# cone(ChainComplexMap) -- mapping cone of a chain map

## Synopsis

• Function: cone
• Usage:
cone f
• Inputs:
• Outputs:
• , the mapping cone of a f

## Description

 i1 : R = ZZ/101[x,y,z] o1 = R o1 : PolynomialRing i2 : m = image vars R o2 = image | x y z | 1 o2 : R-module, submodule of R i3 : m2 = image symmetricPower(2,vars R) o3 = image | x2 xy xz y2 yz z2 | 1 o3 : R-module, submodule of R i4 : M = R^1/m2 o4 = cokernel | x2 xy xz y2 yz z2 | 1 o4 : R-module, quotient of R i5 : N = R^1/m o5 = cokernel | x y z | 1 o5 : R-module, quotient of R i6 : C = cone extend(resolution N,resolution M,id_(R^1)) 1 4 9 9 3 o6 = R <-- R <-- R <-- R <-- R <-- 0 0 1 2 3 4 5 o6 : ChainComplex
Let's check that the homology is correct; for example, HH_0 should be zero.
 i7 : prune HH_0 C o7 = 0 o7 : R-module
Let's check that HH_1 is isomorphic to m/m2.
 i8 : prune HH_1 C o8 = cokernel {1} | z y x 0 0 0 0 0 0 | {1} | 0 0 0 z y x 0 0 0 | {1} | 0 0 0 0 0 0 z y x | 3 o8 : R-module, quotient of R i9 : prune (m/m2) o9 = cokernel {1} | z y x 0 0 0 0 0 0 | {1} | 0 0 0 z y x 0 0 0 | {1} | 0 0 0 0 0 0 z y x | 3 o9 : R-module, quotient of R