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

- Usage:
    `u = higherCIOperators(A,L)`
- Inputs:
  - A, a chain complex, lifted resolution from complete intersection $S\to R$
  - L, a chain complex, Koszul complex resolving R over S
- Outputs:
  - u, a hash table, $u\{n,p,q\}$ is a map $A_p\otimes L_q \to A_{p-n}\otimes L_{q+n-1}$

## 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

$u\{0,p,q\} = (-1)^q(A_p \otimes d_L) :\ A_p\otimes L_q \to A_p\otimes L_{q-1}$

$u\{1,p,q\} = d_A \otimes L_q: A_p\otimes L_q \to A_{p-1}\otimes L_q$

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

and

$u\{n,p,q\}: A_p\otimes L_q \to A_{p-n}\otimes L_{q+n-1} = \mu * u\{n,p,0\}\otimes L_q$,

where

$\mu:L_{n-1}\otimes L_q \to L_{n+q-1}$

is the multiplication in the Koszul algebra.

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

The maps $u\{2,p,q\}$ are thus the classical CI operators from Eisenbud [1981], while the $u\{3,p,q\}$ define maps of the modules $Ext_R^{odd, \geq 3}(M,k) \to E_R^{even}(M,k)$ 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

## See also

- makeALDifferential -- "makes the differential used in ciOperatorResolution"
- ciOperatorResolution -- "lift resolution from complete intersection using higher ci-operators"

## Ways to use higherCIOperators:

- `higherCIOperators(ChainComplex,ChainComplex)`

## For the programmer

The object higherCIOperators is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/HigherCIOperators.m2:343:0.