-- -*- M2-comint -*- {* hash: 1558689331 *} i1 : R = QQ[a..f]; i2 : m = matrix{{a,b,d,e},{b,c,e,f}} o2 = | a b d e | | b c e f | 2 4 o2 : Matrix R <--- R i3 : M = coker m o3 = cokernel | a b d e | | b c e f | 2 o3 : R-module, quotient of R i4 : N = image m o4 = image | a b d e | | b c e f | 2 o4 : R-module, submodule of R i5 : K = kernel m o5 = image {1} | cd-be 0 e2-df ce-bf | {1} | -bd+ae e2-df 0 -be+af | {1} | b2-ac -ce+bf -be+af 0 | {1} | 0 cd-be bd-ae b2-ac | 4 o5 : R-module, submodule of R i6 : presentation M -- this is just the original matrix o6 = | a b d e | | b c e f | 2 4 o6 : Matrix R <--- R i7 : presentation N -- this one requires computation o7 = {1} | cd-be 0 e2-df ce-bf | {1} | -bd+ae e2-df 0 -be+af | {1} | b2-ac -ce+bf -be+af 0 | {1} | 0 cd-be bd-ae b2-ac | 4 4 o7 : Matrix R <--- R i8 : ideal(a,b)*N o8 = image | a2 ab ad ae ab b2 bd be | | ab ac ae af b2 bc be bf | 2 o8 : R-module, submodule of R i9 : a*N + b*N o9 = image | a2 ab ad ae ab b2 bd be | | ab ac ae af b2 bc be bf | 2 o9 : R-module, submodule of R i10 : N0 = image (a**N_{1}|N_{2}-N_{3}) o10 = image | ab d-e | | ac e-f | 2 o10 : R-module, submodule of R i11 : N_{1} o11 = {1} | 0 | {1} | 1 | {1} | 0 | {1} | 0 | o11 : Matrix i12 : a ** N_{1} o12 = {1} | 0 | {1} | a | {1} | 0 | {1} | 0 | o12 : Matrix i13 : a ** N_{1} | N_{2}-N_{3} o13 = {1} | 0 0 | {1} | a 0 | {1} | 0 1 | {1} | 0 -1 | o13 : Matrix i14 : N0 = image(a ** N_{1} | N_{2}-N_{3}) o14 = image | ab d-e | | ac e-f | 2 o14 : R-module, submodule of R i15 : isHomogeneous N0 o15 = true i16 : Nbar = N/N0 o16 = subquotient (| a b d e |, | ab d-e |) | b c e f | | ac e-f | 2 o16 : R-module, subquotient of R i17 : I = ideal(a^2, a*b, c^2) 2 2 o17 = ideal (a , a*b, c ) o17 : Ideal of R i18 : J = module I o18 = image | a2 ab c2 | 1 o18 : R-module, submodule of R i19 : I == ideal J o19 = true i20 : codim I o20 = 2 i21 : codim J o21 = 0 i22 : C = res I 1 3 3 1 o22 = R <-- R <-- R <-- R <-- 0 0 1 2 3 4 o22 : ChainComplex i23 : C.dd 1 3 o23 = 0 : R <---------------- R : 1 | a2 ab c2 | 3 3 1 : R <---------------------- R : 2 {2} | -b -c2 0 | {2} | a 0 -c2 | {2} | 0 a2 ab | 3 1 2 : R <-------------- R : 3 {3} | c2 | {4} | -b | {4} | a | 1 3 : R <----- 0 : 4 0 o23 : ChainComplexMap i24 : betti C 0 1 2 3 o24 = total: 1 3 3 1 0: 1 . . . 1: . 3 1 . 2: . . 2 1 o24 : BettiTally i25 : C = res Nbar 4 6 2 o25 = R <-- R <-- R <-- 0 0 1 2 3 o25 : ChainComplex i26 : betti C 0 1 2 o26 = total: 4 6 2 0: . 1 . 1: 4 1 . 2: . 4 2 o26 : BettiTally i27 : C.dd 4 6 o27 = 0 : R <----------------------------------------------------------- R : 1 {1} | 0 0 ce-bf -cd+be+ce-bf e2-df 0 | {1} | 0 a -be bd-be 0 e2-df | {1} | -1 0 b2-ac 0 bd-ae-be+af cd-be-ce+bf | {1} | 1 0 0 0 0 0 | 6 2 1 : R <--------------------- R : 2 {1} | 0 0 | {2} | e2-df 0 | {3} | -d+e e-f | {3} | -e f | {3} | b -c | {3} | -a b | 2 2 : R <----- 0 : 3 0 o27 : ChainComplexMap i28 : R = QQ[a..h]; i29 : J = ideal(a*c+b*d,a*e+b*f,a*g+b*h) o29 = ideal (a*c + b*d, a*e + b*f, a*g + b*h) o29 : Ideal of R i30 : betti res J 0 1 2 3 o30 = total: 1 3 4 2 0: 1 . . . 