Before invoking the functions in this section, it is necessary to first invoke the function CohomologyGroup(CM, n) for the appropriate n.
Given a cohomology module CM constructed from the K[G]-module M and an element s of the cohomology group H^0(G, M) associated with CM, the function returns the corresponding zero-cocycle. The zero-cocycle is returned as an element of the fixed point submodule of M. The argument s may either be given as an element of the H^0(G, M) or as a sequence of integers defining such an element.
Given a cohomology module CM constructed from the K[G]-module M and an element s of the cohomology group H^0(G, M) associated with CM, this function returns the inverse of the zero-cocycle corresponding to s as an element of the fixed point submodule of M. Hence this is the inverse function to ZeroCocycle. The argument s may either be given as an element of the H^0(G, M) or as a sequence of integers defining such an element.
Given a cohomology module CM constructed from the K[G]-module M and an element s of the cohomology group H^1(G, M) associated with CM, the function returns the corresponding one-cocycle. The one-cocycle is returned as a function from G to the module M. The argument s may either be given as an element of the H^1(G, M) or as a sequence of integers defining such an element.
Given a cohomology module CM constructed from the K[G]-module M and a one-cycle s for CM, specified as a function from G to the submodule of the module M, this function returns the inverse of the corresponding one-cocycle as an element of H^1(G, M). Thus, the function is the inverse to OneCocycle.
Given a cohomology module CM constructed from the K[G]-module M and an element s of the cohomology group H^2(G, M) associated with CM, the function returns the corresponding two-cocycle. The two-cocycle is returned as a function from G x G to the module M. The argument s may either be given as an element of the H^2(G, M) or as a sequence of integers defining such an element.
Given a cohomology module CM constructed from the K[G]-module M and a two-cycle s for CM, specified as a function from G x G to the submodule of the module M, this function returns the inverse of the corresponding two-cocycle as an element of H^2(G, M). Thus, the function is the inverse to TwoCocycle.
> G := PermutationGroup< 4 | (1,2,3,4) >; > invar:=[2,4,4]; > mats := [ Matrix(Integers(),3,3,[1,2,0,0,0,1,0,1,2]) ]; > X := CohomologyModule(G,invar,mats); > C := CohomologyGroup(X,0); > C; Full Quotient RSpace of degree 1 over Integer Ring Column moduli: [ 4 ] > ZeroCocycle(X,[3]); ( 1 -1 1) > IdentifyZeroCocycle(X,-1); (1) > C := CohomologyGroup(X, 1); > C; Full Quotient RSpace of degree 2 over Integer Ring Column moduli: [ 2, 2 ] > z1 := OneCocycle(X, [1, 0]); > z2 := OneCocycle(X, [0, 1]); > z1(G.1); (1 0 0) > z := func< x | z1(x) + z2(x) >; > IdentifyOneCocycle(X, z); (1 1) > C := CohomologyGroup(X, 2); > C; Full Quotient RSpace of degree 1 over Integer Ring Column moduli: [ 4 ] > z1 := TwoCocycle(X, [1]); > z1(G.1, G.1^2); (1 1 3) > z := func< x, y | z1(x, y) + z1(x, y) >; > IdentifyTwoCocycle(X, z); (2)1); (1) > C := CohomologyGroup(X,1); > C; Full Quotient RSpace of degree 2 over Integer Ring Column moduli: [ 2, 2 ] > z1 := OneCocycle(X,[1,0]); > z2 := OneCocycle(X,[0,1]); > z1(G.1); (1 0 0) > z := func< x | z1(x)+z2(x) >; > IdentifyOneCocycle(X,z); (1 1) > C := CohomologyGroup(X,2); > C; Full Quotient RSpace of degree 1 over Integer Ring Column moduli: [ 4 ] > z1 := TwoCocycle(X,[1]); > z1(G.1,G.1^2); (1 1 3) > z := func< x,y | z1(x,y)+z1(x,y) >; > IdentifyTwoCocycle(X,z); (2)