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Kernel: M2

Macaulay2 on CoCalc

Examples picked from http://www2.macaulay2.com/Macaulay2/doc/Macaulay2/share/doc/Macaulay2/Macaulay2Doc/html/index.html

In [1]:

25!

o1 = 15511210043330985984000000

In [2]:

kk=ZZ/101
S=kk[a,b,c,d,e]
M=matrix{{a,b,c},{b,c,d},{c,d,e}}

o4 = | a b c |
     | b c d |
     | c d e |

             3       3
o4 : Matrix S  <--- S

In [3]:

M^2

o5 = | 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
o5 : Matrix S  <--- S

In [4]:

determinant M

        3               2    2
o6 = - c  + 2b*c*d - a*d  - b e + a*c*e

o6 : S

coherent sheaves

http://www2.macaulay2.com/Macaulay2/doc/Macaulay2/share/doc/Macaulay2/Macaulay2Doc/html/___H__H^__Z__Z_sp__Sum__Of__Twists.html

In [2]:

R = QQ[a,b,c,d]/(a^4+b^4+c^4+d^4);
X = Proj R

o3 = X

o3 : ProjectiveVariety

In [3]:

Omega = cotangentSheaf X

o4 = cokernel {2} | c  0  0  d  0   a3  b3 0  |
              {2} | a  d  0  0  b3  -c3 0  0  |
              {2} | -b 0  d  0  a3  0   c3 0  |
              {2} | 0  b  a  0  -d3 0   0  c3 |
              {2} | 0  -c 0  a  0   -d3 0  b3 |
              {2} | 0  0  -c -b 0   0   d3 a3 |

                                         6
o4 : coherent sheaf on X, quotient of OO  (-2)
                                        X

In [4]:

HH^1(Omega)

       20
o5 = QQ

o5 : QQ-module, free

In [5]:

F = sheaf coker matrix {{a,b}}

o6 = cokernel | a b |

                                         1
o6 : coherent sheaf on X, quotient of OO
                                        X

In [6]:

module F

o7 = cokernel | a b |

                            1
o7 : R-module, quotient of R

In [10]:

R = QQ[x_0..x_4];
a = {1,0,0,0,0};
b = {0,1,0,0,1};
c = {0,0,1,1,0};
M1 = matrix table(5,5, (i,j)-> x_((i+j)%5)*a_((i-j)%5))

o20 = | x_0 0   0   0   0   |
      | 0   x_2 0   0   0   |
      | 0   0   x_4 0   0   |
      | 0   0   0   x_1 0   |
      | 0   0   0   0   x_3 |

              5       5
o20 : Matrix R  <--- R

In [11]:

M2 = matrix table(5,5, (i,j)-> x_((i+j)%5)*b_((i-j)%5))

o21 = | 0   x_1 0   0   x_4 |
      | x_1 0   x_3 0   0   |
      | 0   x_3 0   x_0 0   |
      | 0   0   x_0 0   x_2 |
      | x_4 0   0   x_2 0   |

              5       5
o21 : Matrix R  <--- R

In [12]:

M3 = matrix table(5,5, (i,j)-> x_((i+j)%5)*c_((i-j)%5))

o22 = | 0   0   x_2 x_3 0   |
      | 0   0   0   x_4 x_0 |
      | x_2 0   0   0   x_1 |
      | x_3 x_4 0   0   0   |
      | 0   x_0 x_1 0   0   |

              5       5
o22 : Matrix R  <--- R

In [13]:

M = M1 | M2 | M3;

              5       15
o23 : Matrix R  <--- R

In [21]:

betti (C=res coker M)

             0  1  2  3  4 5
o31 = total: 5 15 29 37 20 2
          0: 5 15 10  2  . .
          1: .  .  4  .  . .
          2: .  . 15 35 20 .
          3: .  .  .  .  . 2

o31 : BettiTally

In [25]:

N = transpose submatrix(C.dd_3,{10..28},{2..36});

              35       19
o35 : Matrix R   <--- R

In [26]:

betti (D=res coker N)

              0  1  2  3  4 5
o36 = total: 35 19 19 35 20 2
         -5: 35 15  .  .  . .
         -4:  .  4  .  .  . .
         -3:  .  .  .  .  . .
         -2:  .  .  .  .  . .
         -1:  .  .  .  .  . .
          0:  .  .  4  .  . .
          1:  .  . 15 35 20 .
          2:  .  .  .  .  . 2

o36 : BettiTally

In [27]:

Pfour = Proj(R)

o37 = Pfour

o37 : ProjectiveVariety

In [28]:

HorrocksMumford = sheaf(coker D.dd_3);

In [29]:

T = HH^1(HorrocksMumford(>=-1))

o39 = cokernel {-1} | x_4 x_2  0   0   x_0  0   0    0    x_3 0   0   0    0   0    x_1 |
               {-1} | 0   -x_3 x_1 0   0    x_4 x_2  0    0   0   0   x_0  0   0    0   |
               {-1} | 0   0    0   x_3 -x_2 x_0 0    x_1  0   0   0   0    0   -x_4 0   |
               {-1} | 0   0    0   0   0    0   -x_4 -x_2 x_1 x_0 x_3 0    0   0    0   |
               {-1} | 0   0    0   0   0    0   0    0    0   0   x_4 -x_3 x_2 x_1  x_0 |

                             5
o39 : R-module, quotient of R

In [30]:

apply(-1..2, i-> hilbertFunction(i,T))

o40 = (5, 10, 10, 2)

o40 : Sequence

In [0]: