Given a 1×n matrix f, and a chain complex F, the script attempts to make a family of higher homotopies on F for the elements of f, in the sense described, for example, in Eisenbud "Enriched Free Resolutions and Change of Rings".
The output is a hash table with entries of the form {J,i}=>s, where J is a list of non-negative integers, of length n and H#{J,i}: Fi->Fi+2|J|-1 are maps satisfying the conditions
and
where e0 = {0,…,0} and e is the index of degree 1 with a 1 in the i-th place; and, for each index list I with |I|<=d,
To make the maps homogeneous, H#{J,i} is actually a map from a an appropriate negative twist of F to a shift of S.
i1 : kk=ZZ/101 o1 = kk o1 : QuotientRing |
i2 : S = kk[a,b,c,d] o2 = S o2 : PolynomialRing |
i3 : F = res ideal vars S 1 4 6 4 1 o3 = S <-- S <-- S <-- S <-- S <-- 0 0 1 2 3 4 5 o3 : ChainComplex |
i4 : f = matrix{{a,b,c}} o4 = | a b c | 1 3 o4 : Matrix S <--- S |
i5 : homot = makeHomotopies(f,F,2) o5 = HashTable{{{0, 0, 0}, 0} => 0 } {{0, 0, 0}, 1} => | a b c d | {{0, 0, 0}, 2} => {1} | -b -c 0 -d 0 0 | {1} | a 0 -c 0 -d 0 | {1} | 0 a b 0 0 -d | {1} | 0 0 0 a b c | {{0, 0, 0}, 3} => {2} | c d 0 0 | {2} | -b 0 d 0 | {2} | a 0 0 d | {2} | 0 -b -c 0 | {2} | 0 a 0 -c | {2} | 0 0 a b | {{0, 0, 1}, -1} => 0 {{0, 0, 1}, 0} => {1} | 0 | {1} | 0 | {1} | 1 | {1} | 0 | {{0, 0, 1}, 1} => {2} | 0 0 0 0 | {2} | -1 0 0 0 | {2} | 0 -1 0 0 | {2} | 0 0 0 0 | {2} | 0 0 0 0 | {2} | 0 0 0 1 | {{0, 0, 1}, 2} => {3} | 1 0 0 0 0 0 | {3} | 0 0 0 0 0 0 | {3} | 0 0 0 -1 0 0 | {3} | 0 0 0 0 -1 0 | {{0, 0, 2}, -1} => 0 {{0, 1, 0}, -1} => 0 {{0, 1, 0}, 0} => {1} | 0 | {1} | 1 | {1} | 0 | {1} | 0 | {{0, 1, 0}, 1} => {2} | -1 0 0 0 | {2} | 0 0 0 0 | {2} | 0 0 1 0 | {2} | 0 0 0 0 | {2} | 0 0 0 1 | {2} | 0 0 0 0 | {{0, 1, 0}, 2} => {3} | 0 -1 0 0 0 0 | {3} | 0 0 0 -1 0 0 | {3} | 0 0 0 0 0 0 | {3} | 0 0 0 0 0 1 | {{0, 1, 1}, -1} => 0 {{0, 2, 0}, -1} => 0 {{1, 0, 0}, -1} => 0 {{1, 0, 0}, 0} => {1} | 1 | {1} | 0 | {1} | 0 | {1} | 0 | {{1, 0, 0}, 1} => {2} | 0 1 0 0 | {2} | 0 0 1 0 | {2} | 0 0 0 0 | {2} | 0 0 0 1 | {2} | 0 0 0 0 | {2} | 0 0 0 0 | {{1, 0, 0}, 2} => {3} | 0 0 1 0 0 0 | {3} | 0 0 0 0 1 0 | {3} | 0 0 0 0 0 1 | {3} | 0 0 0 0 0 0 | {{1, 0, 1}, -1} => 0 {{1, 1, 0}, -1} => 0 {{2, 0, 0}, -1} => 0 o5 : HashTable |
In this case the higher homotopies are 0:
i6 : L = sort select(keys homot, k->(homot#k!=0 and sum(k_0)>1)) o6 = {} o6 : List |
