-- -*- M2-comint -*- {* hash: -1501724747 *}
i1 : 2+2
o1 = 4
i2 : 107*431
o2 = 46117
i3 : 25!
o3 = 15511210043330985984000000
i4 : binomial(5,4)
o4 = 5
i5 : factor 32004
2 2
o5 = 2 3 7*127
o5 : Expression of class Product
i6 : ZZ/101
ZZ
o6 = ---
101
o6 : QuotientRing
i7 : QQ
o7 = QQ
o7 : Ring
i8 : GF 2^5
o8 = GF 32
o8 : GaloisField
i9 : k = toField (QQ[i]/(i^2+1))
o9 = k
o9 : PolynomialRing
i10 : 1/i
o10 = -i
o10 : k
i11 : kk=ZZ/101
o11 = kk
o11 : QuotientRing
i12 : S=kk[x_1..x_5]
o12 = S
o12 : PolynomialRing
i13 : S=kk[a,b,c,d,e]
o13 = S
o13 : PolynomialRing
i14 : (3*a^2+1)^5
10 8 6 4 2
o14 = 41a + a - 33a - 11a + 15a + 1
o14 : S
i15 : I=ideal(a^3-b^3, a+b+c+d+e)
3 3
o15 = ideal (a - b , a + b + c + d + e)
o15 : Ideal of S
i16 : R=S/I
o16 = R
o16 : QuotientRing
i17 : use S
o17 = S
o17 : PolynomialRing
i18 : I=ideal"3(a+b)3, 4c5"
3 2 2 3 5
o18 = ideal (3a + 9a b + 9a*b + 3b , 4c )
o18 : Ideal of S
i19 : I^2
6 5 4 2 3 3 2 4 5 6 3 5
o19 = ideal (9a - 47a b + 34a b - 22a b + 34a b - 47a*b + 9b , 12a c +
-----------------------------------------------------------------------
2 5 2 5 3 5 10
36a b*c + 36a*b c + 12b c , 16c )
o19 : Ideal of S
i20 : I*I
6 5 4 2 3 3 2 4 5 6 3 5
o20 = ideal (9a - 47a b + 34a b - 22a b + 34a b - 47a*b + 9b , 12a c +
-----------------------------------------------------------------------
2 5 2 5 3 5 3 5 2 5 2 5 3 5
36a b*c + 36a*b c + 12b c , 12a c + 36a b*c + 36a*b c + 12b c ,
-----------------------------------------------------------------------
10
16c )
o20 : Ideal of S
i21 : I+ideal"a2"
3 2 2 3 5 2
o21 = ideal (3a + 9a b + 9a*b + 3b , 4c , a )
o21 : Ideal of S
i22 : M= matrix{{a,b,c},{b,c,d},{c,d,e}}
o22 = | a b c |
| b c d |
| c d e |
3 3
o22 : Matrix S <--- S
i23 : M^2
o23 = | a2+b2+c2 ab+bc+cd ac+bd+ce |
| ab+bc+cd b2+c2+d2 bc+cd+de |
| ac+bd+ce bc+cd+de c2+d2+e2 |
3 3
o23 : Matrix S <--- S
i24 : determinant M
3 2 2
o24 = - c + 2b*c*d - a*d - b e + a*c*e
o24 : S
i25 : trace M
o25 = a + c + e
o25 : S
i26 : M-transpose M
o26 = 0
3 3
o26 : Matrix S <--- S
i27 : entries M
o27 = {{a, b, c}, {b, c, d}, {c, d, e}}
o27 : List
i28 : flatten entries M
o28 = {a, b, c, b, c, d, c, d, e}
o28 : List
i29 : M_(0,0)
o29 = a
o29 : S
i30 : I_0
3 2 2 3
o30 = 3a + 9a b + 9a*b + 3b
o30 : S
i31 : I_*
3 2 2 3 5
o31 = {3a + 9a b + 9a*b + 3b , 4c }
o31 : List
i32 : coker M
o32 = cokernel | a b c |
| b c d |
| c d e |
3
o32 : S-module, quotient of S
