# higherCIOperators -- "creates the HashTable of higher CI operators on a lifted resolution"

## Synopsis

• Usage:
u = higherCIOperators(A,L)
• Inputs:
• A, , lifted resolution from complete intersection S→R
• L, , Koszul complex resolving R over S
• Outputs:
• u, , u{n,p,q} is a map Ap⊗Lq →Ap-n⊗Lq+n-1

## Description

A is the sequence of maps (generally not really a complex) obtained by lifting the differentials of a free resolution over R back to S.

Definition: u{n,p,q} is determined by induction on n and the rules

u{0,p,q}= (-1)q(Ap ⊗dL) : Ap⊗Lq →Ap⊗Lq-1

u{1,p,q}= dA ⊗Lq: Ap⊗Lq →Ap-1⊗Lq

i+j=n u{j,p-i,q+i-1}* u{i,p,q}= 0

and

u{n,p,q}: Ap⊗Lq →Ap-n⊗Lq+n-1 = μ* u{n,p,0}⊗Lq,

where

μ:Ln-1⊗Lq →Ln+q-1

is the multiplcation in the Koszul algebra.

The output u{n,p,q} will be defined for all keys {n,p,q} such that: length(A) ≥p ≥n ≥0 and if n=0 then length(L)≥q ≥1, else length(L)-n+1 ≥q ≥0.

The maps u{2,p,q} are thus the classical CI operators from Eisenbud , while the u{3,p,q} define maps of the modules ExtRodd, ≥3(M,k) →EReven(M,k) and are obstructions to commutativity of the classic ci operators on the R-free resolution of M.

These maps are used to construct the differentials in the lifted CI resolution