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


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.



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.


  • 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 a package.