Here are some examples of change of rings spectral sequences.
Given a ring map f: R -> S, an R-module M and an R-module S, there is a spectral sequence E with E^2_{p,q} = Tor^S_p(Tor^R_q(M,S),N) that abuts to Tor^R_{p+q}(M,N).
i1 : k=QQ; |
i2 : R=k[a,b,c]; |
i3 : S=k[s,t]; |
i4 : f = map(S,R,{s^2,s*t,t^2}); o4 : RingMap S <--- R |
i5 : N = coker vars S; |
i6 : M = coker vars R --; o6 = cokernel | a b c | 1 o6 : R-module, quotient of R |
i7 : F := complete res N; |
i8 : pushFwdF := pushFwd(f,F); |
i9 : G := complete res M; |
i10 : E := spectralSequence(filteredComplex(G) ** pushFwdF); |
i11 : EE := spectralSequence(G ** (filteredComplex pushFwdF)); |
i12 : e = prune E; |
i13 : ee = prune EE; |
i14 : e^0 +----------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------+ o14 = |cokernel {2} | b2-ac 0 0 | |cokernel {3} | b2-ac 0 0 0 0 0 0 0 0 | |cokernel {4} | b2-ac 0 0 0 0 0 0 0 0 | |cokernel {5} | b2-ac 0 0 | | | {3} | 0 -c -b | | {4} | 0 -c -b 0 0 0 0 0 0 | | {5} | 0 -c -b 0 0 0 0 0 0 | | {6} | 0 -c -b | | | {3} | 0 b a | | {4} | 0 b a 0 0 0 0 0 0 | | {5} | 0 b a 0 0 0 0 0 0 | | {6} | 0 b a | | | | {3} | 0 0 0 b2-ac 0 0 0 0 0 | | {4} | 0 0 0 b2-ac 0 0 0 0 0 | | | |{0, 2} | {4} | 0 0 0 0 -c -b 0 0 0 | | {5} | 0 0 0 0 -c -b 0 0 0 | |{3, 2} | | | {4} | 0 0 0 0 b a 0 0 0 | | {5} | 0 0 0 0 b a 0 0 0 | | | | | {3} | 0 0 0 0 0 0 b2-ac 0 0 | | {4} | 0 0 0 0 0 0 b2-ac 0 0 | | | | | {4} | 0 0 0 0 0 0 0 -c -b | | {5} | 0 0 0 0 0 0 0 -c -b | | | | | {4} | 0 0 0 0 0 0 0 b a | | {5} | 0 0 0 0 0 0 0 b a | | | | | | | | | |{1, 2} |{2, 2} | | +----------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------+ |cokernel {1} | b2-ac 0 0 0 0 0 ||cokernel {2} | b2-ac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||cokernel {3} | b2-ac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||cokernel {4} | b2-ac 0 0 0 0 0 || | {1} | 0 b2-ac 0 0 0 0 || {2} | 0 b2-ac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {3} | 0 b2-ac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {4} | 0 b2-ac 0 0 0 0 || | {2} | 0 0 -c -b 0 0 || {3} | 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {4} | 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {5} | 0 0 -c -b 0 0 || | {2} | 0 0 0 0 -c -b || {3} | 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 || {4} | 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 || {5} | 0 0 0 0 -c -b || | {2} | 0 0 b a 0 0 || {3} | 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {4} | 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {5} | 0 0 b a 0 0 || | {2} | 0 0 0 0 b a || {3} | 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 || {4} | 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 || {5} | 0 0 0 0 b a || | | {2} | 0 0 0 0 0 0 b2-ac 0 0 0 0 0 0 0 0 0 0 0 || {3} | 0 0 0 0 0 0 b2-ac 0 0 0 0 0 0 0 0 0 0 0 || | |{0, 1} | {2} | 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 0 0 0 0 0 || {3} | 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 0 0 0 0 0 ||{3, 1} | | | {3} | 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 || {4} | 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 || | | | {3} | 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 || {4} | 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 || | | | {3} | 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 || {4} | 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 || | | | {3} | 0 0 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 || {4} | 0 0 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 || | | | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 || {3} | 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 || | | | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 || {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 || | | | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 || {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 || | | | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -c -b || {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -c -b || | | | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a 0 0 || {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a 0 0 || | | | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a || {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a || | | | | | | | |{1, 1} |{2, 1} | | +----------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------+ |cokernel {0} | b2-ac 0 0 | |cokernel | b2-ac 0 0 0 0 0 0 0 0 | |cokernel | b2-ac 0 0 0 0 0 0 0 0 | |cokernel | b2-ac 0 0 | | | {1} | 0 -c -b | | | 0 -c -b 0 0 0 0 0 0 | | | 0 -c -b 0 0 0 0 0 0 | | | 0 -c -b | | | {1} | 0 b a | | | 0 b a 0 0 0 0 0 0 | | | 0 b a 0 0 0 0 0 0 | | | 0 b a | | | | | 0 0 0 b2-ac 0 0 0 0 0 | | | 0 0 0 b2-ac 0 0 0 0 0 | | | |{0, 0} | | 0 0 0 0 -c -b 0 0 0 | | | 0 0 0 0 -c -b 0 0 0 | |{3, 0} | | | | 0 0 0 0 b a 0 0 0 | | | 0 0 0 0 b a 0 0 0 | | | | | | 0 0 0 0 0 0 b2-ac 0 0 | | | 0 0 0 0 0 0 b2-ac 0 0 | | | | | | 0 0 0 0 0 0 0 -c -b | | | 0 0 0 0 0 0 0 -c -b | | | | | | 0 0 0 0 0 0 0 b a | | | 0 0 0 0 0 0 0 b a | | | | | | | | | |{1, 0} |{2, 0} | | +----------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------+ o14 : SpectralSequencePage |
i15 : e^1 +------------------+------------------------------+------------------------------+------------------+ o15 = |cokernel | c b a ||cokernel | c b a 0 0 0 0 0 0 ||cokernel | c b a 0 0 0 0 0 0 ||cokernel | c b a || | | | 0 0 0 c b a 0 0 0 || | 0 0 0 c b a 0 0 0 || | |{0, 0} | | 0 0 0 0 0 0 c b a || | 0 0 0 0 0 0 c b a ||{3, 0} | | | | | | | |{1, 0} |{2, 0} | | +------------------+------------------------------+------------------------------+------------------+ o15 : SpectralSequencePage |
i16 : e^2 +------------------+------------------------------+------------------------------+------------------+ o16 = |cokernel | c b a ||cokernel | c b a 0 0 0 0 0 0 ||cokernel | c b a 0 0 0 0 0 0 ||cokernel | c b a || | | | 0 0 0 c b a 0 0 0 || | 0 0 0 c b a 0 0 0 || | |{0, 0} | | 0 0 0 0 0 0 c b a || | 0 0 0 0 0 0 c b a ||{3, 0} | | | | | | | |{1, 0} |{2, 0} | | +------------------+------------------------------+------------------------------+------------------+ o16 : SpectralSequencePage |
i17 : e^infinity +------------------+------------------------------+------------------------------+------------------+ o17 = |cokernel | c b a ||cokernel | c b a 0 0 0 0 0 0 ||cokernel | c b a 0 0 0 0 0 0 ||cokernel | c b a || | | | 0 0 0 c b a 0 0 0 || | 0 0 0 c b a 0 0 0 || | |{0, 0} | | 0 0 0 0 0 0 c b a || | 0 0 0 0 0 0 c b a ||{3, 0} | | | | | | | |{1, 0} |{2, 0} | | +------------------+------------------------------+------------------------------+------------------+ o17 : Page |
i18 : ee^0 +----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+ o18 = |cokernel {3} | b2-ac 0 0 | |cokernel {4} | b2-ac 0 0 0 0 0 | |cokernel {5} | b2-ac 0 0 | | | {4} | 0 -c -b | | {4} | 0 b2-ac 0 0 0 0 | | {6} | 0 -c -b | | | {4} | 0 b a | | {5} | 0 0 -c -b 0 0 | | {6} | 0 b a | | | | {5} | 0 0 0 0 -c -b | | | |{0, 3} | {5} | 0 0 b a 0 0 | |{2, 3} | | | {5} | 0 0 0 0 b a | | | | | | | | |{1, 3} | | +----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+ |cokernel {2} | b2-ac 0 0 0 0 0 0 0 0 ||cokernel {3} | b2-ac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||cokernel {4} | b2-ac 0 0 0 0 0 0 0 0 || | {3} | 0 -c -b 0 0 0 0 0 0 || {3} | 0 b2-ac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {5} | 0 -c -b 0 0 0 0 0 0 || | {3} | 0 b a 0 0 0 0 0 0 || {4} | 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {5} | 0 b a 0 0 0 0 0 0 || | {2} | 0 0 0 b2-ac 0 0 0 0 0 || {4} | 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 || {4} | 0 0 0 b2-ac 0 0 0 0 0 || | {3} | 0 0 0 0 -c -b 0 0 0 || {4} | 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {5} | 0 0 0 0 -c -b 0 0 0 || | {3} | 0 0 0 0 b a 0 0 0 || {4} | 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 || {5} | 0 0 0 0 b a 0 0 0 || | {2} | 0 0 0 0 0 0 b2-ac 0 0 || {3} | 0 0 0 0 0 0 b2-ac 0 0 0 0 0 0 0 0 0 0 0 || {4} | 0 0 0 0 0 0 b2-ac 0 0 || | {3} | 0 0 0 0 0 0 0 -c -b || {3} | 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 0 0 0 0 0 || {5} | 0 0 0 0 0 0 0 -c -b || | {3} | 0 0 0 0 0 0 0 b a || {4} | 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 || {5} | 0 0 0 0 0 0 0 b a || | | {4} | 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 || | |{0, 2} | {4} | 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 ||{2, 2} | | | {4} | 0 0 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 || | | | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 || | | | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 || | | | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 || | | | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -c -b || | | | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a 0 0 || | | | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a || | | | | | | |{1, 2} | | +----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+ |cokernel {1} | b2-ac 0 0 0 0 0 0 0 0 ||cokernel {2} | b2-ac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||cokernel | b2-ac 0 0 0 0 0 0 0 0 | | | {2} | 0 -c -b 0 0 0 0 0 0 || {2} | 0 b2-ac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | 0 -c -b 0 0 0 0 0 0 | | | {2} | 0 b a 0 0 0 0 0 0 || {3} | 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | 0 b a 0 0 0 0 0 0 | | | {1} | 0 0 0 b2-ac 0 0 0 0 0 || {3} | 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 || | 0 0 0 b2-ac 0 0 0 0 0 | | | {2} | 0 0 0 0 -c -b 0 0 0 || {3} | 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | 0 0 0 0 -c -b 0 0 0 | | | {2} | 0 0 0 0 b a 0 0 0 || {3} | 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 || | 0 0 0 0 b a 0 0 0 | | | {1} | 0 0 0 0 0 0 b2-ac 0 0 || {2} | 0 0 0 0 0 0 b2-ac 0 0 0 0 0 0 0 0 0 0 0 || | 0 0 0 0 0 0 b2-ac 0 0 | | | {2} | 0 0 0 0 0 0 0 -c -b || {2} | 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 0 0 0 0 0 || | 0 0 0 0 0 0 0 -c -b | | | {2} | 0 0 0 0 0 0 0 b a || {3} | 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 || | 0 0 0 0 0 0 0 b a | | | | {3} | 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 || | |{0, 1} | {3} | 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 ||{2, 1} | | | {3} | 0 0 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 || | | | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 || | | | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 || | | | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 || | | | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -c -b || | | | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a 0 0 || | | | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a || | | | | | | |{1, 1} | | +----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+ |cokernel {0} | b2-ac 0 0 | |cokernel {1} | b2-ac 0 0 0 0 0 | |cokernel | b2-ac 0 0 | | | {1} | 0 -c -b | | {1} | 0 b2-ac 0 0 0 0 | | | 0 -c -b | | | {1} | 0 b a | | {2} | 0 0 -c -b 0 0 | | | 0 b a | | | | {2} | 0 0 0 0 -c -b | | | |{0, 0} | {2} | 0 0 b a 0 0 | |{2, 0} | | | {2} | 0 0 0 0 b a | | | | | | | | |{1, 0} | | +----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+ o18 : SpectralSequencePage |