By multiplying a chain complex by successive powers of an ideal we obtain a filtered complex.
i1 : B = QQ[a..d] o1 = B o1 : PolynomialRing |
i2 : J = ideal vars B o2 = ideal (a, b, c, d) o2 : Ideal of B |
i3 : C = complete res monomialCurveIdeal(B,{1,3,4})
1 4 4 1
o3 = B <-- B <-- B <-- B <-- 0
0 1 2 3 4
o3 : ChainComplex
|
i4 : K = filteredComplex(J,C,4)
o4 = -5 : image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0
0 1 2 3 4
-4 : image | a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 b4 b3c b3d b2c2 b2cd b2d2 bc3 bc2d bcd2 bd3 c4 c3d c2d2 cd3 d4 | <-- image {2} | a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 0 | <-- image {4} | a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 0 | <-- image {5} | a4 a3b a3c a3d a2b2 a2bc a2bd a2c2 a2cd a2d2 ab3 ab2c ab2d abc2 abcd abd2 ac3 ac2d acd2 ad3 b4 b3c b3d b2c2 b2cd b2d2 bc3 bc2d bcd2 bd3 c4 c3d c2d2 cd3 d4 | <-- image 0
{3} | 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 | {4} | 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 0 |
0 {3} | 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 | {4} | 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 0 | 3 4
{3} | 0 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 | {4} | 0 0 0 a4 0 0 0 a3b 0 0 0 a3c 0 0 0 a3d 0 0 0 a2b2 0 0 0 a2bc 0 0 0 a2bd 0 0 0 a2c2 0 0 0 a2cd 0 0 0 a2d2 0 0 0 ab3 0 0 0 ab2c 0 0 0 ab2d 0 0 0 abc2 0 0 0 abcd 0 0 0 abd2 0 0 0 ac3 0 0 0 ac2d 0 0 0 acd2 0 0 0 ad3 0 0 0 b4 0 0 0 b3c 0 0 0 b3d 0 0 0 b2c2 0 0 0 b2cd 0 0 0 b2d2 0 0 0 bc3 0 0 0 bc2d 0 0 0 bcd2 0 0 0 bd3 0 0 0 c4 0 0 0 c3d 0 0 0 c2d2 0 0 0 cd3 0 0 0 d4 |
1 2
-3 : image | a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 b3 b2c b2d bc2 bcd bd2 c3 c2d cd2 d3 | <-- image {2} | a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 | <-- image {4} | a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 0 | <-- image {5} | a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 b3 b2c b2d bc2 bcd bd2 c3 c2d cd2 d3 | <-- image 0
{3} | 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 | {4} | 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 0 |
0 {3} | 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 | {4} | 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 0 | 3 4
{3} | 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 | {4} | 0 0 0 a3 0 0 0 a2b 0 0 0 a2c 0 0 0 a2d 0 0 0 ab2 0 0 0 abc 0 0 0 abd 0 0 0 ac2 0 0 0 acd 0 0 0 ad2 0 0 0 b3 0 0 0 b2c 0 0 0 b2d 0 0 0 bc2 0 0 0 bcd 0 0 0 bd2 0 0 0 c3 0 0 0 c2d 0 0 0 cd2 0 0 0 d3 |
1 2
-2 : image | a2 ab ac ad b2 bc bd c2 cd d2 | <-- image {2} | a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 | <-- image {4} | a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 0 | <-- image {5} | a2 ab ac ad b2 bc bd c2 cd d2 | <-- image 0
{3} | 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 | {4} | 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 0 |
0 {3} | 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 | {4} | 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 0 | 3 4
{3} | 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 | {4} | 0 0 0 a2 0 0 0 ab 0 0 0 ac 0 0 0 ad 0 0 0 b2 0 0 0 bc 0 0 0 bd 0 0 0 c2 0 0 0 cd 0 0 0 d2 |
1 2
-1 : image | a b c d | <-- image {2} | a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 | <-- image {4} | a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 | <-- image {5} | a b c d | <-- image 0
{3} | 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 | {4} | 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 |
0 {3} | 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 | {4} | 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 | 3 4
{3} | 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d | {4} | 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d |
1 2
1 4 4 1
0 : B <-- B <-- B <-- B <-- 0
0 1 2 3 4
o4 : FilteredComplex
|
Here are higher some pages of the associated spectral sequence:
i5 : e = prune spectralSequence K o5 = e o5 : SpectralSequence |
i6 : e^2
+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------+------------------------------------------------+
o6 = |cokernel {4} | -d -b -d -c 0 0 -d 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c2 -ac 0 0 -cd -ad 0 0 0 -d2 -bd 0 0 0 0 0 0 0 0 0 0 0 0 0 ||0 |0 |0 |0 |
| {4} | c a 0 0 0 -d 0 -d -b 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -bd b2 -d2 0 0 0 0 0 0 0 0 0 0 0 0 0 -d2 -bd 0 0 0 0 0 0 || | | | |
| {4} | 0 0 b a -d 0 0 0 0 0 -d 0 0 0 0 0 -d -c 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c2 cd 0 0 0 0 -d2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||{-3, 4} |{-2, 4} |{-1, 4} |{0, 4} |