1: . 3 . . 2: . . 4 2 o30 : BettiTally i31 : use ring M o31 = QQ[a, b, c, d, e, f] o31 : PolynomialRing i32 : M o32 = cokernel | a b d e | | b c e f | 2 o32 : QQ[a, b, c, d, e, f]-module, quotient of (QQ[a, b, c, d, e, f]) i33 : N = a*M o33 = subquotient (| a 0 |, | a b d e |) | 0 a | | b c e f | 2 o33 : QQ[a, b, c, d, e, f]-module, subquotient of (QQ[a, b, c, d, e, f]) i34 : M/N o34 = cokernel | a 0 a b d e | | 0 a b c e f | 2 o34 : QQ[a, b, c, d, e, f]-module, quotient of (QQ[a, b, c, d, e, f]) i35 : generators N o35 = | a 0 | | 0 a | 2 2 o35 : Matrix (QQ[a, b, c, d, e, f]) <--- (QQ[a, b, c, d, e, f]) i36 : relations N o36 = | a b d e | | b c e f | 2 4 o36 : Matrix (QQ[a, b, c, d, e, f]) <--- (QQ[a, b, c, d, e, f]) i37 : presentation N o37 = {1} | e d b a | {1} | f e c b | 2 4 o37 : Matrix (QQ[a, b, c, d, e, f]) <--- (QQ[a, b, c, d, e, f]) i38 : trim N o38 = subquotient (| a 0 |, | e d b a |) | 0 a | | f e c b | 2 o38 : QQ[a, b, c, d, e, f]-module, subquotient of (QQ[a, b, c, d, e, f]) i39 : minimalPresentation N o39 = cokernel {1} | e d b a | {1} | f e c b | 2 o39 : QQ[a, b, c, d, e, f]-module, quotient of (QQ[a, b, c, d, e, f]) i40 : prune N o40 = cokernel {1} | e d b a | {1} | f e c b | 2 o40 : QQ[a, b, c, d, e, f]-module, quotient of (QQ[a, b, c, d, e, f]) i41 : ambient N 2 o41 = (QQ[a, b, c, d, e, f]) o41 : QQ[a, b, c, d, e, f]-module, free i42 : ambient N == target generators N o42 = true i43 : ambient N == target relations N o43 = true i44 : super N o44 = cokernel | a b d e | | b c e f | 2 o44 : QQ[a, b, c, d, e, f]-module, quotient of (QQ[a, b, c, d, e, f]) i45 : super N == coker relations N o45 = true i46 : cover N 2 o46 = (QQ[a, b, c, d, e, f]) o46 : QQ[a, b, c, d, e, f]-module, free, degrees {1, 1} i47 : cover N == source generators N o47 = true i48 : A = QQ[x,y]/(y^2-x^3) o48 = A o48 : QuotientRing i49 : M = module ideal(x,y) o49 = image | x y | 1 o49 : A-module, submodule of A i50 : F = map(A^1,M,matrix{{y,x^2}}) o50 = | y x2 | o50 : Matrix i51 : source F == M o51 = true i52 : target F == A^1 o52 = true i53 : matrix F o53 = | y x2 | 1 2 o53 : Matrix A <--- A i54 : inducedMap(A^1,M) o54 = | x y | o54 : Matrix i55 : G = F // inducedMap(A^1,M) o55 = {1} | 0 x | {1} | 1 0 | o55 : Matrix i56 : source G o56 = image | x y | 1 o56 : A-module, submodule of A i57 : target G o57 = image | x y | 1 o57 : A-module, submodule of A i58 : isWellDefined G o58 = true i59 : R = QQ[x,y,z,w] o59 = R o59 : PolynomialRing i60 : M = ideal(x,y,z)/ideal(x^2,y^2,z*w) o60 = subquotient (| x y z |, | x2 y2 zw |) 1 o60 : R-module, subquotient of R i61 : N = z*M o61 = subquotient (| xz yz z2 |, | x2 y2 zw |) 1 o61 : R-module, subquotient of R i62 : M/N o62 = subquotient (| x y z |, | xz yz z2 x2 y2 zw |) 1 o62 : R-module, subquotient of R i63 : M o63 = subquotient (| x y z |, | x2 y2 zw |) 1 o63 : R-module, subquotient of R i64 : ambient