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Complex Array -- shift a complex or complex map

Description

The shifted complex $D$ is defined by $D_j = C_{i+j}$ for all $j$ and the sign of the differential is changed if $i$ is odd.

The shifted complex map $g$ is defined by $g_j = f_{i+j}$ for all $j$.

The shift defines a natural automorphism on the category of complexes. Topologists often call the shifted complex $C[1]$ the suspension of $C$.

 i1 : S = ZZ/101[a..d] o1 = S o1 : PolynomialRing i2 : C = freeResolution coker vars S 1 4 6 4 1 o2 = S <-- S <-- S <-- S <-- S 0 1 2 3 4 o2 : Complex i3 : dd^C_3 o3 = {2} | c d 0 0 | {2} | -b 0 d 0 | {2} | a 0 0 d | {2} | 0 -b -c 0 | {2} | 0 a 0 -c | {2} | 0 0 a b | 6 4 o3 : Matrix S <-- S i4 : D = C[1] 1 4 6 4 1 o4 = S <-- S <-- S <-- S <-- S -1 0 1 2 3 o4 : Complex i5 : assert isWellDefined D i6 : assert(dd^D_2 == -dd^C_3)

In order to shift the complex one step, and not change the differential, one can do the following.

 i7 : E = complex(C, Base => -1) 1 4 6 4 1 o7 = S <-- S <-- S <-- S <-- S -1 0 1 2 3 o7 : Complex i8 : assert isWellDefined E i9 : assert(dd^E_2 == dd^C_3)

The shift operator is functorial, as illustrated below.

 i10 : C2 = freeResolution (S^1/(a^2, b^2, c^2, d^2)) 1 4 6 4 1 o10 = S <-- S <-- S <-- S <-- S 0 1 2 3 4 o10 : Complex i11 : C3 = freeResolution (S^1/(a^2, b^3, c^4, d^5)) 1 4 6 4 1 o11 = S <-- S <-- S <-- S <-- S 0 1 2 3 4 o11 : Complex i12 : f2 = extend(C, C2, map(C_0, C2_0, 1)) 1 1 o12 = 0 : S <--------- S : 0 | 1 | 4 4 1 : S <------------------- S : 1 {1} | a 0 0 0 | {1} | 0 b 0 0 | {1} | 0 0 c 0 | {1} | 0 0 0 d | 6 6 2 : S <----------------------------- S : 2 {2} | ab 0 0 0 0 0 | {2} | 0 ac 0 0 0 0 | {2} | 0 0 bc 0 0 0 | {2} | 0 0 0 ad 0 0 | {2} | 0 0 0 0 bd 0 | {2} | 0 0 0 0 0 cd | 4 4 3 : S <--------------------------- S : 3 {3} | abc 0 0 0 | {3} | 0 abd 0 0 | {3} | 0 0 acd 0 | {3} | 0 0 0 bcd | 1 1 4 : S <---------------- S : 4 {4} | abcd | o12 : ComplexMap i13 : f3 = extend(C2, C3, map(C2_0, C3_0, 1)) 1 1 o13 = 0 : S <--------- S : 0 | 1 | 4 4 1 : S <--------------------- S : 1 {2} | 1 0 0 0 | {2} | 0 b 0 0 | {2} | 0 0 c2 0 | {2} | 0 0 0 d3 | 6 6 2 : S <-------------------------------- S : 2 {4} | b 0 0 0 0 0 | {4} | 0 c2 0 0 0 0 | {4} | 0 0 bc2 0 0 0 | {4} | 0 0 0 d3 0 0 | {4} | 0 0 0 0 bd3 0 | {4} | 0 0 0 0 0 c2d3 | 4 4 3 : S <------------------------------ S : 3 {6} | bc2 0 0 0 | {6} | 0 bd3 0 0 | {6} | 0 0 c2d3 0 | {6} | 0 0 0 bc2d3 | 1 1 4 : S <----------------- S : 4 {8} | bc2d3 | o13 : ComplexMap i14 : assert((f2*f3)[1] == (f2[1]) * (f3[1])) i15 : assert(source(f2[1]) == C2[1]) i16 : assert(target(f2[1]) == C[1])