# HigherCIOperators -- "Higher CI operators on a resolution over a complete intersection"

## Description

The "higher CI operators" complete the structure of the ordinary CI operators on (sometimes called "Eisenbud operators") on a resolution over a complete intersection in the same sense that the "higher homotopies" complete the structure of homotopies on with respect to a sequence of elements. Details will appear in a preprint in preparation by Burke, Eisenbud and Schreyer.

The higher CI operators are constructed by the routine higherCIOperators.

Just as a system of higher homotopies for a regular sequence f_1..f_c on a resolution over a ring S allow one to construct the Shamash resolution over R = S/(f_1..f_c), the higher CI operators are involved in a sort of dual construction: from a resolution F over R, lifted to a sequence of maps A over S, and lifted higher CI operators on A\otimes L, where L is the Koszul complex on f, one can construct a nonminimal resolution AL over S using the routine ciOperatorResolution.

## Version

This documentation describes version 0.5 of HigherCIOperators.

## Source code

The source code from which this documentation is derived is in the file HigherCIOperators.m2.

## Exports

• Functions and commands
• Methods
• "ciOperatorResolution(ChainComplex,ChainComplex)" -- see ciOperatorResolution -- "lift resolution from complete intersection using higher ci-operators"
• "exteriorMultiplication(ZZ)" -- see exteriorMultiplication -- "multiplication maps in the exterior algebra"
• "higherCIOperators(ChainComplex,ChainComplex)" -- see higherCIOperators -- "creates the HashTable of higher CI operators on a lifted resolution"
• "makeALDifferential(ZZ,ChainComplex,ChainComplex,HashTable)" -- see makeALDifferential -- "makes the differential used in ciOperatorResolution"
• "trueKoszul(Matrix)" -- see trueKoszul -- "Makes Koszul complex, with bases sorted in lex"

## For the programmer

The object HigherCIOperators is .