# Balancing Tor

To balance Tor we first need to make some modules over a ring.

 i1 : A = QQ[x,y,z,w]; i2 : M = monomialCurveIdeal(A,{1,2,3}); o2 : Ideal of A i3 : N = monomialCurveIdeal(A,{1,3,4}); o3 : Ideal of A

To compute $Tor^A_i(M,N)$ we resolve the modules, tensor appropriately, and then take homology.

 i4 : K = res M 1 3 2 o4 = A <-- A <-- A <-- 0 0 1 2 3 o4 : ChainComplex i5 : J = res N 1 4 4 1 o5 = A <-- A <-- A <-- A <-- 0 0 1 2 3 4 o5 : ChainComplex

The spectral sequence that computes $Tor^A_i(M,N)$ by tensoring $K$ with $N$ and taking homology is given by

 i6 : E = prune spectralSequence((filteredComplex K) ** J) o6 = E o6 : SpectralSequence

The spectral sequence that computes $Tor^A_i(M,N)$ by tensoring $J$ with $M$ and taking homology is given by

 i7 : F = prune spectralSequence((K ** (filteredComplex J))) o7 = F o7 : SpectralSequence

Let's compute some pages and maps of these spectral sequences. The zeroth pages takes the form:

 i8 : E^0 +------+------+------+ | 1 | 3 | 2 | o8 = |A |A |A | | | | | |{0, 3}|{1, 3}|{2, 3}| +------+------+------+ | 4 | 12 | 8 | |A |A |A | | | | | |{0, 2}|{1, 2}|{2, 2}| +------+------+------+ | 4 | 12 | 8 | |A |A |A | | | | | |{0, 1}|{1, 1}|{2, 1}| +------+------+------+ | 1 | 3 | 2 | |A |A |A | | | | | |{0, 0}|{1, 0}|{2, 0}| +------+------+------+ o8 : SpectralSequencePage i9 : E^0 .dd o9 = {2, -3} : 0 <----- 0 : {2, -2} 0 {2, -2} : 0 <----- 0 : {2, -1} 0 2 {2, -1} : 0 <----- A : {2, 0} 0 2 8 {2, 0} : A <------------------------------------------------------------------- A : {2, 1} {3} | yz-xw y3-x2z xz2-y2w z3-yw2 0 0 0 0 | {3} | 0 0 0 0 yz-xw y3-x2z xz2-y2w z3-yw2 | 8 8 {2, 1} : A <------------------------------------------- A : {2, 2} {5} | -y2 -xz -yw -z2 0 0 0 0 | {6} | z w 0 0 0 0 0 0 | {6} | x y -z -w 0 0 0 0 | {6} | 0 0 x y 0 0 0 0 | {5} | 0 0 0 0 -y2 -xz -yw -z2 | {6} | 0 0 0 0 z w 0 0 | {6} | 0 0 0 0 x y -z -w | {6} | 0 0 0 0 0 0 x y | 8 2 {2, 2} : A <----------------- A : {2, 3} {7} | w 0 | {7} | -z 0 | {7} | -y 0 | {7} | x 0 | {7} | 0 w | {7} | 0 -z | {7} | 0 -y | {7} | 0 x | 2 {2, 3} : A <----- 0 : {2, 4} 0 {2, 4} : 0 <----- 0 : {2, 5} 0 {1, -2} : 0 <----- 0 : {1, -1} 0 3 {1, -1} : 0 <----- A : {1, 0} 0 3 12 {1, 0} : A <----------------------------------------------------------------------------------------------------------- A : {1, 1} {2} | -yz+xw -y3+x2z -xz2+y2w -z3+yw2 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 -yz+xw -y3+x2z -xz2+y2w -z3+yw2 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 -yz+xw -y3+x2z -xz2+y2w -z3+yw2 | 12 12 {1, 1} : A <----------------------------------------------- A : {1, 2} {4} | y2 xz yw z2 0 0 0 0 0 0 0 0 | {5} | -z -w 0 0 0 0 0 0 0 0 0 0 | {5} | -x -y z w 0 0 0 0 0 0 0 0 | {5} | 0 0 -x -y 0 0 0 0 0 0 0 0 | {4} | 0 0 0 0 y2 xz yw z2 0 0 0 0 | {5} | 0 0 0 0 -z -w 0 0 0 0 0 0 | {5} | 0 0 0 0 -x -y z w 0 0 0 0 | {5} | 0 0 0 0 0 0 -x -y 0 0 0 0 | {4} | 0 0 0 0 0 0 0 0 y2 xz yw z2 | {5} | 0 0 0 0 0 0 0 0 -z -w 0 0 | {5} | 0 0 0 0 0 0 0 0 -x -y z w | {5} | 0 0 0 0 0 0 0 0 0 0 -x -y | 12 3 {1, 2} : A <-------------------- A : {1, 3} {6} | -w 0 0 | {6} | z 0 0 | {6} | y 0 0 | {6} | -x 0 0 | {6} | 0 -w 0 | {6} | 0 z 0 | {6} | 0 y 0 | {6} | 0 -x 0 | {6} | 0 0 -w | {6} | 0 0 z | {6} | 0 0 y | {6} | 0 0 -x | 3 {1, 3} : A <----- 0 : {1, 4} 0 {1, 4} : 0 <----- 0 : {1, 5} 0 {1, 5} : 0 <----- 0 : {1, 6} 0 1 {0, -1} : 0 <----- A : {0, 0} 0 1 4 {0, 0} : A <----------------------------------- A : {0, 1} | yz-xw y3-x2z xz2-y2w z3-yw2 | 4 4 {0, 1} : A <--------------------------- A : {0, 2} {2} | -y2 -xz -yw -z2 | {3} | z w 0 0 | {3} | x y -z -w | {3} | 0 0 x y | 4 1 {0, 2} : A <-------------- A : {0, 3} {4} | w | {4} | -z | {4} | -y | {4} | x | 1 {0, 3} : A <----- 0 : {0, 4} 0 {0, 4} : 0 <----- 0 : {0, 5} 0 {0, 5} : 0 <----- 0 : {0, 6} 0 {0, 6} : 0 <----- 0 : {0, 7} 0 {-1, 0} : 0 <----- 0 : {-1, 1} 0 {-1, 1} : 0 <----- 0 : {-1, 2} 0 {-1, 2} : 0 <----- 0 : {-1, 3} 0 {-1, 3} : 0 <----- 0 : {-1, 4} 0 {-1, 4} : 0 <----- 0 : {-1, 5} 0 {-1, 5} : 0 <----- 0 : {-1, 6} 0 {-1, 6} : 0 <----- 0 : {-1, 7} 0 {-1, 7} : 0 <----- 0 : {-1, 8} 0 o9 : SpectralSequencePageMap i10 : F^0 +------+------+------+------+ | 2 | 8 | 8 | 2 | o10 = |A |A |A |A | | | | | | |{0, 2}|{1, 2}|{2, 2}|{3, 2}| +------+------+------+------+ | 3 | 12 | 12 | 3 | |A |A |A |A | | | | | | |{0, 1}|{1, 1}|{2, 1}|{3, 1}| +------+------+------+------+ | 1 | 4 | 4 | 1 | |A |A |A |A | | | | | | |{0, 0}|{1, 0}|{2, 0}|{3, 0}| +------+------+------+------+ o10 : SpectralSequencePage

