# I-adic filtrations of chain complexes and their spectral sequences

By multiplying a chain complex by sucessive 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 some higher pages of the associated spectral sequence:

 ```i5 : E = prune spectralSequence K o5 = E o5 : SpectralSequence``` ```i6 : E^4 +-------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+------------------------------------------------+--------------------+ o6 = |cokernel {4} | -d -b 0 -c 0 0 -d -c 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 | | {4} | c a -d 0 -b 0 0 0 0 -d 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 || | | | | | {4} | 0 0 c b a -d 0 0 -d 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 ||{-3, 4} |{-2, 4} |{-1, 4} |{0, 4} | | {4} | 0 0 0 0 0 c b a 0 0 0 -d 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 || | | | | | {4} | 0 0 0 0 0 0 0 0 b a c 0 0 0 -d 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 || | | | | | {4} | 0 0 0 0 0 0 0 0 0 0 -d c b a 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 || | | | | | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a -d 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 || | | | | | {4} | 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 -d 0 0 0 -d 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 c b a -d 0 0 0 0 0 0 -d 0 0 0 -d 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 c b a -d 0 0 0 0 0 0 -d 0 0 0 -d 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 c b a -d 0 0 0 0 0 0 -d 0 0 0 -d 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 c b a -d 0 0 0 0 0 0 -d 0 0 0 -d 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 c b a -d 0 0 0 0 0 0 -d 0 0 0 -d || | | | | | {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 c b a -d 0 0 0 0 0 0 -d 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 c b a -d 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 -d 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 c b a || | | | | | | | | | | |{-4, 4} | | | | | +-------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+------------------------------------------------+--------------------+ |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 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 0 0 0 || | | | |{-4, 3} | {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 0 0 0 0 ||{-2, 3} |{-1, 3} |{0, 3} | | | {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 0 0 0 0 || | | | | | {3} | 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 || | | | | | {3} | 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 || | | | | | {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 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 || | | | | | {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 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 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 || | | | | | {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 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 || | | | | | {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 || | | | | | {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 d c b a 0 0 0 0 || | | | | | {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 || | | | | | | | | | | |{-3, 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 | | | | {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 || | | |{-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 ||{-1, 2} |{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 || | | | | | {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 || | | | | | {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 || | | | | | {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 || | | | | | {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 || | | | | | {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 || | | | | | | | | | | |{-2, 2} | | | +-------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+------------------------------------------------+--------------------+ |0 |0 |0 |cokernel {1} | d c b a 0 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 || | |{-4, 1} |{-3, 1} |{-2, 1} | {1} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 ||{0, 1} | | | | | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a || | | | | | | | | | | |{-1, 1} | | +-------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+------------------------------------------------+--------------------+ |0 |0 |0 |0 |cokernel | d c b a || | | | | | | |{-4, 0} |{-3, 0} |{-2, 0} |{-1, 0} |{0, 0} | +-------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+------------------------------------------------+--------------------+ o6 : SpectralSequencePage``` ```i7 : E^4 .dd o7 = {-9, 8} : 0 <----- 0 : {-5, 5} 0 {-9, 9} : 0 <----- 0 : {-5, 6} 0 {-9, 10} : 0 <----- 0 : {-5, 7} 0 {-9, 11} : 0 <----- 0 : {-5, 8} 0 {-9, 12} : 0 <----- 0 : {-5, 9} 0 {-4, 3} : 0 <----- cokernel | d c b a | : {0, 0} 0 {-4, 4} : cokernel {4} | -d -b 0 -c 0 0 -d -c 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, 1} {4} | c a -d 0 -b 0 0 0 0 -d 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 {4} | 0 0 c b a -d 0 0 -d 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 | {4} | 0 0 0 0 0 c b a 0 0 0 -d 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 | {4} | 0 0 0 0 0 0 0 0 b a c 0 0 0 -d 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 | {4} | 0 0 0 0 0 0 0 0 0 0 -d c b a 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 | {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c b a -d 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 | {4} | 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 -d 0 0 0 -d 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 c b a -d 0 0 0 0 0 0 -d 0 0 0 -d 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 c b a -d 0 0 0 0 0 0 -d 0 0 0 -d 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 c b a -d 0 0 0 0 0 0 -d 0 0 0 -d 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 c b a -d 0 0 0 0 0 0 -d 0 0 0 -d 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 c b a -d 0 0 0 0 0 0 -d 0 0 0 -d | {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 c b a -d 0 0 0 0 0 0 -d 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 c b a -d 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 -d 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 c b a | {-4, 5} : 0 <----- 0 : {0, 2} 0 {-4, 6} : 0 <----- 0 : {0, 3} 0 {-4, 7} : 0 <----- 0 : {0, 4} 0 {-5, 4} : 0 <----- cokernel {1} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 | : {-1, 1} 0 {1} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 | {1} | 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 | {-5, 5} : 0 <----- 0 : {-1, 2} 0 {-5, 6} : 0 <----- 0 : {-1, 3} 0 {-5, 7} : 0 <----- 0 : {-1, 4} 0 {-5, 8} : 0 <----- 0 : {-1, 5} 0 {-6, 5} : 0 <----- cokernel {2} | d c b a 0 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