On the other hand, if we take a complete intersection and something contained in it in a more complicated situation, the program gives nonzero higher homotopies:
i7 : kk= ZZ/32003; |
i8 : S = kk[a,b,c,d]; |
i9 : M = S^1/(ideal"a2,b2,c2,d2"); |
i10 : F = res M 1 4 6 4 1 o10 = S <-- S <-- S <-- S <-- S <-- 0 0 1 2 3 4 5 o10 : ChainComplex |
i11 : setRandomSeed 0 o11 = 0 |
i12 : f = random(S^1,S^{2:-5}); 1 2 o12 : Matrix S <--- S |
i13 : homot = makeHomotopies(f,F,5) o13 = HashTable{{{0, 0}, 0} => 0 } {{0, 0}, 1} => | a2 b2 c2 d2 | {{0, 0}, 2} => {2} | -b2 -c2 0 -d2 0 0 | {2} | a2 0 -c2 0 -d2 0 | {2} | 0 a2 b2 0 0 -d2 | {2} | 0 0 0 a2 b2 c2 | {{0, 0}, 3} => {4} | c2 d2 0 0 | {4} | -b2 0 d2 0 | {4} | a2 0 0 d2 | {4} | 0 -b2 -c2 0 | {4} | 0 a2 0 -c2 | {4} | 0 0 a2 b2 | {{0, 0}, 4} => {6} | -d2 | {6} | c2 | {6} | -b2 | {6} | a2 | {{0, 0}, 5} => 0 {{0, 0}, 6} => 0 {{0, 1}, -1} => 0 {{0, 1}, 0} => {2} | 1236a3+8922a2b-3589ab2+8971b3-5006a2c-5599abc-14165b2c+8880a2d+4259abd-3002b2d-13892acd-10521bcd | {2} | 10370ab2-7092b3-9702abc-6627b2c+8886abd+4700b2d-15acd+5969bcd | {2} | 13707a3+2177a2b-7028ab2+9797b3-7021a2c+6377abc+5874b2c-7600ac2-11726bc2+1140c3+1206a2d-11435abd+9074b2d-6040acd+8022bcd+3968c2d | {2} | -994a3-1946a2b-6723ab2-5483b3-6453a2c-1192abc-15250b2c-3164ac2+295bc2-7650c3-11045a2d-15333abd+10567b2d-3560acd+2292bcd+14388c2d-7194ad2+6245bd2-8639cd2+9426d3 | {{0, 1}, 1} => {4} | -10370ab2+7092b3+9702abc+6627b2c+7028ac2-9797bc2-5874c3-8886abd-4700b2d+15acd-5969bcd-9074c2d+6723ad2+5483bd2+15250cd2-10567d3 1236a3+8922a2b-3589ab2+8971b3-5006a2c-5599abc-14165b2c+8880a2d+4259abd-3002b2d-13892acd-10521bcd 0 0 | {4} | -13707a3-2177a2b+7021a2c-6377abc+7600ac2+11726bc2-1140c3-1206a2d+11435abd+6040acd-8022bcd-3968c2d+3164ad2-295bd2+7650cd2-14388d3 0 1236a3+8922a2b-3589ab2+8971b3-5006a2c-5599abc-14165b2c+8880a2d+4259abd-3002b2d-13892acd-10521bcd 0 | {4} | 7028a3-9797a2b-5874a2c-9074a2d -13707a3-2177a2b+7028ab2-9797b3+7021a2c-6377abc-5874b2c+7600ac2+11726bc2-1140c3-1206a2d+11435abd-9074b2d+6040acd-8022bcd-3968c2d+3164ad2-295bd2+7650cd2-14388d3 10370ab2-7092b3-9702abc-6627b2c+8886abd+4700b2d-15acd+5969bcd 0 | {4} | 994a3+1946a2b+6453a2c+1192abc+11045a2d+15333abd+3560acd-2292bcd+7194ad2-6245bd2+8639cd2-9426d3 0 0 1236a3+8922a2b-3589ab2+8971b3-5006a2c-5599abc-14165b2c+8880a2d+4259abd-3002b2d-13892acd-10521bcd | {4} | 6723a3+5483a2b+15250a2c-10567a2d 