M 1 o64 = R o64 : R-module, free i65 : N = z*M o65 = subquotient (| xz yz z2 |, | x2 y2 zw |) 1 o65 : R-module, subquotient of R i66 : ambient(M/N) 1 o66 = R o66 : R-module, free i67 : super M o67 = cokernel | x2 y2 zw | 1 o67 : R-module, quotient of R i68 : super N o68 = cokernel | x2 y2 zw | 1 o68 : R-module, quotient of R i69 : image generators M o69 = image | x y z | 1 o69 : R-module, submodule of R i70 : inducedMap(M,M) == id_M o70 = true i71 : inducedMap(super M,M) == map(super id_M) -- the map (P+Q)/Q --> R^n/Q, where M=(P+Q)/Q. o71 = true i72 : inducedMap(super M,ambient M) -- the quotient map R^n --> R^n/Q o72 = | 1 | o72 : Matrix i73 : inducedMap(M,N) -- the inclusion map o73 = {1} | z 0 0 | {1} | 0 z 0 | {1} | 0 0 z | o73 : Matrix i74 : inducedMap(M/N,M) -- the projection map o74 = {1} | 1 0 0 | {1} | 0 1 0 | {1} | 0 0 1 | o74 : Matrix i75 : inducedMap(M/N,N) -- the zero map o75 = 0 o75 : Matrix i76 : inducedMap(M,M/N,Verify => false) o76 = {1} | 1 0 0 | {1} | 0 1 0 | {1} | 0 0 1 | o76 : Matrix i77 : inducedMap(M/N,x*M) o77 = {1} | 0 y 0 | {1} | 0 0 0 | {1} | 0 0 0 | o77 : Matrix i78 : inducedMap(M/N,M) * inducedMap(M,x*M) == inducedMap(M/N,x*M) o78 = true i79 : A = QQ[x,y,Degrees=>{2,3}]/(y^2-x^3) o79 = A o79 : QuotientRing i80 : M = module ideal(x,y) o80 = image | x y | 1 o80 : A-module, submodule of A i81 : H = Hom(M,M) o81 = subquotient ({0} | 1 0 |, {0} | 0 y 0 x2 |) {1} | 0 1 | {1} | 0 -x 0 -y | {-1} | 0 x | {-1} | y 0 x2 0 | {0} | 1 0 | {0} | -x 0 -y 0 | 4 o81 : A-module, subquotient of A i82 : F = homomorphism(H_{0}) o82 = {2} | 1 0 | {3} | 0 1 | o82 : Matrix i83 : G = homomorphism(H_{1}) o83 = {2} | 0 x | {3} | 1 0 | o83 : Matrix i84 : source F == M o84 = true i85 : target F == M o85 = true i86 : ker F o86 = image 0 1 o86 : A-module, submodule of A i87 : coker F o87 = subquotient (| x y |, | x y |) 1 o87 : A-module, subquotient of A i88 : m = matrix{{x,y},{y,x}} o88 = | x y | | y x | 2 2 o88 : Matrix A <--- A i89 : Hom(m,A^2) o89 = {-3} | x 0 y 0 | {-3} | 0 x 0 y | {-3} | y 0 x 0 | {-3} | 0 y 0 x | 4 4 o89 : Matrix A <--- A i90 : Hom(A^2,m) o90 = | x y 0 0 | | y x 0 0 | | 0 0 x y | | 0 0 y x | 4 4 o90 : Matrix A <--- A i91 : m ** m o91 = | x2 xy xy y2 | | xy x2 y2 xy | | xy y2 x2 xy | | y2 xy xy x2 | 4 4 o91 : Matrix A <--- A i92 : (coker m) ** (coker m) o92 = cokernel | x y 0 0 x y 0 0 | | y x 0 0 0 0 x y | | 0 0 x y y x 0 0 | | 0 0 y x 0 0 y x | 4 o92 : A-module, quotient of A i93 : M = coker m o93 = cokernel | x y | | y x | 2 o93 : A-module, quotient of A i94 : M2 = prune(M ** M) o94 = cokernel | 0 -x y x x 0 | | -x 0 x y 0 x | | x 0 0 0 y 0 | | 0 x 0 0 0 y | 4 o94 : A-module, quotient of A i95 : A = QQ[a,b,c] o95 = A o95 : PolynomialRing i96 : A ** A o96 = QQ[a, b, c, a, b, c] o96 : PolynomialRing i97 : B = oo o97 = B o97 : PolynomialRing i98 : a == B_3 o98 = true i99 : a == B_0 o99 = false i100 : tensor(A,A,Variables=>{a,b,c,d,e,f}) o100 = QQ[a, b, c, d, e, f] o100 : PolynomialRing i101 :