-- -*- M2-comint -*- {* hash: 2045688505 *}
i1 : 2+2
o1 = 4
i2 : 3/5 + 7/11
68
o2 = --
55
o2 : QQ
i3 : 1*2*3*4
o3 = 24
i4 : 2^200
o4 = 1606938044258990275541962092341162602522202993782792835301376
i5 : 40!
o5 = 815915283247897734345611269596115894272000000000
i6 : 100!
o6 = 933262154439441526816992388562667004907159682643816214685929638952175999
932299156089414639761565182862536979208272237582511852109168640000000000
00000000000000
i7 : 1;2;3*4
o9 = 12
i10 : 4*5;
i11 : oo
o11 = 20
i12 : o5 + 1
o12 = 815915283247897734345611269596115894272000000001
i13 : "hi there"
o13 = hi there
i14 : s = "hi there"
o14 = hi there
i15 : s | " - " | s
o15 = hi there - hi there
i16 : s || " - " || s
o16 = hi there
-
hi there
i17 : {1, 2, s}
o17 = {1, 2, hi there}
o17 : List
i18 : 10*{1,2,3} + {1,1,1}
o18 = {11, 21, 31}
o18 : List
i19 : f = i -> i^3
o19 = f
o19 : FunctionClosure
i20 : f 5
o20 = 125
i21 : g = (x,y) -> x * y
o21 = g
o21 : FunctionClosure
i22 : g(6,9)
o22 = 54
i23 : apply({1,2,3,4}, i -> i^2)
o23 = {1, 4, 9, 16}
o23 : List
i24 : apply({1,2,3,4}, f)
o24 = {1, 8, 27, 64}
o24 : List
i25 : apply(1 .. 4, f)
o25 = (1, 8, 27, 64)
o25 : Sequence
i26 : apply(5, f)
o26 = {0, 1, 8, 27, 64}
o26 : List
i27 : scan(5, i -> print (i, i^3))
(0, 0)
(1, 1)
(2, 8)
(3, 27)
(4, 64)
i28 : j=1; scan(10, i -> j = 2*j); j
o30 = 1024
i31 : R = ZZ/5[x,y,z];
i32 : (x+y)^5
5 5
o32 = x + y
o32 : R
i33 : R
o33 = R
o33 : PolynomialRing
i34 : describe R
ZZ
o34 = --[x..z, Degrees => {3:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1]
5 {GRevLex => {3:1} }
{Position => Up }
i35 : F = R^3
3
o35 = R
o35 : R-module, free
i36 : F_1
o36 = | 0 |
| 1 |
| 0 |
3
o36 : R
i37 : F_{1,2}
o37 = | 0 0 |
| 1 0 |
| 0 1 |
3 2
o37 : Matrix R <--- R
i38 : F_{2,1,1}
o38 = | 0 0 0 |
| 0 1 1 |
| 1 0 0 |
3 3
o38 : Matrix R <--- R
i39 : f = matrix {{x,y,z}}
--warning: function f redefined
o39 = | x y z |
1 3
o39 : Matrix R <--- R
i40 : image f
o40 = image | x y z |
1
o40 : R-module, submodule of R
i41 : ideal (x,y,z)
o41 = ideal (x, y, z)
o41 : Ideal of R
i42 : kernel f
o42 = image {1} | -y 0 -z |
{1} | x -z 0 |
{1} | 0 y x |
3
o42 : R-module, submodule of R
i43 : generators oo
o43 = {1} | -y 0 -z |
{1} | x -z 0 |
{1} | 0 y x |
3 3
o43 : Matrix R <--- R
i44 : poincare kernel f
2 3
o44 = 3T - T
o44 : ZZ[T]
i45 : rank kernel f
o45 = 2
i46 : presentation kernel f
o46 = {2} | z |
{2} | x |
{2} | -y |
3 1
o46 : Matrix R <--- R
i47 : cokernel f
o47 = cokernel | x y z |
1
o47 : R-module, quotient of R
i48 : N = kernel f ++ cokernel f
o48 = subquotient ({1} | -y 0 -z 0 |, {1} | 0 0 0 |)
{1} | x -z 0 0 | {1} | 0 0 0 |
{1} | 0 y x 0 | {1} | 0 0 0 |
{0} | 0 0 0 1 | {0} | x y z |
4
o48 : R-module, subquotient of R
i49 : generators N
o49 = {1} | -y 0 -z 0 |
{1} | x -z 0 0 |
{1} | 0 y x 0 |
{0} | 0 0 0 1 |
4 4
o49 : Matrix R <--- R
