# beilinsonContraction -- compute a Beilinson contraction

## Synopsis

• Usage:
beilinsonContraction(e, rowdeg, coldeg)
• Inputs:
• e, , an element of the exterior algebra E
• rowdeg, a list,
• coldeg, a list, two multidegrees
• Optional inputs:
• BundleType => , default value PrunedQuotient, with values PrunedQuotient or SubBundle
• Outputs:
• , a map between modules over the symmetric algebra S

## Description

Returns a map between two Beilinson generators (i.e. the beilinson generators beilinsonBundle(a, E) of the derived category). Note:

(1) (S,E) is the result of productOfProjectiveSpaces

(2) e is a homogeneous element of E giving a map from E(-coldeg) --> E(-rowdeg).

 i1 : (S,E) = productOfProjectiveSpaces {2,1} o1 = (S, E) o1 : Sequence i2 : gens S, gens E o2 = ({x , x , x , x , x }, {e , e , e , e , e }) 0,0 0,1 0,2 1,0 1,1 0,0 0,1 0,2 1,0 1,1 o2 : Sequence i3 : f=e_(0,0)*e_(0,1)*e_(1,0) o3 = e e e 0,0 0,1 1,0 o3 : E i4 : beilinsonContraction(f,{0,0},{2,1}) o4 = | x_(0,2)x_(1,1) | 1 1 o4 : Matrix S <--- S i5 : m=beilinsonContraction(e_(0,0)*e_(1,0),{0,0},{1,1}) o5 = | x_(0,1)x_(1,1) x_(0,2)x_(1,1) 0 | 1 3 o5 : Matrix S <--- S

## Caveat

E is positively graded, in contrast to the paper!

• beilinson -- apply the beilinson functor
• beilinsonBundle -- compute a basic Beilinson bundle
• BundleType -- Option in beilinson with values PrunedQuotient, QuotientBundle, DummyQuotientBundle, SubBundle, FreeBundle, or MapsBetweenFreeBundles
• SubBundle -- value for the option BundleType in beilinson
• PrunedQuotient -- value for the option BundleType in beilinson

## Ways to use beilinsonContraction :

• "beilinsonContraction(RingElement,List,List)"

## For the programmer

The object beilinsonContraction is .