i33 : image M
o33 = image | a b c |
| b c d |
| c d e |
3
o33 : S-module, submodule of S
i34 : kernel matrix"a,b,0;0,a,b"
o34 = image {1} | b2 |
{1} | -ab |
{1} | a2 |
3
o34 : S-module, submodule of S
i35 : N = matrix{{a,b},{b,c},{c,d}}
o35 = | a b |
| b c |
| c d |
3 2
o35 : Matrix S <--- S
i36 : (image M)/(image N)
o36 = subquotient (| a b c |, | a b |)
| b c d | | b c |
| c d e | | c d |
3
o36 : S-module, subquotient of S
i37 : subquotient(M,N)
o37 = subquotient (| a b c |, | a b |)
| b c d | | b c |
| c d e | | c d |
3
o37 : S-module, subquotient of S
i38 : kk = ZZ/32003
o38 = kk
o38 : QuotientRing
i39 : R = kk[x,y,z,w]
o39 = R
o39 : PolynomialRing
i40 : I = ideal(x^2*y,x*y^2+x^3)
2 3 2
o40 = ideal (x y, x + x*y )
o40 : Ideal of R
i41 : J = groebnerBasis I
o41 = | x2y x3+xy2 xy3 |
1 3
o41 : Matrix R <--- R
i42 : I= intersect (ideal"x2,y3", ideal"y2,z3", (ideal"x,y,z")^4)
3 4 3 2 2 2 3
o42 = ideal (y z, y , x*y , x y , x z )
o42 : Ideal of R
i43 : primaryDecomposition I
2 3 2 3 4
o43 = {ideal (x , y ), ideal (y , z ), ideal (x, y , z)}
o43 : List
i44 : decompose I
o44 = {ideal (y, x), ideal (y, z)}
o44 : List
i45 : R = kk[a..d]
o45 = R
o45 : PolynomialRing
i46 : phi = map(kk[s,t],R,{s^4, s^3*t, s*t^3, t^4})
4 3 3 4
o46 = map(kk[s, t],R,{s , s t, s*t , t })
o46 : RingMap kk[s, t] <--- R
i47 : I = ker phi
3 2 2 2 3 2
o47 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c)
o47 : Ideal of R
i48 : I = monomialCurveIdeal(R,{1,3,4})
3 2 2 2 3 2
o48 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c)
o48 : Ideal of R
i49 : dim I
o49 = 2
i50 : codim I
o50 = 2
i51 : degree I
o51 = 4
i52 : hilbertPolynomial(R/I)
o52 = - 3*P + 4*P
0 1
o52 : ProjectiveHilbertPolynomial
i53 : hilbertPolynomial(R/I, Projective => false)
o53 = 4i + 1
o53 : QQ[i]
i54 : hilbertPolynomial(R/I, Projective => true)
o54 = - 3*P + 4*P
0 1
o54 : ProjectiveHilbertPolynomial
i55 : hilbertSeries (R/I)
2 3 4 5
1 - T - 3T + 4T - T
o55 = -----------------------
4
(1 - T)
o55 : Expression of class Divide
i56 : reduceHilbert hilbertSeries (R/I)
2 3
1 + 2T + 2T - T
o56 = -----------------
2
(1 - T)
o56 : Expression of class Divide
i57 : M=R^1/I
o57 = cokernel | bc-ad c3-bd2 ac2-b2d b3-a2c |
1
o57 : R-module, quotient of R
i58 : Mres = res M
1 4 4 1
o58 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o58 : ChainComplex
i59 : betti Mres
0 1 2 3
o59 = total: 1 4 4 1
0: 1 . . .
1: . 1 . .