994a3+1946a2b+6723ab2+5483b3+6453a2c+1192abc+15250b2c+11045a2d+15333abd-10567b2d+3560acd-2292bcd+7194ad2-6245bd2+8639cd2-9426d3 0 10370ab2-7092b3-9702abc-6627b2c+8886abd+4700b2d-15acd+5969bcd | {4} | 3164a3-295a2b+7650a2c-14388a2d 3164ab2-295b3+7650b2c-14388b2d 994a3+1946a2b+6723ab2+5483b3+6453a2c+1192abc+15250b2c+3164ac2-295bc2+7650c3+11045a2d+15333abd-10567b2d+3560acd-2292bcd-14388c2d+7194ad2-6245bd2+8639cd2-9426d3 13707a3+2177a2b-7028ab2+9797b3-7021a2c+6377abc+5874b2c-7600ac2-11726bc2+1140c3+1206a2d-11435abd+9074b2d-6040acd+8022bcd+3968c2d | {{0, 1}, 2} => {6} | 13707a3+2177a2b-7021a2c+6377abc-7600ac2-11726bc2+1140c3+1206a2d-11435abd-6040acd+8022bcd+3968c2d-3164ad2+295bd2-7650cd2+14388d3 -10370ab2+7092b3+9702abc+6627b2c+7028ac2-9797bc2-5874c3-8886abd-4700b2d+15acd-5969bcd-9074c2d+5483bd2+15250cd2-10567d3 1236a3+8922a2b-3589ab2+8971b3-5006a2c-5599abc-14165b2c+8880a2d+4259abd-3002b2d-13892acd-10521bcd 7028ad2 0 0 | {6} | -994a3-1946a2b-6453a2c-1192abc-11045a2d-15333abd-3560acd+2292bcd-7194ad2+6245bd2-8639cd2+9426d3 6723ac2 0 -10370ab2+7092b3+9702abc+6627b2c-9797bc2-5874c3-8886abd-4700b2d+15acd-5969bcd-9074c2d+6723ad2+5483bd2+15250cd2-10567d3 1236a3+8922a2b-3589ab2+8971b3-5006a2c-5599abc-14165b2c+8880a2d+4259abd-3002b2d-13892acd-10521bcd 0 | {6} | 0 -994a3-1946a2b-6723ab2-6453a2c-1192abc-11045a2d-15333abd-3560acd+2292bcd-7194ad2+6245bd2-8639cd2+9426d3 0 -13707a3-2177a2b+7028ab2+7021a2c-6377abc+7600ac2+11726bc2-1140c3-1206a2d+11435abd+6040acd-8022bcd-3968c2d+3164ad2-295bd2+7650cd2-14388d3 0 1236a3+8922a2b-3589ab2+8971b3-5006a2c-5599abc-14165b2c+8880a2d+4259abd-3002b2d-13892acd-10521bcd | {6} | 0 -5483a2b-15250a2c+10567a2d -994a3-1946a2b-6723ab2-5483b3-6453a2c-1192abc-15250b2c-11045a2d-15333abd+10567b2d-3560acd+2292bcd-7194ad2+6245bd2-8639cd2+9426d3 -9797a2b-5874a2c-9074a2d -13707a3-2177a2b+7028ab2-9797b3+7021a2c-6377abc-5874b2c+7600ac2+11726bc2-1140c3-1206a2d+11435abd-9074b2d+6040acd-8022bcd-3968c2d+3164ad2-295bd2+7650cd2-14388d3 10370ab2-7092b3-9702abc-6627b2c+8886abd+4700b2d-15acd+5969bcd | {{0, 1}, 3} => {8} | 994a3+1946a2b+6723ab2+6453a2c+1192abc+11045a2d+15333abd+3560acd-2292bcd+7194ad2-6245bd2+8639cd2-9426d3 13707a3+2177a2b-7028ab2-7021a2c+6377abc-7600ac2-11726bc2+1140c3+1206a2d-11435abd-6040acd+8022bcd+3968c2d-3164ad2+295bd2-7650cd2+14388d3 -10370ab2+7092b3+9702abc+6627b2c-9797bc2-5874c3-8886abd-4700b2d+15acd-5969bcd-9074c2d+5483bd2+15250cd2-10567d3 1236a3+8922a2b-3589ab2+8971b3-5006a2c-5599abc-14165b2c+8880a2d+4259abd-3002b2d-13892acd-10521bcd | {{0, 1}, 4} => 0 {{0, 1}, 5} => 0 {{0, 2}, -1} => 0 {{0, 2}, 0} => {6} | 13795a4-2019a3b-13769a2b2-7586ab3-8649b4-6454a3c+10187a2bc+1783ab2c-9219b3c-5513a2c2-10558abc2-2590b2c2-11624a3d+5603a2bd-14058ab2d+12615b3d-7869a2cd+2052abcd+1831b2cd-6042ac2d+2561bc2d+8709a2d2+13219abd2-4209b2d2-12225acd2+2605bcd2+92c2d2-15968ad3-14860bd3+8829cd3+11274d4 | {6} | -11152a4+1336a3b-11846a2b2-10264ab3-618b4+11051a3c-12129a2bc-5927ab2c-489b3c+15383a2c2-507abc2+13804b2c2+8416ac3-92c4+11057a3d+5113a2bd+2762ab2d-14095b3d+1588a2cd-2000abcd+2080b2cd-9175ac2d+649bc2d-8829c3d-2164a2d2-8635abd2+7161b2d2-997acd2-3015bcd2-11274c2d2 | {6} | 6338a4-10025a3b-14987a3c+9959a2bc+11691a2c2-12336abc2+7786a3d+1156a2bd-4960a2cd+5589abcd+8163ac2d+1895bc2d-9464a2d2+7253abd2-12642acd2+1958bcd2 | {6} | -2275a4+239a3b-14594a2b2+8153ab3+11945b4+8416a3c-6251a2bc+3023ab2c-5933b3c-92a2c2-5343abc2-3798b2c2+15968a3d-473a2bd-13293ab2d+3761b3d+7717a2cd+7389abcd-4723b2cd+13262ac2d-5431bc2d-11274a2d2+217abd2-1261b2d2-8201acd2+14080bcd2 | {{0, 2}, 1} => {8} | 5026a4+10435a3b-6989a2b2-15697ab3-8113b4+2741a3c+6357a2bc+6046ab2c-11866b3c-14083a2c2-177abc2-7596b2c2-15503ac3-12196a3d-15730a2bd+5417ab2d+7522b3d-5961a2cd-13409abcd-9446b2cd+8159ac2d-10862bc2d+15614a2d2+15082abd2-2522b2d2+12529acd2-3843bcd2+14263ad3 -2275a2b2+239ab3-6989b4+8416ab2c-12502b3c-92b2c2+15968ab2d-15806b3d-7740b2cd-11274b2d2 7648a3b+13261a2b2-3358ab3+618b4+767a3c+5580a2bc+15560ab2c+489b3c+12070a2c2+985abc2-10989b2c2+8416ac3-12502bc3-92c4+3588a3d-14541a2bd+8685ab2d+14095b3d-12750a2cd-4053abcd-2080b2cd+9108ac2d-1595bc2d-7740c3d+2164a2d2+8635abd2-7161b2d2+997acd2+3015bcd2-11274c2d2 -11958a3b-8641a2b2-9864ab3-8649b4+4417a3c-3731a2bc-7930ab2c-9219b3c-5513a2c2-10558abc2-2590b2c2-14414a3d-3844a2bd-5979ab2d+12615b3d+15957a2cd-15293abcd+1831b2cd-6042ac2d+2561bc2d+4159a2d2+13697abd2-1394b2d2+4607acd2-9897bcd2-92c2d2+15968ad3-15806bd3-7740cd3-11274d4 | {{0, 2}, 2} => 0 {{0, 3}, -1} => 0 {{0, 3}, 0} => 0 {{0, 4}, -1} => 0 {{0, 5}, -1} => 0 {{1, 0}, -1} => 0 {{1, 0}, 0} => {2} | 107a3+4376a2b+3783ab2+10359b3-5570a2c-5307abc-7464b2c+3187a2d+8570abd-8251b2d+8444acd+5071bcd | {2} | 8231ab2+13177b3+5864abc+13990b2c+5026abd-11521b2d-7501acd-1779bcd | {2} | -15344a3+2653a2b+10259ab2-1309b3+12365a2c-7216abc+5398b2c+6230ac2-5326bc2+1031c3-13508a2d-10125abd+5549b2d+9033acd+2998bcd-2036c2d | {2} | -10480a3-6203a2b+9534ab2+10866b3-9480a2c+7256abc-7061b2c+5107ac2+5679bc2-6325c3-11950a2d-5321abd+2627b2d-3996acd-7152bcd-11740c2d+9398ad2+15317bd2-6922cd2-5080d3 | {{1, 0}, 1} => {4} | -8231ab2-13177b3-5864abc-13990b2c-10259ac2+1309bc2-5398c3-5026abd+11521b2d+7501acd+1779bcd-5549c2d-9534ad2-10866bd2+7061cd2-2627d3 107a3+4376a2b+3783ab2+10359b3-5570a2c-5307abc-7464b2c+3187a2d+8570abd-8251b2d+8444acd+5071bcd 0 0 | {4} | 15344a3-2653a2b-12365a2c+7216abc-6230ac2+5326bc2-1031c3+13508a2d+10125abd-9033acd-2998bcd+2036c2d-5107ad2-5679bd2+6325cd2+11740d3 0 107a3+4376a2b+3783ab2+10359b3-5570a2c-5307abc-7464b2c+3187a2d+8570abd-8251b2d+8444acd+5071bcd 0 | {4} | -10259a3+1309a2b-5398a2c-5549a2d 15344a3-2653a2b-10259ab2+1309b3-12365a2c+7216abc-5398b2c-6230ac2+5326bc2-1031c3+13508a2d+10125abd-5549b2d-9033acd-2998bcd+2036c2d-5107ad2-5679bd2+6325cd2+11740d3 8231ab2+13177b3+5864abc+13990b2c+5026abd-11521b2d-7501acd-1779bcd 0 | {4} | 10480a3+6203a2b+9480a2c-7256abc+11950a2d+5321abd+3996acd+7152bcd-9398ad2-15317bd2+6922cd2+5080d3 0 0 107a3+4376a2b+3783ab2+10359b3-5570a2c-5307abc-7464b2c+3187a2d+8570abd-8251b2d+8444acd+5071bcd | {4} | -9534a3-10866a2b+7061a2c-2627a2d 10480a3+6203a2b-9534ab2-10866b3+9480a2c-7256abc+7061b2c+11950a2d+5321abd-2627b2d+3996acd+7152bcd-9398ad2-15317bd2+6922cd2+5080d3 0 8231ab2+13177b3+5864abc+13990b2c+5026abd-11521b2d-7501acd-1779bcd | {4} | -5107a3-5679a2b+6325a2c+11740a2d -5107ab2-5679b3+6325b2c+11740b2d 10480a3+6203a2b-9534ab2-10866b3+9480a2c-7256abc+7061b2c-5107ac2-5679bc2+6325c3+11950a2d+5321abd-2627b2d+3996acd+7152bcd+11740c2d-9398ad2-15317bd2+6922cd2+5080d3 -15344a3+2653a2b+10259ab2-1309b3+12365a2c-7216abc+5398b2c+6230ac2-5326bc2+1031c3-13508a2d-10125abd+5549b2d+9033acd+2998bcd-2036c2d | {{1, 0}, 2} => {6} | -15344a3+2653a2b+12365a2c-7216abc+6230ac2-5326bc2+1031c3-13508a2d-10125abd+9033acd+2998bcd-2036c2d+5107ad2+5679bd2-6325cd2-11740d3 -8231ab2-13177b3-5864abc-13990b2c-10259ac2+1309bc2-5398c3-5026abd+11521b2d+7501acd+1779bcd-5549c2d-10866bd2+7061cd2-2627d3 107a3+4376a2b+3783ab2+10359b3-5570a2c-5307abc-7464b2c+3187a2d+8570abd-8251b2d+8444acd+5071bcd -10259ad2 0 0 | {6} | -10480a3-6203a2b-9480a2c+7256abc-11950a2d-5321abd-3996acd-7152bcd+9398ad2+15317bd2-6922cd2-5080d3 -9534ac2 0 -8231ab2-13177b3-5864abc-13990b2c+1309bc2-5398c3-5026abd+11521b2d+7501acd+1779bcd-5549c2d-9534ad2-10866bd2+7061cd2-2627d3 107a3+4376a2b+3783ab2+10359b3-5570a2c-5307abc-7464b2c+3187a2d+8570abd-8251b2d+8444acd+5071bcd 0 | {6} | 0 -10480a3-6203a2b+9534ab2-9480a2c+7256abc-11950a2d-5321abd-3996acd-7152bcd+9398ad2+15317bd2-6922cd2-5080d3 0 15344a3-2653a2b-10259ab2-12365a2c+7216abc-6230ac2+5326bc2-1031c3+13508a2d+10125abd-9033acd-2998bcd+2036c2d-5107ad2-5679bd2+6325cd2+11740d3 0 107a3+4376a2b+3783ab2+10359b3-5570a2c-5307abc-7464b2c+3187a2d+8570abd-8251b2d+8444acd+5071bcd | {6} | 0 10866a2b-7061a2c+2627a2d -10480a3-6203a2b+9534ab2+10866b3-9480a2c+7256abc-7061b2c-11950a2d-5321abd+2627b2d-3996acd-7152bcd+9398ad2+15317bd2-6922cd2-5080d3 1309a2b-5398a2c-5549a2d 15344a3-2653a2b-10259ab2+1309b3-12365a2c+7216abc-5398b2c-6230ac2+5326bc2-1031c3+13508a2d+10125abd-5549b2d-9033acd-2998bcd+2036c2d-5107ad2-5679bd2+6325cd2+11740d3 8231ab2+13177b3+5864abc+13990b2c+5026abd-11521b2d-7501acd-1779bcd | {{1, 0}, 3} => {8} | 10480a3+6203a2b-9534ab2+9480a2c-7256abc+11950a2d+5321abd+3996acd+7152bcd-9398ad2-15317bd2+6922cd2+5080d3 -15344a3+2653a2b+10259ab2+12365a2c-7216abc+6230ac2-5326bc2+1031c3-13508a2d-10125abd+9033acd+2998bcd-2036c2d+5107ad2+5679bd2-6325cd2-11740d3 -8231ab2-13177b3-5864abc-13990b2c+1309bc2-5398c3-5026abd+11521b2d+7501acd+1779bcd-5549c2d-10866bd2+7061cd2-2627d3 107a3+4376a2b+3783ab2+10359b3-5570a2c-5307abc-7464b2c+3187a2d+8570abd-8251b2d+8444acd+5071bcd | {{1, 0}, 4} => 0 {{1, 0}, 5} => 0 {{1, 1}, -1} => 0 {{1, 1}, 0} => {6} | 8991a4-8982a3b-13320a2b2+7229ab3-7672b4+13689a3c-10065a2bc+8787ab2c+1726b3c-9033a2c2+14966abc2+6929b2c2-11993a3d-13593a2bd-8103ab2d+14054b3d+4240a2cd+9187abcd+14685b2cd+10581ac2d-5228bc2d+15866a2d2+1793abd2+12245b2d2-12283acd2-6730bcd2-8527c2d2-5942ad3-14925bd3+296cd3-8084d4 | {6} | 8375a4+225a3b+11039a2b2+10196ab3-4676b4-2695a3c+8977a2bc+15950ab2c+7421b3c+14919a2c2+11346abc2-14172b2c2-1910ac3-11656bc3+8527c4+8979a3d+8104a2bd-8734ab2d-5218b3d+12802a2cd+12618abcd+5728b2cd-5347ac2d-13394bc2d-296c3d-7446a2d2-10046abd2-6342b2d2+7480acd2+7843bcd2+8084c2d2 | {6} | 10857a4-14166a3b+858a3c-5335a2bc+5513a2c2-5100abc2+10723a3d+1870a2bd+9822a2cd-2760abcd-4119ac2d-5574bc2d-8831a2d2+14373abd2-13676acd2-11671bcd2 | {6} | 5508a4-13660a3b-15336a2b2+13943ab3+842b4-1910a3c-12230a2bc+9252ab2c+6448b3c+8527a2c2+7998abc2+13623b2c2+5942a3d+11150a2bd+12791ab2d+12401b3d-6638a2cd-13439abcd+12371b2cd-137ac2d+14313bc2d+8084a2d2+4552abd2-6564b2d2-5813acd2+15345bcd2 | {{1, 1}, 1} => {8} | 5792a4+13013a3b+14484a2b2-4117ab3+1684b4+15977a3c-5595a2bc-13499ab2c+12896b3c+12556a2c2+2132abc2-4757b2c2+978ac3+15946a3d+10318a2bd-6421ab2d-7201b3d+9255a2cd-4930abcd-7261b2cd-6164ac2d-3377bc2d-6528a2d2-2633abd2-13128b2d2-12963acd2-1313bcd2+11793ad3 