 i1 : kk= ZZ/101 o1 = kk o1 : QuotientRing i2 : S = kk[x,y,z] o2 = S o2 : PolynomialRing i3 : I2 = ideal"x3,yz" 3 o3 = ideal (x , y*z) o3 : Ideal of S i4 : R2 = S/I2 o4 = R2 o4 : QuotientRing i5 : M2 = R2^1/ideal"x2,y,z" o5 = cokernel | x2 y z | 1 o5 : R2-module, quotient of R2 i6 : betti res (M2, LengthLimit =>10) 0 1 2 3 4 5 6 7 8 9 10 o6 = total: 1 3 5 7 9 11 13 15 17 19 21 0: 1 2 2 2 2 2 2 2 2 2 2 1: . 1 3 4 4 4 4 4 4 4 4 2: . . . 1 3 4 4 4 4 4 4 3: . . . . . 1 3 4 4 4 4 4: . . . . . . . 1 3 4 4 5: . . . . . . . . . 1 3 o6 : BettiTally i7 : E = ExtModule M2 8 o7 = (kk[X ..X ]) 0 1 o7 : kk[X ..X ]-module, free, degrees {0..1, 2:1, 3:2, 3} 0 1 i8 : apply(toList(0..10), i->hilbertFunction(i, E)) o8 = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21} o8 : List i9 : Eeven = evenExtModule M2 4 o9 = (kk[X ..X ]) 0 1 o9 : kk[X ..X ]-module, free, degrees {0..1, 2:1} 0 1 i10 : apply(toList(0..5), i->hilbertFunction(i, Eeven)) o10 = {1, 5, 9, 13, 17, 21} o10 : List