The first pages take the form:

 i11 : E^1 +----------------------------------------+----------------------------------------------------------------------------------------------------+------------------------------------------------------------------------+ o11 = |cokernel | yz-xw z3-yw2 xz2-y2w y3-x2z ||cokernel {2} | yz-xw 0 0 z3-yw2 xz2-y2w y3-x2z 0 0 0 0 0 0 ||cokernel {3} | yz-xw 0 z3-yw2 xz2-y2w y3-x2z 0 0 0 || | | {2} | 0 yz-xw 0 0 0 0 z3-yw2 xz2-y2w y3-x2z 0 0 0 || {3} | 0 yz-xw 0 0 0 z3-yw2 xz2-y2w y3-x2z || |{0, 0} | {2} | 0 0 yz-xw 0 0 0 0 0 0 z3-yw2 xz2-y2w y3-x2z || | | | |{2, 0} | | |{1, 0} | | +----------------------------------------+----------------------------------------------------------------------------------------------------+------------------------------------------------------------------------+ o11 : SpectralSequencePage i12 : F^1 +------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------+ o12 = |cokernel | z2-yw yz-xw y2-xz ||cokernel {2} | z2-yw yz-xw y2-xz 0 0 0 0 0 0 0 0 0 ||cokernel {4} | z2-yw yz-xw y2-xz 0 0 0 0 0 0 0 0 0 ||cokernel {5} | z2-yw yz-xw y2-xz || | | {3} | 0 0 0 z2-yw yz-xw y2-xz 0 0 0 0 0 0 || {4} | 0 0 0 z2-yw yz-xw y2-xz 0 0 0 0 0 0 || | |{0, 0} | {3} | 0 0 0 0 0 0 z2-yw yz-xw y2-xz 0 0 0 || {4} | 0 0 0 0 0 0 z2-yw yz-xw y2-xz 0 0 0 ||{3, 0} | | | {3} | 0 0 0 0 0 0 0 0 0 z2-yw yz-xw y2-xz || {4} | 0 0 0 0 0 0 0 0 0 z2-yw yz-xw y2-xz || | | | | | | | |{1, 0} |{2, 0} | | +------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------+ o12 : SpectralSequencePage

The second pages take the form:

 i13 : E^2 +------------------------------------------------------+--------------------------------------------------------------------------+ o13 = |cokernel | z2-yw yz-xw y2-xz xw2-yw2 xyw-xzw x2z-x2w ||cokernel {2} | z2-yw yz-xw y2-xz xw2-yw2 xzw-yw2 xyw-yw2 x2w-yw2 x2z-yw2 || | | | |{0, 0} |{1, 0} | +------------------------------------------------------+--------------------------------------------------------------------------+ o13 : SpectralSequencePage i14 : F^2 +------------------------------------------------------+--------------------------------------------------------------------------+ o14 = |cokernel | z2-yw yz-xw y2-xz xw2-yw2 xyw-xzw x2z-x2w ||cokernel {2} | z2-yw yz-xw y2-xz xw2-yw2 xzw-yw2 xyw-yw2 x2w-yw2 x2z-yw2 || | | | |{0, 0} |{1, 0} | +------------------------------------------------------+--------------------------------------------------------------------------+ o14 : SpectralSequencePage

Observe that $E^2$ and $F^2$ are equal as they should.