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# ZZdFactorization _ ZZ -- access individual object in a ZZ/d-graded factorization

- Operator: _
- Usage:
    `C_i`
    `C^i`
- Inputs:
  - C, a ZZ/d graded factorization,
  - i, an integer, either the homological or cohomological index
- Outputs:
  - a module, the i-th object

## Description

ZZ/d-graded factorizations can be either homologically or cohomologically graded. Subscripts refer to homological indices, and superscripts refer to cohomological indices.

In this package homological indices are used by default. We always have the equation $C^i = C_{-i}$.

    i1 : S = ZZ/101[x_1..x_3]

    o1 = S

    o1 : PolynomialRing

    i2 : C = koszulMF({x_1,x_2,x_3}, x_1^2+x_2^2+x_3^2)

          4      4      4
    o2 = S  <-- S  <-- S

         0      1      0

    o2 : ZZdFactorization

    i3 : C_2

          4
    o3 = S

    o3 : S-module, free

    i4 : C^(-2)

          4
    o4 = S

    o4 : S-module, free

    i5 : C_2 == C^(-2)

    o5 = true

Indices that are outside of the period are taken modulo the period.

    i6 : C_-7

          4
    o6 = S

    o6 : S-module, free

## See also

- Making ZZdFactorizations -- Information about the basic constructors

## Ways to use this method:

- ZZdFactorization _ ZZ -- access individual object in a ZZ/d-graded factorization

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MatrixFactorizations/MatrixFactorizationsDOC.m2:820:0.