Examples of change of rings Spectral Sequences

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 | 0 0 b2-ac | |cokernel {1} | 0 0 0 0 0 0 b2-ac 0 0 | |cokernel {2} | 0 0 0 0 0 0 b2-ac 0 0 | |cokernel {3} | 0 0 b2-ac | | | | -c -b 0 | | {1} | -c -b 0 0 0 0 0 0 0 | | {2} | -c -b 0 0 0 0 0 0 0 | | {3} | -c -b 0 | | | | b a 0 | | {1} | b a 0 0 0 0 0 0 0 | | {2} | b a 0 0 0 0 0 0 0 | | {3} | b a 0 | | | | {1} | 0 0 0 0 0 0 0 b2-ac 0 | | {2} | 0 0 0 0 0 0 0 b2-ac 0 | | | |{0, 2} | {1} | 0 0 -c -b 0 0 0 0 0 | | {2} | 0 0 -c -b 0 0 0 0 0 | |{3, 2} | | | {1} | 0 0 b a 0 0 0 0 0 | | {2} | 0 0 b a 0 0 0 0 0 | | | | | {1} | 0 0 0 0 0 0 0 0 b2-ac | | {2} | 0 0 0 0 0 0 0 0 b2-ac | | | | | {1} | 0 0 0 0 -c -b 0 0 0 | | {2} | 0 0 0 0 -c -b 0 0 0 | | | | | {1} | 0 0 0 0 b a 0 0 0 | | {2} | 0 0 0 0 b a 0 0 0 | | | | | | | | | |{1, 2} |{2, 2} | | +------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------+ |cokernel | 0 0 0 0 b2-ac 0 ||cokernel {1} | 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 ||cokernel {2} | 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 ||cokernel {3} | 0 0 0 0 b2-ac 0 || | | 0 0 0 0 0 b2-ac || {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 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 b2-ac || | | -c -b 0 0 0 0 || {1} | -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {3} | -c -b 0 0 0 0 || | | 0 0 -c -b 0 0 || {1} | 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {3} | 0 0 -c -b 0 0 || | | b a 0 0 0 0 || {1} | b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {3} | b a 0 0 0 0 || | | 0 0 b a 0 0 || {1} | 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {3} | 0 0 b a 0 0 || | | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 || {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 || | |{0, 1} | {1} | 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 0 0 0 0 0 0 0 0 b2-ac 0 0 ||{3, 1} | | | {1} | 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 || | | | {1} | 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 || | | | {1} | 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 || | | | {1} | 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 || | | | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 || {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 || | | | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac || {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac || | | | {1} | 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 || | | | {1} | 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 || {2} | 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 || | | | {1} | 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 || | | | {1} | 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 b a 0 0 0 0 0 0 || | | | | | | | |{1, 1} |{2, 1} | | +------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------+ |cokernel | 0 0 b2-ac | |cokernel | 0 0 0 0 0 0 b2-ac 0 0 | |cokernel | 0 0 0 0 0 0 b2-ac 0 0 | |cokernel | 0 0 b2-ac | | | | -c -b 0 | | | -c -b 0 0 0 0 0 0 0 | | | -c -b 0 0 0 0 0 0 0 | | | -c -b 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 0 0 0 0 0 b2-ac 0 | | | 0 0 0 0 0 0 0 b2-ac 0 | | | |{0, 0} | | 0 0 -c -b 0 0 0 0 0 | | | 0 0 -c -b 0 0 0 0 0 | |{3, 0} | | | | 0 0 b a 0 0 0 0 0 | | | 0 0 b a 0 0 0 0 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 -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 0 0 0 | | | | | | | | | |{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} | 0 0 b2-ac | |cokernel {3} | 0 0 0 0 b2-ac 0 | |cokernel {3} | 0 0 b2-ac | | | {3} | -c -b 0 | | {3} | 0 0 0 0 0 b2-ac | | {3} | -c -b 0 | | | {3} | b a 0 | | {3} | -c -b 0 0 0 0 | | {3} | b a 0 | | | | {3} | 0 0 -c -b 0 0 | | | |{0, 3} | {3} | b a 0 0 0 0 | |{2, 3} | | | {3} | 0 0 b a 0 0 | | | | | | | | |{1, 3} | | +----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+ |cokernel {2} | 0 0 0 0 0 0 b2-ac 0 0 ||cokernel {2} | 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 ||cokernel {2} | 0 0 0 0 0 0 b2-ac 0 0 || | {2} | -c -b 0 0 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 || {2} | -c -b 0 0 0 0 0 0 0 || | {2} | b a 0 0 0 0 0 0 0 || {2} | -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | b a 0 0 0 0 0 0 0 || | {2} | 0 0 0 0 0 0 0 b2-ac 0 || {2} | 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 0 0 0 b2-ac 0 || | {2} | 0 0 -c -b 0 0 0 0 0 || {2} | b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 -c -b 0 0 0 0 0 || | {2} | 0 0 b a 0 0 0 0 0 || {2} | 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 b a 0 0 0 0 0 || | {2} | 0 0 0 0 0 0 0 0 b2-ac || {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 || {2} | 0 0 0 0 0 0 0 0 b2-ac || | {2} | 0 0 0 0 -c -b 0 0 0 || {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 || {2} | 0 0 0 0 -c -b 0 0 0 || | {2} | 0 0 0 0 b a 0 0 0 || {2} | 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 b a 0 0 0 || | | {2} | 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 || | |{0, 2} | {2} | 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 ||{2, 2} | | | {2} | 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 || | | | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 || | | | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac || | | | {2} | 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 || | | | {2} | 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 || | | | {2} | 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 || | | | {2} | 0 0 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 || | | | | | | |{1, 2} | | +----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+ |cokernel {1} | 0 0 0 0 0 0 b2-ac 0 0 ||cokernel {1} | 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 ||cokernel | 0 0 0 0 0 0 b2-ac 0 0 | | | {1} | -c -b 0 0 0 0 0 0 0 || {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 || | -c -b 0 0 0 0 0 0 0 | | | {1} | b a 0 0 0 0 0 0 0 || {1} | -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | b a 0 0 0 0 0 0 0 | | | {1} | 0 0 0 0 0 0 0 b2-ac 0 || {1} | 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | 0 0 0 0 0 0 0 b2-ac 0 | | | {1} | 0 0 -c -b 0 0 0 0 0 || {1} | 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 0 0 | | | {1} | 0 0 b a 0 0 0 0 0 || {1} | 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 0 0 | | | {1} | 0 0 0 0 0 0 0 0 b2-ac || {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 || | 0 0 0 0 0 0 0 0 b2-ac | | | {1} | 0 0 0 0 -c -b 0 0 0 || {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 || | 0 0 0 0 -c -b 0 0 0 | | | {1} | 0 0 0 0 b a 0 0 0 || {1} | 0 0 0 0 -c -b 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 -c -b 0 0 0 0 0 0 0 0 0 0 || | |{0, 1} | {1} | 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 ||{2, 1} | | | {1} | 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 || | | | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 || | | | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac || | | | {1} | 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 || | | | {1} | 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 || | | | {1} | 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 || | | | {1} | 0 0 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 || | | | | | | |{1, 1} | | +----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+ |cokernel | 0 0 b2-ac | |cokernel | 0 0 0 0 b2-ac 0 | |cokernel | 0 0 b2-ac | | | | -c -b 0 | | | 0 0 0 0 0 b2-ac | | | -c -b 0 | | | | b a 0 | | | -c -b 0 0 0 0 | | | b a 0 | | | | | 0 0 -c -b 0 0 | | | |{0, 0} | | b a 0 0 0 0 | |{2, 0} | | | | 0 0 b a 0 0 | | | | | | | | |{1, 0} | | +----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+ o18 : SpectralSequencePage