Macaulay2 » Documentation
Packages » Schubert2 » AbstractSheaf RingElement
next | previous | forward | backward | up | index | toc

# AbstractSheaf RingElement -- twist by a divisor class

- Operator: SPACE
- Usage:
    `F(n)`
- Inputs:
  - F, an abstract sheaf,
  - n, a ring element, in (or promotable to) the intersection ring of the variety of $F$, homogeneous of degree 1 or 0
- Outputs:
  - an abstract sheaf, which depends on the degree of n. In the case where n has degree 0, the sheaf returned is the tensor product of F with the n-th (tensor) power of the tautological line bundle on the variety of F. In the case where n has degree 1, the sheaf returned is the tensor product of F with the line bundle whose first Chern class is n.

## Description

    i1 : X = abstractProjectiveSpace' 4

    o1 = X

    o1 : a flag bundle with subquotient ranks {4, 1}

    i2 : OO_X(3)

    o2 = a sheaf

    o2 : an abstract sheaf of rank 1 on X

    i3 : chi oo

    o3 = 35

    i4 : pt = base n

    o4 = pt

    o4 : an abstract variety of dimension 0

    i5 : Y = abstractProjectiveSpace'(4,pt)

    o5 = Y

    o5 : a flag bundle with subquotient ranks {4, 1}

    i6 : OO_Y(n)

    o6 = a sheaf

    o6 : an abstract sheaf of rank 1 on Y

    i7 : chi oo

          1 4    5 3   35 2   25
    o7 = --n  + --n  + --n  + --n + 1
         24     12     24     12

    o7 : QQ[n]

The notation OO(n) is an abbreviation for OO_X(n), where X is the variety whose intersection ring n is in. By default, the first Chern class of the tautological line bundle on a projective space or projective bundle is called h, so we may use OO(h) as alternative notation for OO_Y(1):

    i8 : A = intersectionRing Y

    o8 = A

    o8 : QuotientRing

    i9 : chern OO_Y(1)

    o9 = 1 + h

    o9 : A

    i10 : OO(h)

    o10 = a sheaf

    o10 : an abstract sheaf of rank 1 on Y

    i11 : chern oo

    o11 = 1 + h

    o11 : A

## Caveat

Beware the low parsing precedence of the adjacency operator SPACE.

## See also

- tautologicalLineBundle -- get the tautological line bundle of a flag bundle or abstract variety

## Ways to use this method:

- `AbstractSheaf QQ`
- AbstractSheaf RingElement -- twist by a divisor class
- `AbstractSheaf ZZ`

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Schubert2.m2:1857:0.