2: . 3 4 1
o59 : BettiTally
i60 : pdim M
o60 = 3
i61 : regularity M
o61 = 2
i62 : R = kk[a..i]
o62 = R
o62 : PolynomialRing
i63 : M = genericMatrix(R,a,3,3)
o63 = | a d g |
| b e h |
| c f i |
3 3
o63 : Matrix R <--- R
i64 : I = ideal M^3
3 2 2
o64 = ideal (a + 2a*b*d + b*d*e + 2a*c*g + b*f*g + c*d*h + c*g*i, a b + b d
-----------------------------------------------------------------------
2 2
+ a*b*e + b*e + b*c*g + a*c*h + c*e*h + b*f*h + c*h*i, a c + b*c*d +
-----------------------------------------------------------------------
2 2 2 2
a*b*f + b*e*f + c g + c*f*h + a*c*i + b*f*i + c*i , a d + b*d + a*d*e
-----------------------------------------------------------------------
2 3
+ d*e + c*d*g + a*f*g + e*f*g + d*f*h + f*g*i, a*b*d + 2b*d*e + e +
-----------------------------------------------------------------------
2
b*f*g + c*d*h + 2e*f*h + f*h*i, a*c*d + c*d*e + b*d*f + e f + c*f*g +
-----------------------------------------------------------------------
2 2 2 2
f h + c*d*i + e*f*i + f*i , a g + b*d*g + c*g + a*d*h + d*e*h + f*g*h
-----------------------------------------------------------------------
2 2 2
+ a*g*i + d*h*i + g*i , a*b*g + b*e*g + b*d*h + e h + c*g*h + f*h +
-----------------------------------------------------------------------
2
b*g*i + e*h*i + h*i , a*c*g + b*f*g + c*d*h + e*f*h + 2c*g*i + 2f*h*i +
-----------------------------------------------------------------------
3
i )
o64 : Ideal of R
i65 : Tr = trace M
o65 = a + e + i
o65 : R
i66 : for p from 1 to 10 do print (Tr^p % I)
a + e + i
2 2 2
a + 2a*e + e + 2a*i + 2e*i + i
2 2 3 2 2 2 2 3
3a e + 3b*d*e + 3a*e + 3e + 3b*f*g + 3c*d*h + 6e*f*h + 3a i + 6a*e*i + 3e i + 3c*g*i + 6f*h*i + 3a*i + 3e*i + 3i
2 2 2 3 4 2 2 2 2 2 3 2 2 2 2 2 2 2 3 3 4
6a e + 6b*d*e + 6a*e + 6e + 6b*e*f*g + 6c*d*e*h - 6b*d*f*h + 6a*e*f*h + 12e f*h - 6c*f*g*h - 6f h + 12a e*i + 12b*d*e*i + 12a*e i + 12e i + 12c*e*g*i + 6b*f*g*i + 6c*d*h*i + 6a*f*h*i + 36e*f*h*i + 6a i + 12a*e*i + 6e i + 6c*g*i + 12f*h*i + 6a*i + 12e*i + 6i
2 2 2 3 4 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 3 3 3 2 3 3 3 4 4 5
30a e i + 30b*d*e i + 30a*e i + 30e i + 30c*e g*i + 30a f*h*i + 30b*d*f*h*i + 60a*e*f*h*i + 90e f*h*i + 30c*f*g*h*i + 30f h i + 30a e*i + 30b*d*e*i + 30a*e i + 30e i + 30c*e*g*i + 60a*f*h*i + 120e*f*h*i + 30a i + 30b*d*i + 30a*e*i + 30e i + 30c*g*i + 90f*h*i + 30a*i + 30e*i + 30i
2 2 2 2 2 3 2 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 2 3 3 3 3 3 3 2 4 4 4 2 4 4 4 5 5 6
90a e i + 90b*d*e i + 90a*e i + 90e i + 90c*e g*i + 90a f*h*i + 90b*d*f*h*i + 180a*e*f*h*i + 270e f*h*i + 90c*f*g*h*i + 90f h i + 90a e*i + 90b*d*e*i + 90a*e i + 90e i + 90c*e*g*i + 180a*f*h*i + 360e*f*h*i + 90a i + 90b*d*i + 90a*e*i + 90e i + 90c*g*i + 270f*h*i + 90a*i + 90e*i + 90i
0
0
0
0
i67 : Tr^7//(gens I)
o67 = {3} | a4+7a3e+3abde+21a2e2+21bde2+37ae3+49e4+6aefh+36e2fh-6f2h2+7a3i+42
{3} | 0
{3} | 0
{3} | -63abe2-81be3-288bcdh-297ce2h+636befh-264cfh2-189abei-315be2i-276
{3} | -2a4-14a3e-6abde+21a2e2+30bde2+7ae3+e4+288abfg+939befg+45cdeh-6a2
{3} | -288b2dg+324abeg+81be2g-288bcg2+3a3h+288abdh+18a2eh+288bdeh+279ae
{3} | 288bcde-252ace2+282abef-378be2f+288bcfg-264acfh-378cefh+24bf2h-28
{3} | 3a3f-288abdf-264a2ef-1212bdef+453ae2f-891e3f+333cefg-258cdfh-276a
{3} | -2a4+288b2d2-14a3e-618abde+210a2e2-132bde2-74ae3-98e4+288bcdg+252
-----------------------------------------------------------------------
a2ei+21bdei+105ae2i+154e3i+6afhi+75efhi+21a2i2+105aei2+210e2i2+30fhi2+3
cehi+1323bfhi-252abi2-504bei2+105chi2-288bi3
fh-672aefh+687e2fh+258cfgh-12f2h2-14a3i+105a2ei+147bdei+42ae2i+7e3i+147
2h+300e3h+45cegh-546bfgh+276afh2+684efh2+612abgi+834begi+15a2hi+171bdhi
8bcdi-168acei-504ce2i+540abfi+18befi-1347cfhi-48aci2-336cei2+1482bfi2-9
f2h-1200ef2h-147cdei-525a2fi-2070bdfi+426aefi-1122e2fi+666cfgi-2169f2hi
ce2g-288abfg-939befg-45cdeh+258a2fh+810bdfh+612aefh-255e2fh+264cfgh+540
-----------------------------------------------------------------------
7ai3+154ei3+46i4
cegi+1530bfgi+153cdhi-1365afhi+1581efhi+210a2i2+252bdi2+105aei2+21e2i2+
+159aehi+510e2hi+153cghi+1101fh2i+624bgi2-324ahi2+360ehi2+210hi3
6ci3
-240cdi2-1497afi2+528efi2-567fi3
f2h2-11a3i-324abdi+105a2ei-510bdei+105ae2i-197e3i+168cegi-1530bfgi-153c
-----------------------------------------------------------------------
240cgi2+639fhi2+214ai3+133ei3+232i4
dhi+1593afhi-1143efhi+21a2i2+42aei2+21e2i2+48cgi2+123fhi2+7ai3+7ei3+i4
-----------------------------------------------------------------------
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9 1
o67 : Matrix R <--- R
i68 : x = symbol x
o68 = x
o68 : Symbol
i69 : R = kk[x_0..x_3]
o69 = R
o69 : PolynomialRing
i70 : M = map(R^2, 3, (i,j)->x_(i+j))
o70 = | x_0 x_1 x_2 |
| x_1 x_2 x_3 |
2 3
o70 : Matrix R <--- R
i71 : I = minors(2,M)
2 2
o71 = ideal (- x + x x , - x x + x x , - x + x x )
1 0 2 1 2 0 3 2 1 3
o71 : Ideal of R
i72 : pideal = ideal(x_0+x_3, x_1, x_2)
o72 = ideal (x + x , x , x )
0 3 1 2
o72 : Ideal of R
i73 : Rbar = R/I
o73 = Rbar
o73 : QuotientRing
i74 : pideal = substitute(pideal, Rbar)
o74 = ideal (x + x , x , x )
0 3 1 2
o74 : Ideal of Rbar
i75 : S = kk[u,v,w]
o75 = S
o75 : PolynomialRing
i76 : J=kernel map (Rbar, S, gens pideal)
3 3
o76 = ideal(v - u*v*w + w )
o76 : Ideal of S
i77 : K = ideal singularLocus(J)
3 3 2 2
o77 = ideal (v - u*v*w + w , -v*w, 3v - u*w, - u*v + 3w )
o77 : Ideal of S
i78 : saturate K
o78 = ideal (w, v)
o78 : Ideal of S
i79 : saturate (ideal"u3w,uv", ideal"u")
o79 = ideal (w, v)
o79 : Ideal of S
i80 : ideal"u3w,uv":ideal"u"
2
o80 = ideal (v, u w)
o80 : Ideal of S
i81 : Collatz = n ->
while n != 1 list if n%2 == 0 then n=n//2 else n=3*n+1
o81 = Collatz
o81 : FunctionClosure
i82 : Collatz 27
o82 = {82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242,
-----------------------------------------------------------------------
121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700,
-----------------------------------------------------------------------
350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668,
-----------------------------------------------------------------------
334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319,
-----------------------------------------------------------------------
958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644,
-----------------------------------------------------------------------
1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154,
-----------------------------------------------------------------------
577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92,
-----------------------------------------------------------------------
