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# ComplexMap * ComplexMap -- composition of homomorphisms of complexes

## Synopsis

• Operator: *
• Usage:
f = h * g
• Inputs:
• h, , if a ring element or integer, then we multiply the ring element by the appropriate identity map
• g, ,
• Outputs:
• f, , the composition of $g$ followed by $h$

## Description

If $g_i : C_i \rightarrow D_{d+i}$, and $h_j : D_j \rightarrow E_{e+j}$, then the composition corresponds to $f_i := h_{d+i} * g_i : C_i \rightarrow E_{i+d+e}$. In particular, the degree of the composition $f$ is the sum of the degrees of $g$ and $h$.

 i1 : R = ZZ/101[a..d] o1 = R o1 : PolynomialRing i2 : C = freeResolution coker vars R 1 4 6 4 1 o2 = R <-- R <-- R <-- R <-- R 0 1 2 3 4 o2 : Complex i3 : 3 * dd^C 1 4 o3 = 0 : R <------------------- R : 1 | 3a 3b 3c 3d | 4 6 1 : R <----------------------------------- R : 2 {1} | -3b -3c 0 -3d 0 0 | {1} | 3a 0 -3c 0 -3d 0 | {1} | 0 3a 3b 0 0 -3d | {1} | 0 0 0 3a 3b 3c | 6 4 2 : R <--------------------------- R : 3 {2} | 3c 3d 0 0 | {2} | -3b 0 3d 0 | {2} | 3a 0 0 3d | {2} | 0 -3b -3c 0 | {2} | 0 3a 0 -3c | {2} | 0 0 3a 3b | 4 1 3 : R <--------------- R : 4 {3} | -3d | {3} | 3c | {3} | -3b | {3} | 3a | o3 : ComplexMap i4 : 0 * dd^C o4 = 0 o4 : ComplexMap i5 : dd^C * dd^C o5 = 0 o5 : ComplexMap