5508a2b2-13660ab3+14484b4-1910ab2c-12804b3c+8527b2c2+5942ab2d+7375b3d-12980b2cd+8084b2d2 10492a3b-13996a2b2+9338ab3+4676b4+9826a3c-4731a2bc+12735ab2c-7421b3c-3903a2c2-6663abc2+15503b2c2-1910ac3-12804bc3+8527c4-6770a3d+9107a2bd+8846ab2d+5218b3d+802a2cd+6332abcd-5728b2cd-14772ac2d+3691bc2d-12980c3d+7446a2d2+10046abd2+6342b2d2-7480acd2-7843bcd2+8084c2d2 -6409a3b+10151a2b2+3863ab3-7672b4-3762a3c+9280a2bc-12253ab2c+1726b3c-9033a2c2+14966abc2+6929b2c2+11253a3d-4921a2bd+11860ab2d+14054b3d-15299a2cd+718abcd+14685b2cd+10581ac2d-5228bc2d-5121a2d2+6476abd2+13576b2d2+15900acd2+813bcd2+8527c2d2+5942ad3+7375bd3-12980cd3+8084d4 | {{1, 1}, 2} => 0 {{1, 2}, -1} => 0 {{1, 2}, 0} => 0 {{1, 3}, -1} => 0 {{1, 4}, -1} => 0 {{2, 0}, -1} => 0 {{2, 0}, 0} => {6} | -9611a4-13127a3b+9489a2b2+464ab3-9341b4+15743a3c+9530a2bc-14935ab2c+13901b3c-15960a2c2+4501abc2-1105b2c2-6056a3d+11857a2bd-13748ab2d+11752b3d+12345a2cd+14908abcd+9894b2cd-8440ac2d-10693bc2d-12062a2d2+13626abd2-4247b2d2+12703acd2-2957bcd2+5375c2d2+10305ad3-8297bd3-10663cd3-6341d4 | {6} | 3958a4+574a3b+7101a2b2-15674ab3-6343b4-1142a3c-9820a2bc+4821ab2c-5737b3c-5028a2c2-9312abc2-1610b2c2+15936ac3-5385bc3-5375c4-7073a3d-12092a2bd-8241ab2d+4420b3d-5234a2cd-7006abcd+14921b2cd-8810ac2d+3271bc2d+10663c3d+12537a2d2-15281abd2+9346b2d2-4309acd2-8269bcd2+6341c2d2 | {6} | -2398a4-9734a3b+35a3c+4979a2bc+5053a2c2+4372abc2-10422a3d+5261a2bd+2750a2cd+9680abcd-4707ac2d+7069bc2d+3873a2d2-5632abd2-12734acd2+7960bcd2 | {6} | -6945a4-12098a3b+8978a2b2-12099ab3-9169b4+15936a3c+1975a2bc-10573ab2c-9051b3c-5375a2c2-1677abc2-1545b2c2-10305a3d+6921a2bd+3135ab2d+9105b3d+10679a2cd+14065abcd-6297b2cd-15379ac2d+12881bc2d+6341a2d2-8292abd2-11862b2d2+10516acd2+12499bcd2 | {{2, 0}, 1} => {8} | 5315a4+13847a3b-6800a2b2+7805ab3+13665b4+2997a3c-13888a2bc+10857ab2c+13901b3c-6123a2c2+7323abc2-3090b2c2+4633ac3+8055a3d-14136a2bd+6270ab2d-13793b3d+10365a2cd-10131abcd-12594b2cd+15842ac2d-6241bc2d-1130a2d2+7556abd2+8279b2d2+10275acd2-7005bcd2+167ad3 -6945a2b2-12098ab3-6800b4+15936ab2c+9335b3c-5375b2c2-10305ab2d+5545b3d+10695b2cd+6341b2d2 10554a3b-6842a2b2+14226ab3+6343b4+12545a3c+10015a2bc+14286ab2c+5737b3c-8862a2c2-14884abc2-12437b2c2+15936ac3+9335bc3-5375c4-6938a3d+9371a2bd+9901ab2d-4420b3d-12317a2cd-15378abcd-14921b2cd-11800ac2d+10571bc2d+10695c3d-12537a2d2+15281abd2-9346b2d2+4309acd2+8269bcd2+6341c2d2 