# Hom(Module,ChainComplex)

## Synopsis

• Function: Hom
• Usage:
Hom(M,C)
Hom(C,M)
• Inputs:
• M,
• C,
• Outputs:
• , The chain complex whose i-th spot is Hom(M,C_i), in the first case, or Hom(C_(-i),M) in the second case

## Description

 i1 : R = QQ[a..d]; i2 : C = res coker vars R 1 4 6 4 1 o2 = R <-- R <-- R <-- R <-- R <-- 0 0 1 2 3 4 5 o2 : ChainComplex i3 : M = R^1/(a,b) o3 = cokernel | a b | 1 o3 : R-module, quotient of R i4 : C' = Hom(C,M) o4 = 0 <-- cokernel {-4} | b a | <-- cokernel {-3} | b a 0 0 0 0 0 0 | <-- cokernel {-2} | b a 0 0 0 0 0 0 0 0 0 0 | <-- cokernel {-1} | b a 0 0 0 0 0 0 | <-- cokernel | b a | {-3} | 0 0 b a 0 0 0 0 | {-2} | 0 0 b a 0 0 0 0 0 0 0 0 | {-1} | 0 0 b a 0 0 0 0 | -5 -4 {-3} | 0 0 0 0 b a 0 0 | {-2} | 0 0 0 0 b a 0 0 0 0 0 0 | {-1} | 0 0 0 0 b a 0 0 | 0 {-3} | 0 0 0 0 0 0 b a | {-2} | 0 0 0 0 0 0 b a 0 0 0 0 | {-1} | 0 0 0 0 0 0 b a | {-2} | 0 0 0 0 0 0 0 0 b a 0 0 | -3 {-2} | 0 0 0 0 0 0 0 0 0 0 b a | -1 -2 o4 : ChainComplex i5 : C'.dd_-1 o5 = {-2} | 0 0 0 0 | {-2} | c 0 0 0 | {-2} | 0 c 0 0 | {-2} | d 0 0 0 | {-2} | 0 d 0 0 | {-2} | 0 0 d -c | o5 : Matrix i6 : C'.dd^2 == 0 o6 = true

## Caveat

Hom of two chain complexes is not yet implemented