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.