12051a3b-353a2b2-8518ab3-9341b4-1532a3c+2120a2bc-4932ab2c+13901b3c-15960a2c2+4501abc2-1105b2c2+14314a3d-12757a2bd-12822ab2d+11752b3d+7220a2cd+1419abcd+9894b2cd-8440ac2d-10693bc2d+6051a2d2-10570abd2+13709b2d2+12572acd2+993bcd2-5375c2d2-10305ad3+5545bd3+10695cd3+6341d4 | {{2, 0}, 2} => 0 {{2, 1}, -1} => 0 {{2, 1}, 0} => 0 {{2, 2}, -1} => 0 {{2, 3}, -1} => 0 {{3, 0}, -1} => 0 {{3, 0}, 0} => 0 {{3, 1}, -1} => 0 {{3, 2}, -1} => 0 {{4, 0}, -1} => 0 {{4, 1}, -1} => 0 {{5, 0}, -1} => 0 o13 : HashTable |
We can see that all 6 potential higher homotopies are nontrivial:
i14 : L = sort select(keys homot, k->(homot#k!=0 and sum(k_0)>1)) o14 = {{{0, 2}, 0}, {{0, 2}, 1}, {{1, 1}, 0}, {{1, 1}, 1}, {{2, 0}, 0}, {{2, ----------------------------------------------------------------------- 0}, 1}} o14 : List |
i15 : #L o15 = 6 |
i16 : netList L +------+-+ o16 = |{0, 2}|0| +------+-+ |{0, 2}|1| +------+-+ |{1, 1}|0| +------+-+ |{1, 1}|1| +------+-+ |{2, 0}|0| +------+-+ |{2, 0}|1| +------+-+ |
For example we have:
i17 : homot#(L_0) o17 = {6} | 13795a4-2019a3b-13769a2b2-7586ab3-8649b4-6454a3c+10187a2bc+1783ab {6} | -11152a4+1336a3b-11846a2b2-10264ab3-618b4+11051a3c-12129a2bc-5927 {6} | 6338a4-10025a3b-14987a3c+9959a2bc+11691a2c2-12336abc2+7786a3d+115 {6} | -2275a4+239a3b-14594a2b2+8153ab3+11945b4+8416a3c-6251a2bc+3023ab2 ----------------------------------------------------------------------- 2c-9219b3c-5513a2c2-10558abc2-2590b2c2-11624a3d+5603a2bd-14058ab2d+1261 ab2c-489b3c+15383a2c2-507abc2+13804b2c2+8416ac3-92c4+11057a3d+5113a2bd+ 6a2bd-4960a2cd+5589abcd+8163ac2d+1895bc2d-9464a2d2+7253abd2-12642acd2+1 c-5933b3c-92a2c2-5343abc2-3798b2c2+15968a3d-473a2bd-13293ab2d+3761b3d+7 ----------------------------------------------------------------------- 5b3d-7869a2cd+2052abcd+1831b2cd-6042ac2d+2561bc2d+8709a2d2+13219abd2-42 2762ab2d-14095b3d+1588a2cd-2000abcd+2080b2cd-9175ac2d+649bc2d-8829c3d-2 958bcd2 717a2cd+7389abcd-4723b2cd+13262ac2d-5431bc2d-11274a2d2+217abd2-1261b2d2 ----------------------------------------------------------------------- 09b2d2-12225acd2+2605bcd2+92c2d2-15968ad3-14860bd3+8829cd3+11274d4 | 164a2d2-8635abd2+7161b2d2-997acd2-3015bcd2-11274c2d2 | | -8201acd2+14080bcd2 | 4 1 o17 : Matrix S <--- S |
But all the homotopies are minimal in this case:
i18 : k1 = S^1/(ideal vars S); |
i19 : select(keys homot,k->(k1**homot#k)!=0) o19 = {} o19 : List |