Path: m2.ipynb
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Kernel: M2

# Macaulay2 on CoCalc

```25!
```
``` o1 = 15511210043330985984000000 ```
```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 ```
```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 ```
```determinant M
```
``` 3 2 2 o6 = - c + 2b*c*d - a*d - b e + a*c*e o6 : S ```

## coherent sheaves

```R = QQ[a,b,c,d]/(a^4+b^4+c^4+d^4);
X = Proj R
```
``` o3 = X o3 : ProjectiveVariety ```
```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 ```
```HH^1(Omega)
```
``` 20 o5 = QQ o5 : QQ-module, free ```
```F = sheaf coker matrix {{a,b}}
```
``` o6 = cokernel | a b | 1 o6 : coherent sheaf on X, quotient of OO X ```
```module F
```
``` o7 = cokernel | a b | 1 o7 : R-module, quotient of R ```
```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 ```
```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 ```
```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 ```
```M = M1 | M2 | M3;
```
``` 5 15 o23 : Matrix R <--- R ```
```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 ```
```N = transpose submatrix(C.dd_3,{10..28},{2..36});
```
``` 35 19 o35 : Matrix R <--- R ```
```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 ```
```Pfour = Proj(R)
```
``` o37 = Pfour o37 : ProjectiveVariety ```
```HorrocksMumford = sheaf(coker D.dd_3);
```
```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 ```
```apply(-1..2, i-> hilbertFunction(i,T))
```
``` o40 = (5, 10, 10, 2) o40 : Sequence ```
```
```