46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1}
o82 : List
i83 : tally for n from 1 to 30 list length Collatz n
o83 = Tally{0 => 1 }
1 => 1
2 => 1
3 => 1
4 => 1
5 => 1
6 => 1
7 => 3
8 => 1
9 => 2
10 => 2
12 => 1
14 => 1
15 => 2
16 => 1
17 => 2
18 => 3
19 => 1
20 => 2
23 => 1
111 => 1
o83 : Tally
i84 : record = length Collatz 1
o84 = 0
i85 : L = for n from 2 to 1000 list (
l = length Collatz n;
if l > record
then (record = l; (n,l))
else continue)
o85 = {(2, 1), (3, 7), (6, 8), (7, 16), (9, 19), (18, 20), (25, 23), (27,
-----------------------------------------------------------------------
111), (54, 112), (73, 115), (97, 118), (129, 121), (171, 124), (231,
-----------------------------------------------------------------------
127), (313, 130), (327, 143), (649, 144), (703, 170), (871, 178)}
o85 : List
i86 : L/last
o86 = {1, 7, 8, 16, 19, 20, 23, 111, 112, 115, 118, 121, 124, 127, 130, 143,
-----------------------------------------------------------------------
144, 170, 178}
o86 : List
i87 : loadPackage "EdgeIdeals"
o87 = EdgeIdeals
o87 : Package
i88 : R = kk[vars(0..10)]
o88 = R
o88 : PolynomialRing
i89 : G=randomGraph (R,20)
o89 = Graph{edges => {{a, c}, {c, d}, {d, g}, {d, i}, {g, j}, {g, k}, {c, e}, {d, j}, {f, g}, {f, i}, {c, k}, {g, i}, {i, j}, {c, h}, {e, g}, {a, g}, {e, i}, {e, h}, {b, f}, {d, h}}}
ring => R
vertices => {a, b, c, d, e, f, g, h, i, j, k}
o89 : Graph
i90 : K=edgeIdeal G
o90 = monomialIdeal (a*c, c*d, c*e, b*f, a*g, d*g, e*g, f*g, c*h, d*h, e*h,
-----------------------------------------------------------------------
d*i, e*i, f*i, g*i, d*j, g*j, i*j, c*k, g*k)
o90 : MonomialIdeal of R
i91 : hilbertSeries K
2 3 4 5 6 7 8 9 10
1 - 20T + 60T - 63T - 14T + 105T - 120T + 70T - 22T + 3T
o91 = ------------------------------------------------------------------
11
(1 - T)
o91 : Expression of class Divide
i92 : betti res K
0 1 2 3 4 5 6 7 8
o92 = total: 1 20 77 153 190 153 78 23 3
0: 1 . . . . . . . .
1: . 20 60 80 59 26 7 1 .
2: . . 17 73 131 127 71 22 3
o92 : BettiTally
i93 : R = ZZ/2[vars(0..10)]
o93 = R
o93 : PolynomialRing
i94 : L=for j from 1 to 100 list(
I = edgeIdeal randomGraph (R,5);
(codim I, degree I, betti res I));
i95 : tally L
0 1 2 3 4
o95 = Tally{(4, 12, total: 1 5 9 7 2) => 16 }
0: 1 . . . .
1: . 5 2 . .
2: . . 7 4 .
3: . . . 3 2
0 1 2 3 4 5
(2, 1, total: 1 5 10 10 5 1) => 6
0: 1 . . . . .
1: . 5 4 1 . .
2: . . 6 9 5 1
0 1 2 3 4 5
(2, 2, total: 1 5 10 10 5 1) => 1
0: 1 . . . . .
1: . 5 6 4 1 .
2: . . 4 6 4 1
0 1 2 3 4 5
(3, 2, total: 1 5 10 10 5 1) => 9
0: 1 . . . . .
1: . 5 2 . . .
2: . . 8 6 1 .
3: . . . 4 4 1
0 1 2 3 4 5
(3, 4, total: 1 5 10 10 5 1) => 5
0: 1 . . . . .
1: . 5 3 1 . .
2: . . 7 6 2 .
3: . . . 3 3 1
0 1 2 3 4 5
(4, 8, total: 1 5 10 10 5 1) => 7
0: 1 . . . . .
1: . 5 1 . . .
2: . . 9 3 . .
3: . . . 7 3 .
4: . . . . 2 1
0 1 2 3
(2, 1, total: 1 5 6 2) => 3
0: 1 . . .
1: . 5 6 2
0 1 2 3
(3, 5, total: 1 5 6 2) => 4
0: 1 . . .
1: . 5 5 1
2: . . 1 1
0 1 2 3 4
(2, 1, total: 1 5 7 4 1) => 2
0: 1 . . . .
1: . 5 5 1 .
2: . . 2 3 1
0 1 2 3 4
(2, 2, total: 1 5 7 4 1) => 1
0: 1 . . . .
1: . 5 7 4 1
0 1 2 3 4
(3, 2, total: 1 5 8 5 1) => 13
0: 1 . . . .
1: . 5 3 . .
2: . . 5 4 .
3: . . . 1 1
0 1 2 3 4
(3, 3, total: 1 5 9 7 2) => 21
0: 1 . . . .
1: . 5 3 . .
2: . . 6 7 2
0 1 2 3 4
(3, 4, total: 1 5 7 4 1) => 2
0: 1 . . . .