i50 : relations N
o50 = {1} | 0 0 0 |
{1} | 0 0 0 |
{1} | 0 0 0 |
{0} | x y z |
4 3
o50 : Matrix R <--- R
i51 : prune N
o51 = cokernel {2} | 0 0 0 z |
{2} | 0 0 0 x |
{2} | 0 0 0 -y |
{0} | z y x 0 |
4
o51 : R-module, quotient of R
i52 : C = resolution cokernel f
1 3 3 1
o52 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o52 : ChainComplex
i53 : C.dd
1 3
o53 = 0 : R <------------- R : 1
| x y z |
3 3
1 : R <-------------------- R : 2
{1} | -y -z 0 |
{1} | x 0 -z |
{1} | 0 x y |
3 1
2 : R <-------------- R : 3
{2} | z |
{2} | -y |
{2} | x |
1
3 : R <----- 0 : 4
0
o53 : ChainComplexMap
i54 : C.dd^2 == 0
o54 = true
i55 : betti C
0 1 2 3
o55 = total: 1 3 3 1
0: 1 3 3 1
o55 : BettiTally
i56 : R = ZZ/101[a .. r];
i57 : g = genericMatrix(R,a,3,6)
o57 = | a d g j m p |
| b e h k n q |
| c f i l o r |
3 6
o57 : Matrix R <--- R
i58 : M = cokernel g
o58 = cokernel | a d g j m p |
| b e h k n q |
| c f i l o r |
3
o58 : R-module, quotient of R
i59 : time C = resolution M
-- used 0.001 seconds
3 6 15 18 6
o59 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o59 : ChainComplex
i60 : betti C
0 1 2 3 4
o60 = total: 3 6 15 18 6
0: 3 6 . . .
1: . . . . .
2: . . 15 18 6
o60 : BettiTally
i61 : S = ZZ/101[t_1 .. t_9, u_1 .. u_9];
i62 : m = genericMatrix(S, t_1, 3, 3)
o62 = | t_1 t_4 t_7 |
| t_2 t_5 t_8 |
| t_3 t_6 t_9 |
3 3
o62 : Matrix S <--- S
i63 : n = genericMatrix(S, u_1, 3, 3)
o63 = | u_1 u_4 u_7 |
| u_2 u_5 u_8 |
| u_3 u_6 u_9 |
3 3
o63 : Matrix S <--- S
i64 : m*n
o64 = | t_1u_1+t_4u_2+t_7u_3 t_1u_4+t_4u_5+t_7u_6 t_1u_7+t_4u_8+t_7u_9 |
| t_2u_1+t_5u_2+t_8u_3 t_2u_4+t_5u_5+t_8u_6 t_2u_7+t_5u_8+t_8u_9 |
| t_3u_1+t_6u_2+t_9u_3 t_3u_4+t_6u_5+t_9u_6 t_3u_7+t_6u_8+t_9u_9 |
3 3
o64 : Matrix S <--- S
i65 : j = flatten(m*n - n*m)
o65 = | t_4u_2+t_7u_3-t_2u_4-t_3u_7 t_2u_1-t_1u_2+t_5u_2+t_8u_3-t_2u_5-t_3u_8
-----------------------------------------------------------------------
t_3u_1+t_6u_2-t_1u_3+t_9u_3-t_2u_6-t_3u_9
-----------------------------------------------------------------------
-t_4u_1+t_1u_4-t_5u_4+t_4u_5+t_7u_6-t_6u_7 -t_4u_2+t_2u_4+t_8u_6-t_6u_8
-----------------------------------------------------------------------
-t_4u_3+t_3u_4+t_6u_5-t_5u_6+t_9u_6-t_6u_9
-----------------------------------------------------------------------
-t_7u_1-t_8u_4+t_1u_7-t_9u_7+t_4u_8+t_7u_9
-----------------------------------------------------------------------
-t_7u_2-t_8u_5+t_2u_7+t_5u_8-t_9u_8+t_8u_9 -t_7u_3-t_8u_6+t_3u_7+t_6u_8
-----------------------------------------------------------------------
|
1 9
o65 : Matrix S <--- S
i66 : gb j
o66 = GroebnerBasis[status: done; S-pairs encountered up to degree 5]
o66 : GroebnerBasis
i67 : generators gb j;
1 26
o67 : Matrix S <--- S
i68 : betti gb j
0 1
o68 = total: 1 26
0: 1 .
1: . 8
2: . 12
3: . 5
4: . 1
o68 : BettiTally
i69 :
