Seminormalization -- a package used to seminormalize rings

Description

Seminormalization is a package which can be used to seminormalize rings or more generally check if a ring is seminormal. Roughly speaking a ring is non-normal if in the Spec, points are identified and tangent spaces killed. Seminormal rings are those where only points are identified. See the following for more discussion: MathOverflow question

Core functions

seminormalize computes the seminormalization of a ring
isSeminormal checks if a ring is seminormal

To accomplish this, this we first normalize the ring via integralClosure and then glue points together via the Pullback package. There are some other functions exported which people may also find useful.

Other useful functions

ringProduct finds a ring equal to a product of rings
betterNormalizationMap computes the integral closure of reduced rings (that are not necessarily domains)

Authors

Karl Schwede <[email protected]>
Bernard Serbinowski <[email protected]>

Certification

Version 0.21 of this package was accepted for publication in volume 10 of The Journal of Software for Algebra and Geometry on 5 December 2019, in the article Seminormalization package for Macaulay2 (DOI: 10.2140/jsag.2020.10.1). That version can be obtained from the journal.

Version

This documentation describes version 0.22 of Seminormalization.

Citation

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

@misc{SeminormalizationSource,
  title = {{Seminormalization: seminormalization of rings. Version~0.22}},
  author = {Karl Schwede and Bernard Serbinowski},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

@article{SeminormalizationArticle,
  title = {{Seminormalization package for \emph{Macaulay2}}},
  author = {Karl Schwede and Bernard Serbinowski},
  journal = {The Journal of Software for Algebra and Geometry},
  volume = {10},
  year = {2019},
}

Exports

Functions and commands
- betterNormalizationMap -- normalizes non domains
- conductorOfRingMap -- given a ring map this finds the conductor of the map
- findElementMappingToTarget -- given a ring map and an element of the target, this find an element from the source whose image is the given one
- flattenVarDegrees -- turns a multigraded ring into a singly graded ring
- isSeminormal -- checks if a ring is seminormal
- ringProduct -- compute the product of a list of rings
- ringToAlgebraMap -- presents the target of a map of rings as an algebra over the source
- seminormalize -- seminormalize a reduced ring
Methods
- betterNormalizationMap(Ring) -- see betterNormalizationMap -- normalizes non domains
- conductorOfRingMap(RingMap) -- see conductorOfRingMap -- given a ring map this finds the conductor of the map
- findElementMappingToTarget(RingMap,RingElement) -- see findElementMappingToTarget -- given a ring map and an element of the target, this find an element from the source whose image is the given one
- flattenVarDegrees(Ring) -- see flattenVarDegrees -- turns a multigraded ring into a singly graded ring
- isSeminormal(Ring) -- see isSeminormal -- checks if a ring is seminormal
- ringProduct(List) -- see ringProduct -- compute the product of a list of rings
- ringToAlgebraMap(RingMap) -- see ringToAlgebraMap -- presents the target of a map of rings as an algebra over the source
- ringToAlgebraMap(RingMap,ZZ) -- see ringToAlgebraMap -- presents the target of a map of rings as an algebra over the source
- seminormalize(Ring) -- see seminormalize -- seminormalize a reduced ring
Symbols
- Yy -- default symbol for new variables in the seminormalization method
Mutable symbols
- Yy -- default symbol for new variables in the seminormalization method

For the programmer

The object Seminormalization is a package, defined in Seminormalization.m2.

The source of this document is in Seminormalization.m2:694:0.