1: . 5 4 . .
2: . . 3 4 1
0 1 2 3 4
(3, 4, total: 1 5 8 5 1) => 10
0: 1 . . . .
1: . 5 4 1 .
2: . . 4 4 1
o95 : Tally
i96 : #tally L
o96 = 14
i97 : S^{-2,-3}
2
o97 = S
o97 : S-module, free, degrees {2, 3}
i98 : S = kk[x,y]
o98 = S
o98 : PolynomialRing
i99 : phi1 = map(S^1, S^{-2,-3}, matrix"x2,xy2")
o99 = | x2 xy2 |
1 2
o99 : Matrix S <--- S
i100 : phi2 = map(S^{-2,-3}, S^{-5}, matrix"xy2;-x2")
o100 = {2} | xy2 |
{3} | -x2 |
2 1
o100 : Matrix S <--- S
i101 : phi1*phi2
o101 = 0
1 1
o101 : Matrix S <--- S
i102 : (ker phi1)/(image phi2)
o102 = subquotient ({2} | y2 |, {2} | xy2 |)
{3} | -x | {3} | -x2 |
2
o102 : S-module, subquotient of S
i103 : FF = chainComplex(phi1,phi2)
1 2 1
o103 = S <-- S <-- S
0 1 2
o103 : ChainComplex
i104 : FF.dd
1 2
o104 = 0 : S <-------------- S : 1
| x2 xy2 |
2 1
1 : S <--------------- S : 2
{2} | xy2 |
{3} | -x2 |
o104 : ChainComplexMap
i105 : homology FF
o105 = 0 : cokernel | x2 xy2 |
1 : subquotient ({2} | y2 |, {2} | xy2 |)
{3} | -x | {3} | -x2 |
2 : image 0
o105 : GradedModule
i106 : presentation (homology FF)_1
o106 = {4} | x |
1 1
o106 : Matrix S <--- S
i107 : FF = chainComplex matrix {{x^2}} ** chainComplex matrix {{x*y^2}}
1 2 1
o107 = S <-- S <-- S
0 1 2
o107 : ChainComplex
i108 : FF.dd
1 2
o108 = 0 : S <-------------- S : 1
| xy2 x2 |
2 1
1 : S <---------------- S : 2
{3} | x2 |
{2} | -xy2 |
o108 : ChainComplexMap
i109 : FF = koszul matrix {{x^2, x*y^2}}
1 2 1
o109 = S <-- S <-- S
0 1 2
o109 : ChainComplex
i110 : FF.dd
1 2
o110 = 0 : S <-------------- S : 1
| x2 xy2 |
2 1
1 : S <---------------- S : 2
{2} | -xy2 |
{3} | x2 |
o110 : ChainComplexMap
i111 : S=kk[a,b,c,d]
o111 = S
o111 : PolynomialRing
i112 : IX = intersect(ideal(a,b), ideal(c,d))
o112 = ideal (b*d, a*d, b*c, a*c)
o112 : Ideal of S
i113 : IY = ideal(a-c, b-d)
o113 = ideal (a - c, b - d)
o113 : Ideal of S
i114 : degree ((S^1/IX) ** (S^1/IY))
o114 = 3
i115 : for j from 0 to 4 list degree Tor_j(S^1/IX, S^1/IY)
o115 = {3, 1, 0, 0, 0}
o115 : List
i116 : sum(0..4, j-> (-1)^j * degree Tor_j(S^1/IX, S^1/IY))
o116 = 2
i117 : Hom(IX, S^1/IX)
o117 = subquotient ({-2} | b d 0 0 |, {-2} | bd ad bc ac 0 0 0 0 0 0 0 0 0 0 0 0 |)
{-2} | a 0 d 0 | {-2} | 0 0 0 0 bd ad bc ac 0 0 0 0 0 0 0 0 |
{-2} | 0 c 0 b | {-2} | 0 0 0 0 0 0 0 0 bd ad bc ac 0 0 0 0 |
{-2} | 0 0 c a | {-2} | 0 0 0 0 0 0 0 0 0 0 0 0 bd ad bc ac |
4
o117 : S-module, subquotient of S
i118 : Ext^1(IX, S^1/IX)
o118 = cokernel {-2} | b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{-2} | 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{-2} | 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 |
{-2} | 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 |
{-2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 |
{-2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c |
6
o118 : S-module, quotient of S
i119 : HH^1 (sheaf (S^{-2}**(S^1/IX)))
2
o119 = kk
o119 : kk-module, free
i120 :