| {4} | 0 0 0 0 c b a 0 0 -d 0 0 0 -d 0 0 0 0 0 -d 0 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 -d 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d2 0 0 0 0 0 0 0 -d2 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 c a 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 -d -b 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d2 bd b2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 -d 0 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 0 0 0 -d 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 -d -c 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cd c2 0 0 0 -d2 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a -d 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 -d 0 0 0 0 0 0 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d2 0 0 0 -d2 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d2 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c a 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d -b 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d2 bd b2 0 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d2 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d2 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b a -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 -d -c 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d2 cd c2 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d2 0 0 -d2 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d2 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -d2 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c a 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d2 bd b2 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d2 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d2 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
| {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | | | |
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| | | | | |
|{-4, 4} | | | | |
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| | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | |
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| | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | |
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| | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | |
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| |{-3, 3} | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | |
| | | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 || | |
| | | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 || | |
| | | {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a || | |
| | | | | |
| | |{-2, 3} | | |
+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------+------------------------------------------------+
|0 |0 |cokernel {2} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | |cokernel {3} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||0 |
| | | {2} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || |
|{-4, 2} |{-3, 2} | {2} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | {3} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||{0, 2} |
| | | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | {3} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 | | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 | | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 | | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a | | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 || |
| | | | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 || |
| | |{-2, 2} | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a || |
| | | | | |
| | | |{-1, 2} | |
+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------+------------------------------------------------+
|0 |0 |0 |cokernel {1} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 | |cokernel {2} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 ||
| | | | {1} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 | | {3} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 ||
|{-4, 1} |{-3, 1} |{-2, 1} | {1} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 | | {3} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 ||
| | | | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a | | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a ||
| | | | | |
| | | |{-1, 1} |{0, 1} |
+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------+------------------------------------------------+
|0 |0 |0 |0 |cokernel | d c b a | |
| | | | | |
|{-4, 0} |{-3, 0} |{-2, 0} |{-1, 0} |{0, 0} |
+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------+------------------------------------------------+
o6 : SpectralSequencePage
|
i7 : assert(all(keys support e^0, j -> isIsomorphism homologyIsomorphism(e,j#0,j#1,0))) |
i8 : assert(all(keys support e^1, j -> isIsomorphism homologyIsomorphism(e,j#0,j#1,1))) |
i9 : assert(all(keys support e^2, j -> isIsomorphism homologyIsomorphism(e,j#0,j#1,2))) |
i10 : assert(all(keys support e^3, j -> isIsomorphism homologyIsomorphism(e,j#0,j#1,3))) |
i11 : assert(all(keys support e^4, j -> isIsomorphism homologyIsomorphism(e,j#0,j#1,4))) |