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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.

Author

Version

This documentation describes version 0.5 of HigherCIOperators, released May 10, 2015.

Citation

If you have used this package in your research, please cite it as follows:

@misc{HigherCIOperatorsSource,
  title = {{HigherCIOperators: higher CI operators. Version~0.5}},
  author = {David Eisenbud},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}

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 a package, defined in HigherCIOperators.m2.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/HigherCIOperators.m2:465:0.