DGIdeal ^ ZZ -- Nonnegative integer power of a DG ideal

Operator: ^
Usage:

K = I^n
Inputs:
- I, an instance of the type DGIdeal,
- n, an integer, A nonnegative integer.
Outputs:
- K, an instance of the type DGIdeal, I^0 is the unit DGIdeal (1); I^n for n >= 1 is the n-fold product I * I * ... * I.

Description

Distributes with multiplication: I^2 * I == I^3.

i1 : R = ZZ/101[x, y]

o1 = R

o1 : PolynomialRing

i2 : A = koszulComplexDGA R

o2 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,2
      Differential => {x, y}

o2 : DGAlgebra

i3 : Anat = A.natural

o3 = Anat

o3 : PolynomialRing, 2 skew commutative variable(s)

i4 : I = dgIdeal(A, {x_Anat, y_Anat})

o4 = DGIdeal of Anat
     generators => | y x |

o4 : DGIdeal

i5 : isWellDefined(I^2)

o5 = true

i6 : isWellDefined(I^3)

o6 = true

i7 : (I^2 * I) == I^3

o7 = true

I^0 always returns the unit DGIdeal (1). Taking powers of the zero DG ideal is well-defined at every exponent.

i8 : I^0

o8 = DGIdeal of Anat
     generators => | 1 |

o8 : DGIdeal

i9 : Z = dgIdeal(A, {})

o9 = DGIdeal of Anat
     generators => 0

o9 : DGIdeal

i10 : isWellDefined(Z^0) and isWellDefined(Z^5)

o10 = true

Caveat

Negative exponents are rejected with an error.

Ways to use this method:

DGIdeal ^ ZZ -- Nonnegative integer power of a DG ideal

The source of this document is in DGAlgebras/doc.m2:2549:0.

DGIdeal ^ ZZ -- Nonnegative integer power of a DG ideal

Description

Caveat

See also

Ways to use this method: