Macaulay2Doc : Table of Contents

• Macaulay2Doc
  • Copyright and license
    • Singular-Factory
    • Singular-Libfac
    • GNU MP
    • MPFR
    • GC garbage collector
    • LAPACK
    • BLAS
    • NTL library
  • Acknowledgements
  • The authors
    • Daniel R. Grayson
    • Michael E. Stillman
    • David Eisenbud
  • Other sources of information about Macaulay 2
  • getting started
    • checking your Macaulay2 installation -- and bookmarking the Macaulay2 documentation
    • finding the Macaulay 2 files
    • teaching M2 how to find its shared libraries
    • teaching your shell how to find M2
    • moving or copying the Macaulay 2 files
    • running Macaulay 2 in a terminal window
    • using Macaulay2 with emacs
      • setting up the Macaulay2 emacs interface
      • teaching emacs how to find M2-init.el
      • teaching emacs how to find M2
      • running Macaulay 2 in emacs
      • using Macaulay2 with emacs after it has been set up
      • editing Macaulay 2 code with emacs
    • using Macaulay2 with TeXmacs
    • a first Macaulay 2 session
    • reading the documentation
    • getting help or reporting bugs
  • rings
    • basic rings of numbers
    • integers modulo a prime
    • finite fields
    • polynomial rings
    • monoid -- make or retrieve a monoid
    • monomial orderings
      • examples of specifying alternate monomial orders
      • monomial orders for free modules
        • Schreyer orders -- induced monomial order on a free module
      • packing monomials for efficiency
      • GRevLex -- graded reverse lexicographical monomial order.
      • Lex -- lexicographical monomial order.
      • GLex -- graded lexicographic ordering
      • Weights -- assigning weights to the variables
      • Eliminate -- elimination order
      • GroupLex -- defines a ring where some variables are inverted
      • GroupRevLex
      • definition of product (block) orders
      • RevLex -- reverse lexicographic ordering
      • NCLex -- Non-commutative lexicographical order.
      • MonomialOrder -- monomial ordering
      • obtaining the monomial order of a ring
    • graded and multigraded polynomial rings
    • quotient rings
    • manipulating polynomials
    • factoring polynomials
    • fraction fields
    • finite field extensions
    • exterior algebras
    • symmetric algebras
    • tensor products of rings
    • Weyl algebras
    • associative algebras
  • ideals
    • creating an ideal
    • ideals to and from matrices
    • ideals to and from modules
    • sums, products, and powers of ideals
    • equality and containment
    • extracting generators of an ideal
    • dimension, codimension, and degree
    • intersection of ideals
    • ideal quotients and saturation
    • radical of an ideal
    • minimal primes of an ideal
    • associated primes of an ideal
    • primary decomposition
  • matrices
    • inputting a matrix
    • projections, inclusions, and permutations
    • random and generic matrices
    • extracting information about a matrix
    • basic arithmetic of matrices
    • concatenating matrices
    • differentiation
    • rank of a matrix
    • determinants and minors
    • Pfaffians
    • exterior power of a matrix
    • format and display of matrices in Macaulay 2
    • importing and exporting matrices
  • substitution and maps between rings
    • substitute -- substituting values for variables
    • working with multiple rings
    • basic construction, source and target of a ring map
    • evaluation and composition of ring maps
    • kernel and image of a ring map
    • graphIdeal(RingMap) -- the ideal of the graph of the regular map corresponding to a ring map
    • graphRing(RingMap) -- the coordinate ring of the graph of the regular map corresponding to a ring map
  • modules
    • free modules
    • matrices to and from modules -- including kernel, cokernel and image
    • submodules and quotients
    • subquotient modules -- the way Macaulay 2 represents modules
    • module homomorphisms
    • right modules or left modules?
    • Hilbert functions and free resolutions -- including degree and betti numbers
    • basis -- basis of all or part of a module or ring
    • exterior power of a module
  • chain complexes
    • free resolutions of modules
    • extracting information from chain complexes
    • making chain complexes by hand
    • manipulating chain complexes
    • maps between chain complexes
  • varieties
    • coherent sheaves
    • Variety -- the class of all algebraic varieties
  • Groebner bases
    • what is a Groebner basis?
    • Groebner basis examples and applications
      • simple Groebner basis computations over various rings
    • computing Groebner bases
  • normal forms
  • elimination of variables
  • primaryDecomposition -- irredundant primary decomposition of an ideal
  • combinatorial functions
  • The Macaulay2 language
    • variables
      • valid names
      • assigning values
      • local variables in a file
      • viewing the symbols defined so far
      • subscripted variables
      • numbered variables
    • numeric types
    • strings and nets
      • /// -- delineate a string
      • concatenate -- join strings
      • format -- format a string or a real number
      • lines -- split a string into lines
      • separate -- split a string into pieces
      • regular expressions
        • match -- regular expression matching
          • match(String,String) -- regular expression matching
        • regex -- regular expression matching
        • replace -- regular expression replacement
          • replace(String,String,String) -- regular expression replacement
        • select(String,String) -- select substrings matching a regular expression from a string
        • select(String,String,String) -- select substrings matching a regular expression from a string
      • horizontalJoin -- join nets or strings horizontally
      • stack -- join nets or string vertically
      • unstack -- list the rows of a net
      • String -- the class of all strings
      • Net -- the class of all nets and strings
    • lists and sequences
      • ranges and repetitions
      • # BasicList -- length
      • BasicList # ZZ -- get element from list
      • BasicList #? ZZ -- check for element in list
      • VisibleList _ ZZ -- get element from list
      • VisibleList _ List -- get some entries of a list
      • first -- first element of a list
      • last -- last element of a list
      • toList -- list of elements
      • toSequence -- convert to sequence
      • sequence -- make a sequence
      • unsequence -- extract the single element from a sequence of length 1
      • append -- append an element to a list
      • between -- insert something between elements of a list
      • delete -- delete elements of a list
      • drop -- drop some elements from a list or sequence
      • flatten -- flatten a list by unnesting lists
      • fold -- apply binary operator repeatedly
      • join -- join lists
      • List | List -- join lists, sequences or arrays
      • mingle -- mingle elements of several lists
      • pack -- pack elements of a list into shorter ones
      • prepend -- add an element to the beginning of a list
      • reverse(BasicList) -- reverse a list
      • rsort -- sort a list or matrix in reverse order
      • sort -- sort a list or columns of a matrix
      • subtable -- extract a subtable from a table
      • table -- make a table (nested list)
      • take -- take some elements from a list
      • unique -- eliminate duplicates from a list
      • apply(BasicList,Function) -- apply a function to each element of a list
      • scan(BasicList,Function) -- apply a function to each element of a list
      • all(BasicList,Function) -- whether all elements of a list satisfy a specified condition
      • any(BasicList,Function) -- whether any elements of a list satisfy a specified condition
      • position -- find first element of a list satisfying a condition
      • positions -- which elements of a list satisfy a condition
      • select(BasicList,Function) -- select elements from a list
      • select(ZZ,BasicList,Function) -- select a limited number of elements from a list
      • VisibleList -- the class of all visible lists
        • List -- the class of all lists -- {...}
        • Sequence -- the class of all sequences -- (...)
        • Array -- the class of all arrays -- [...]
      • BasicList -- the class of all basic lists
    • hash tables
      • copy -- copy an object
      • hashTable -- make a hash table
      • keys -- keys used in a hash table
      • values -- values in a hash table
      • pairs -- list the pairs in a hash table
      • mutable -- whether something may be modified
      • remove -- remove an entry from a hash table
      • applyKeys -- apply a function to each key in a hash table
      • applyValues -- apply a function to each value
      • applyPairs -- apply a function to each pair in a hash table
      • merge(HashTable,HashTable,Function) -- merge hash tables
      • combine -- combine hash tables
      • hashing
      • HashTable -- the class of all hash tables
      • MutableHashTable -- the class of all mutable hash tables
    • operators
      • = -- assignment
      • := -- assignment of method or new local variable
      • <- -- assignment with left side evaluated
      • == -- equality
      • != -- inequality
      • === -- strict equality
      • =!= -- strict inequality
      • < -- less than
      • <= -- less than or equal
      • > -- greater than
      • >= -- greater than or equal
      • ? -- comparison operator
      • and -- conjunction
      • or -- disjunction
      • not -- negation
      • SPACE -- blank operator; often used for function application, making polynomial rings
      • -> -- make a function
      • @@ -- a binary operator
      • \\ -- a binary operator
      • \ -- a binary operator
      • _ -- a binary operator, used for subscripting and access to elements
      • . -- access to elements whose key is a symbol
      • .? -- check for presence of elements whose key is a symbol
      • # -- length, or access to elements
      • #? -- check for presence of elements
      • ! -- factorial
      • + -- a unary or binary operator, usually used for addition
      • - -- a unary or binary operator, usually used for negation or subtraction
      • * -- a binary operator, usually used for multiplication
      • / -- a binary operator, usually used for division
      • // -- a binary operator, usually used for quotient
      • % -- a binary operator, usually used for remainder and reduction
      • ^ -- a binary operator, usually used for powers
      • & -- a binary operator
      • ++ -- a binary operator, usually used for direct sum
      • ** -- a binary operator, usually used for tensor product or Cartesian product
      • ^** -- a binary operator, usually used for tensor or Cartesian power
      • ~ -- a unary postfix operator
      • (*) -- a unary postfix operator, used for indicating a graded object
      • : -- a binary operator, uses include repetition; ideal quotients
      • .. -- a binary operator, uses include ranges; sequence of consecutive items
      • << -- a binary operator (file output, ...)
      • >> -- a binary operator, uses include bit shifting, or attaching optional inputs to functions
      • | -- a binary operator, often used for horizontal concatenation
      • || -- a binary operator, often used for vertical concatenation
      • @ -- a binary operator
      • ^^ -- a binary operator
      • |- -- a binary operator
      • <==> -- a binary operator
      • ===> -- a binary operator
      • ==> -- a binary operator
      • => -- produce an Option
      • , -- the comma, used for separating entries in a list or sequence
      • ; -- statement separator
      • precedence of operators
      • operatorAttributes
    • conditional execution -- if-then-else statements
    • while -- while loops
    • for -- for loops
    • mapping over lists -- apply a function to each element of a list
    • mapping over hash tables -- apply a function to each element of a hash table
    • error handling -- signalling and trapping errors
      • error -- deliver error message
      • try -- catch an error
    • catch -- catch a thrown exception
    • using functions
    • using functions with optional inputs
    • making functions
    • local variables in a function
    • making functions with a variable number of arguments
    • making functions with multiple return values
    • making new functions with optional arguments
    • what a class is
      • class -- class of an object
      • parent -- parent type of an object
      • instance -- whether something has a certain type
      • ancestor(Type,Type) -- whether one type is an ancestor of another
    • installing methods
    • binary methods
    • inheritance
    • making new classes
    • new -- new objects and new types
      • newClass -- set the class and parent of an object
    • printing and formatting for new classes
    • making a new method function
      • method -- make a new method function
    • printing to the screen
    • reading files
    • getting input from the user
    • creating and writing files
    • saving polynomials and matrices in files
    • two dimensional formatting
    • file manipulation -- Unix file manipulation functions
      • fileExists -- whether a file exists
      • isRegularFile -- whether a file is a regular file
      • isDirectory -- whether a file is a directory
      • fileTime -- get or set file modified time
      • copyFile(String,String)
      • moveFile(String,String)
      • removeFile -- unlink a file
      • findFiles(String) -- find files recursively
      • symlinkFile -- make a symbolic link to a file
      • linkFile(String,String) -- make a new link to a file
      • readlink -- readlink fn
      • mkdir
      • makeDirectory(String) -- make a directory
      • copyDirectory(String,String)
      • removeDirectory -- remove a directory
      • symlinkDirectory(String,String) -- make symbolic links for all files in a directory tree
      • readDirectory -- read the contents of a directory
      • realpath -- convert a filename to one passing through no symbolic links
      • temporaryFileName -- make a temporary file name
    • communicating with programs
    • using sockets
    • packages
      • needsPackage -- load a package if not already loaded
      • loadPackage -- load a package
      • installPackage -- load and install a package and its documentation
      • loadedPackages -- the list of loaded packages
      • creating a package
        • an example of a package
        • newPackage(String) -- package item: start a new package
        • export -- package item: export functions
        • exportMutable -- package item: export writable variables
        • beginDocumentation -- package item: start documentation section
        • document -- package item: documentation node
        • TEST -- package item: register a test of the package
        • writing documentation
        • check -- perform tests of a package
        • informing others about your package
      • getPackage(String) -- download a package from the repository
    • system facilities
      • autoload(Symbol,String) -- arrange for a function to be loaded automatically
      • initialization file
      • input -- read Macaulay 2 commands and echo
      • load -- read Macaulay 2 commands
      • needs -- read Macaulay 2 commands if necessary
      • clearEcho -- turn off echoing
      • setEcho -- turn on echoing
      • dumpdata -- dump state of the system to a file
      • loaddata -- load state of the system from a file
      • restart -- restart Macaulay 2
      • addStartFunction -- add a startup function
      • addEndFunction -- add an ending function
      • top level loop
      • alarm -- set an alarm
      • currentDirectory -- current working directory
      • exec -- execute another program
      • exit -- exit the program
      • fork -- fork the process
      • getenv -- get value of environment variable
      • processID -- the process identifier
      • path -- list of directories to look in
      • quit -- quit the program
      • run -- run an external command
      • sleep -- sleep for a while
      • time -- time a computation
      • timing -- time a computation
      • wait -- wait for child process
      • commandLine -- the command line arguments
      • environment -- the environment variables
      • version -- information about this version of the program
      • collectGarbage -- collect the garbage in memory
    • debugging
  • packages provided with Macaulay 2
  • handling hypertext
  • mathematical examples
    • Tutorial: Elementary uses of Groebner bases
    • Tutorial: Canonical Embeddings of Plane Curves and Gonality
    • Tutorial: Fano varieties
    • Tutorial: Divisors
  • basic commutative algebra
    • Elementary uses of Groebner bases I. Math 634 Fall 2005
    • modules in Macaulay2
    • M2SingularBook -- Macaulay2 examples for the Singular book
      • Singular Book 1.1.8 -- computation in fields
      • Singular Book 1.1.9 -- computation in polynomial rings
      • Singular Book 1.1.10 -- methods for creating ring maps
      • Singular Book 1.2.3 -- leading data
      • Singular Book 1.2.13 -- monomial orderings
      • Singular Book 1.3.3 -- properties of ring maps
      • Singular Book 1.3.13 -- computation in quotient rings
      • Singular Book 1.3.15 -- computing with radicals
      • Singular Book 1.4.9 -- global versus local rings
      • Singular Book 1.5.10 -- realization of rings
      • Singular Book 1.6.13 -- normal form
      • Singular Book 1.7.10 -- standard bases
      • Singular Book 1.7.12 -- highest corner
      • Singular Book 1.8.1 -- ideal membership
      • Singular Book 1.8.2 -- linear combination of ideal members
      • Singular Book 1.8.4 -- elimination of variables
      • Singular Book 1.8.6 -- Zariski closure of the image
      • Singular Book 1.8.7 -- solving equations
      • Singular Book 1.8.9 -- radical membership
      • Singular Book 1.8.11 -- intersection of ideals
      • Singular Book 1.8.13 -- quotient of ideals
      • Singular Book 1.8.15 -- saturation
      • Singular Book 1.8.18 -- kernel of a ring map
      • Singular Book 1.8.19 -- algebraic dependence
      • Singular Book 1.8.20 -- subalgebra membership
      • Singular Book 2.1.6 -- matrix operations
      • Singular Book 2.1.7 -- maps induced by Hom
      • Singular Book 2.1.10 -- Submodules of A^n
      • Singular Book 2.1.13 -- kernel, image and cokernel of a module homomorphism
      • Singular Book 2.1.20 -- sum, intersection, module quotient
      • Singular Book 2.1.24 -- submodules, presentation of a module
      • Singular Book 2.1.26 -- computation of Hom
      • Singular Book 2.1.34 -- minimal presentations, prune
      • Singular Book 2.2.15 -- graded rings and modules
      • Singular Book 2.3.10 -- normal form
      • Singular Book 2.3.12 -- standard bases
      • Singular Book 2.4.12 -- resolution and Betti numbers
      • Singular Book 2.4.15 -- homogeneous resolution and graded Betti numbers
      • Singular Book 2.5.5 -- syzygies
      • Singular Book 2.5.18 -- Schreyer resolution
      • Singular Book 2.6.3 -- diagonal form
      • Singular Book 2.6.11 -- cyclic decomposition
      • Singular Book 2.6.15 -- Jordan normal form
      • Singular Book 2.7.5 -- tensor product of maps
      • Singular Book 2.7.9 -- tensor product of modules
      • Singular Book 2.7.14 -- tensor product of rings
      • Singular Book 2.8.1 -- module membership
      • Singular Book 2.8.3 -- elimination of module components
      • Singular Book 2.8.5 -- intersection of modules
      • Singular Book 2.8.6 -- quotient of modules
      • Singular Book 2.8.7 -- radical, zerodivisors of modules
      • Singular Book 2.8.8 -- annihilator and Fitting ideal
      • Singular Book 2.8.9 -- kernel of a module homomorphism
      • Singular Book 2.8.10 -- solving linear equations
  • replacements for commands and scripts from Macaulay
  • how Macaulay 2 finds its files
    • LAYOUT -- relative locations of Macaulay 2 files
    • prefixDirectory -- the prefix directory
    • buildHomeDirectory -- the directory where the Macaulay 2 program is being built
    • sourceHomeDirectory -- the directory where the source code is while Macaulay 2 is being built
  • Type -- the class of all types
  • Function -- the class of all functions
• # HashTable -- length
• # Set -- cardinality
• # String -- length
• - List -- negation of a vector
• -- -- comment
• > ZZ -- greater than
• >= ZZ -- greater than or equal
• ^* -- a unary postfix operator, used for indicating pullback maps
• _* -- a unary postfix operator, used for indicating pushforward maps
• abs -- absolute value function
• AbsoluteLinks -- name for an optional argument
• accumulate -- apply binary operator repeatedly
• accumulate(Function,Thing,VisibleList)
• accumulate(Function,VisibleList)
• accumulate(VisibleList,Function)
• accumulate(VisibleList,Thing,Function)
• acos -- arccosine
• acosh -- inverse hyperbolic cosine
• acot -- compute the inverse cotangent
• addHook -- add a hook function to an object for later processing
• addHook(HashTable,Thing,Function) -- add a hook function to an object for later processing
• addHook(Symbol,Function) -- add a hook function to an object for later processing
• Adjacent -- the class of all adjacent expression pairs
• adjoint(Matrix,Module,Module) -- an adjoint map
• adjoint1(Matrix,Module,Module) -- an adjoint map
• AffineVariety -- the class of all affine varieties
• AfterEval -- top level method applied after evaluation
• AfterNoPrint -- top level method applied after not printing
• AfterPrint -- top level method applied after printing
• agm -- arithmetic-geometric mean
• Algorithm -- name for an optional argument
• Alignment -- name for an optional argument
• all -- whether all elements satisfy a specified condition
• all(BasicList,BasicList,Function) -- whether all corresponding elements of a pair of lists satisfy a condition
• all(HashTable,Function) -- whether all key/value pairs in a hash table satisfy a specified condition
• ambient -- ambient free module of a subquotient, or ambient ring
• ambient(AffineVariety)
• ambient(CoherentSheaf)
• ambient(GaloisField) -- corresponding quotient ring
• ambient(GradedModule)
• ambient(Matrix)
• ambient(Module) -- ambient free module
• ambient(ProjectiveVariety)
• ambient(Ring) -- ambient polynomial ring
• ancestors -- the ancestors of something
• ANCHOR -- hypertext anchor
• annihilator -- the annihilator ideal
• annihilator of a module
• any -- whether any elements satisfy a specified condition
• any(BasicList,BasicList,Function) -- whether any corresponding elements of a pair of lists satisfy a condition
• any(HashTable,Function) -- whether all key/value pairs in a hash table satisfy a specified condition
• append(BasicList,Thing) -- append an element to a list
• applicationDirectory -- the path to the user's application directory
• applicationDirectorySuffix -- suffix that determines the user's application directory
• apply -- apply a function to each element
• apply(BasicList,BasicList,Function) -- apply a function to pairs of elements, one from each list
• apply(ZZ,Function) -- apply a function to {0,..., n-1}
• applyTable -- apply a function to elements of a table
• apropos(String) -- symbols matching a pattern
• argument -- specify the function in a scripted functor for an argument
• Ascending
• ascii -- ASCII character conversion
• ascii(List) -- ASCII character conversion
• ascii(String) -- ASCII character conversion
• asin -- arcsine
• asinh -- inverse hyperbolic sine
• assert -- assert something is true
• associatedPrimes -- find the associated primes of an ideal
• AssociativeExpression -- the class of associative expressions
• atan -- compute the arctangent
• atan(RR) -- compute the arctangent of a number
• atan2(RR,RR) -- compute an angle of a certain triangle
• atEndOfFile(File) -- test for end of file
• backtrace -- whether a backtrace is displayed following an error message
• backupFileRegexp -- a regular expression for recognizing names of backup files
• Bag -- the class of all bags
• Bareiss
• baseFilename -- the base part of a filename or path
• baseName -- the base name of a generator
• baseRings -- store the list of base rings of a ring
• BaseRow -- name for an optional argument
• BasisElementLimit -- stop when this number of basis elements is obtained
• Bayer -- a Strategy option value for saturate
• BeforePrint -- top level method applied before printing results
• benchmark -- accurate timing of execution
• BesselJ -- Bessel function of the first kind
• BesselY -- Bessel function of the second kind
• betti -- display degrees
• betti(..., Weights => ...)
• betti(BettiTally) -- view and set the weights of a betti display
• betti(ChainComplex) -- display of degrees in a chain complex
• betti(GroebnerBasis) -- diagram of the degrees of a groebner basis
• betti(Ideal) -- gives the degrees of generators.
• betti(Matrix) -- display of the degrees of a map
• betti(Module) -- show the degrees of the generators and relations of a module or a coherent sheaf
• BettiTally -- the class of all Betti tallies
• between(Thing,VisibleList) -- insert something between elements of a list
• BinaryOperation -- the class of all binary operation expressions
• Binomial -- name for an optional argument
• binomial -- binomial coefficient
• BLOCKQUOTE -- hypertext BLOCKQUOTE item
• BODY -- hypertext BODY item
• BOLD -- hypertext BOLD item
• Boolean -- the class of Boolean values
• borel(Matrix) -- make a Borel fixed submodule
• Boxes -- name for an optional argument
• BR -- line break
• break -- break from a loop
• cache -- a key under which to store cache tables
• CacheFunction -- the class of cache functions
• CacheTable -- hash tables for caching
• cacheValue -- cache values of functions in their arguments
• capture(String) -- evaluate Macaulay 2 code and capture the output (under development)
• CC -- the class of all complex numbers
• CDATA -- hypertext CDATA element
• ceiling -- ceiling function
• Center -- value for an optional argument
• centerString -- center a string or net
• ChainComplex -- the class of all chain complexes
• chainComplex -- make a chain complex
• ChainComplex ** ChainComplex -- tensor product
• ChainComplex ** ChainComplexMap -- tensor product
• ChainComplex ** GradedModule -- tensor product
• ChainComplex ++ ChainComplex -- direct sum
• ChainComplex ^ ZZ -- access member, cohomological degree
• ChainComplex _ ZZ -- component
• ChainComplex _ ZZ = Thing -- install component of chain complex
• ChainComplex Array -- degree shift
• chainComplex(GradedModule) -- make a chain complex from a graded module
• chainComplex(List) -- make a chain complex
• chainComplex(Matrix) -- make a small chain complex
• chainComplex(Ring) -- make an empty chain complex over a ring
• ChainComplexMap -- the class of all maps between chain complexes
• ChainComplexMap ** ChainComplex -- tensor product
• ChainComplexMap ** ChainComplexMap -- tensor product
• ChainComplexMap ^ ZZ -- iterated composition
• ChainComplexMap _ ZZ -- component map
• ChainComplexMap _ ZZ = Thing -- install component of chain complex map
• ChangeMatrix -- whether to produce the change of basis matrix
• char -- computes the characteristic of the ring or field
• characters -- get characters from a string
• CheckDocumentation -- name for an optional argument
• clean -- Set to zero elements which are approximately zero
• clearAll -- forget everything
• clearOutput -- forget output values
• close -- close a file
• closeIn -- close an input file
• closeOut -- close an output file
• ClosestFit -- name for an optional argument
• CODE -- hypertext CODE item
• code -- display source code
• codim -- compute the codimension
• codim(CoherentSheaf) -- codimension of the support of a coherent sheaf on a projective variety
• codim(Ideal) -- compute the codimension
• codim(Ideal, Generic => ...)
• codim(Module) -- codimension of the support of a module
• codim(MonomialIdeal) -- compute the codimension
• codim(ProjectiveVariety) -- codimension of the projective variety
• codim(QuotientRing) -- compute the codimension
• CodimensionLimit -- stop when this codimension is reached
• coefficient -- coefficient of a monomial
• CoefficientRing -- name for an optional argument
• coefficientRing -- get the coefficient ring
• coefficients -- monomials and their coefficients
• coefficients(..., Monomials => ...) -- specify monomials
• coefficients(..., Variables => ...) -- take coefficients using these variables
• Cofactor
• CoherentSheaf -- the class of all coherent sheaves
• CoherentSheaf (*) -- sum of twists
• CoherentSheaf ** CoherentSheaf -- tensor product of coherent sheaves
• CoherentSheaf ++ CoherentSheaf -- direct sum of coherent sheaves
• CoherentSheaf / CoherentSheaf -- quotient of coherent sheaves
• CoherentSheaf ^ ZZ -- direct sum
• CoherentSheaf ^** ZZ -- tensor power
• CoherentSheaf ZZ -- canonical twist of a coherent sheaf
• cohomology -- general cohomology functor
• cohomology(..., Degree => ...)
• coimage -- coimage of a map
• cokernel -- cokernel of a map of modules, graded modules, or chaincomplexes
• columnAdd -- add a multiple of one column to another
• columnate -- arrange strings in columns
• columnMult -- multiply a column by a ring element
• columnPermute -- permute some columns
• columnSwap -- interchange columns
• Command -- the class of all commands
• commandInterpreter -- the top level command interpreter
• COMMENT -- hypertext COMMENT element
• commonest -- the most common elements of a list or tally
• commonRing -- find smallest containing ring
• comodule -- submodule to quotient module
• compactMatrixForm -- global flag for compact printing
• comparison operators
• CompiledFunction -- the class of all compiled functions
• CompiledFunctionBody -- the class of all compiled function bodies
• CompiledFunctionClosure -- the class of all compiled function closures
• Complement
• complement(Matrix) -- find the minimal generators for cokernel of a matrix (low level form)
• complete
• complete(ChainComplex) -- complete the internal parts
• CompleteIntersection -- name for an optional argument
• ComplexField
• component example
• components -- list the components of a direct sum
• components(ChainComplex) -- list the components of a direct sum
• compositions -- compositions of an integer
• compress -- extract nonzero columns from a matrix
• computing resolutions
• computing syzygies
• cone -- mapping cone of a chain map
• cone(ChainComplexMap) -- mapping cone of a chain map
• conjugate -- complex conjugate
• conjugate(CC) -- complex conjugate
• conjugate(Partition) -- conjugate a partition
• connectionCount -- the number of connections
• Constant
• constructing maps between modules
• content(RingElement) -- the content of a polynomial
• continue -- continue with the next iteration of a loop
• contract -- contract one matrix by another
• contract'(Matrix,Matrix) -- contract a matrix by a matrix, the dual notion
• contract(Matrix,Matrix) -- contract a matrix by a matrix
• conventions for documentation
• copyDirectory(..., UpdateOnly => ...)
• COPYING -- the Macaulay 2 license agreement
• copyright -- a string containing the copyright notice for Macaulay 2
• Core -- the core part of Macaulay 2
• cos -- compute the cosine
• cosh -- compute the hyperbolic cosine
• cot -- cotangent
• cotangentSheaf -- cotangent sheaf of a projective variety
• cotangentSheaf(ProjectiveVariety) -- cotangent sheaf of a projective variety
• cotangentSheaf(ZZ,ProjectiveVariety) -- exterior powers of the cotangent sheaf of a projective variety
• coth -- hyperbolic cotangent
• cover -- get the covering free module
• cover(Matrix) -- get the covering free module
• cover(Module) -- get the covering free module
• coverMap(Module) -- the surjective map from a free module to a module corresponding to the generators
• cpuTime -- seconds of cpu time used since Macaulay2 began
• csc -- cosecant
• csch -- hyperbolic cosecant
• currentFileDirectory -- the directory containing the current input file
• currentFileName -- the current source file
• currentLineNumber -- current line number of the current input file
• currentPackage -- the current package
• currentTime -- get the current time
• Database -- the class of all database files
• Database # String -- get value from database
• Database #? String -- check for value in database
• DD -- hypertext DD element
• dd -- differential in a chain complex
• debug -- open private dictionary of a package
• debugError -- a function to debug
• DebuggingMode -- name for an optional argument
• debuggingMode -- whether to enter the debugger when an error occurs
• debugLevel -- current level debugging
• Decompose -- name for an optional argument
• deepSplice -- remove subsequences
• default -- default member of a family
• defaultPrecision -- default precision of numbers
• Degree
• degree
• degree(Ideal)
• degree(Matrix)
• degree(Module)
• degree(ProjectiveHilbertPolynomial)
• degree(ProjectiveVariety)
• degree(Ring)
• degree(RingElement)
• degree(RingElement,RingElement) -- degree with respect to a variable
• degreeLength -- the number of degrees
• DegreeLimit -- compute up to a certain degree
• DegreeMap
• DegreeOrder -- sort primarily by degree
• degrees -- degrees of generators
• degrees(Matrix) -- degrees of target and source
• degrees(Ring) -- degrees of generators
• degreesMonoid -- get the monoid of degrees
• degreesRing -- the ring of degrees
• degreesRing(Ring) -- the ring of degrees
• degreesRing(ZZ) -- the ring of degrees
• demark -- insert a string between elements of a list of strings
• denominator -- denominator of a fraction
• Dense -- name for an optional argument
• Density -- name for an optional argument
• depth -- depth of a net
• Descending -- specify descending order
• Descent
• describe -- real description
• Description -- name for an optional argument
• determinant -- determinant of a matrix
• determinant(..., Strategy => ...) -- choose between Bareiss and Cofactor algorithms
• Developer's Corner
  • engine
• diagonalMatrix -- make a diagonal matrix
• diagonalMatrix(Matrix) -- make a diagonal matrix from entries of a matrix
• diagonalMatrix(Ring,ZZ,ZZ,List) -- make a diagonal matrix from a list
• Dictionary -- the class of all dictionaries
• dictionary -- determine the dictionary to which a symbol belongs
• dictionaryPath
• diff -- differentiate or take difference
  • diff(RingElement,RingElement) -- differentiation
  • diff(RingElement,Matrix) -- differentiate each entry of a matrix
  • diff(Matrix,RingElement) -- differentiation
  • diff(Matrix,Matrix) -- differentiate a matrix by a matrix
  • diff(ProjectiveHilbertPolynomial)
  • diff(ProjectiveHilbertPolynomial,ZZ)
• diff and contract
• diff'(Matrix,Matrix) -- differentiate a matrix by a matrix, the dual notion
• difference -- difference
• dim -- compute the Krull dimension
• dim(AffineVariety) -- dimension of the affine variety
• dim(Ideal) -- compute the Krull dimension
• dim(Module) -- compute the Krull dimension
• dim(ProjectiveHilbertPolynomial) -- the degree of the Hilbert polynomial
• dim(ProjectiveVariety) -- dimension of the projective variety
• dim(Ring) -- compute the Krull dimension
• directSum -- direct sum of modules or maps
• disassemble -- disassemble pseudocode or a function
• dismiss -- dismiss a package
• DIV -- a hypertext division
• DIV1 -- a single-spaced paragraph separator
• Divide -- the class of all divide expressions
• divideByVariable -- divide all columns by a (power of a) variable
• DivideConquer -- name for an optional argument
• DL -- hypertext DL element
• do -- loop control
• document(..., BaseFunction => ...) -- function for a documented feature
• document(..., Caveat => ...) -- warnings
• document(..., Consequences => ...) -- side-effects of a function
• document(..., Inputs => ...) -- inputs for a function
• document(..., Key => ...) -- key of a documentation node
• document(..., Outputs => ...) -- outputs for a function
• document(..., SeeAlso => ...) -- crossreferences in documentation
• document(..., Subnodes => ...) -- a menu of documentation nodes
• documentation keys
• DocumentTag -- the class of all document tags
• drop(BasicList,List)
• drop(BasicList,ZZ)
• DT -- hypertext DT element
• dual -- dual module or map
• dual(ChainComplex) -- dual
• dual(ChainComplexMap) -- dual of a chain complex
• dual(CoherentSheaf) -- dual coherent sheaf
• dual(Matrix) -- dual of a map
• dual(Module) -- dual module
• dual(MonomialIdeal) -- the Alexander dual of a monomial ideal
• dual(MonomialIdeal,List) -- the Alexander dual
• dual(MonomialIdeal,RingElement) -- the Alexander dual
• eagonNorthcott(Matrix) -- Eagon-Northcott complex of a matrix of linear forms
• echoOff -- turn off echoing
• echoOn -- turn on echoing
• edit -- edit source code
• eigenvalues -- find eigenvalues of a matrix
• eigenvalues(..., Hermitian => ...) -- Hermitian=>true means assume the matrix is symmetric or Hermitian
• eigenvectors -- find eigenvectors of a matrix over RR or CC
• eigenvectors(..., Hermitian => ...) -- Hermitian=>true means assume the matrix is symmetric or Hermitian
• eint -- exponential integral
• elements -- list of elements
• EliminationOrder -- use the natural elmination order in a pushForward1 computation
• else -- condition testing
• EM -- hypertext EM item
• encapDirectory -- encapsulated installation directory
• Encapsulate -- name for an optional argument
• EncapsulateDirectory -- name for an optional argument
• End -- module of endomorphisms
• end -- stop loading a file
• endl -- end an output line
• endPackage(String) -- end a new package
• Engine -- specify whether a ring is handled by the engine
• engineDebugLevel -- current engine debugging level
• EngineRing -- the class of rings handled by the engine
• entries -- lists the entries of a matrix
• entries(Vector) -- lists the entries of a vector
• epkg
• equality and containment of modules
• equality versus strict equality
• Equation -- the class of all equation expressions
• erase -- remove a global symbol
• erf -- error function
• erfc -- complementary error function
• errorCode -- the pseudocode that produced an error
• errorDepth -- set the error printing depth
• euler -- Euler characteristic
• euler(CoherentSheaf) -- Euler characteristic of coherent sheaf
• euler(Ideal) -- Euler characteristic
• euler(Module) -- Euler characteristic
• euler(ProjectiveHilbertPolynomial) -- constant term of the Hilbert polynomial
• euler(ProjectiveVariety) -- topological Euler characteristic of a (smooth) projective variety
• euler(Ring) -- Euler characteristic
• EulerConstant -- the Euler-Mascheroni constant
• eulers -- list the sectional Euler characteristics
• eulers(CoherentSheaf) -- list the sectional Euler characteristics
• eulers(Ideal) -- list the sectional Euler characteristics
• eulers(Ring) -- list the sectional Euler characteristics
• even -- tell whether an integer is even
• EXAMPLE -- hypertext EXAMPLE item
• ExampleItem -- a type of hypertext for holding example inputs awaiting outputs
• examples -- list the examples in documentation
• Exclude -- name for an optional argument
• exp -- exponential function
• expm1 -- exponential minus 1
• exponents -- list the exponents in a polynomial
• Expression -- the class of all expressions
• expression -- convert to an expression
• Ext -- compute an Ext module
• Ext(Module,Module) -- total Ext module
• Ext^ZZ(CoherentSheaf,CoherentSheaf) -- global Ext
• Ext^ZZ(CoherentSheaf,SumOfTwists) -- global Ext
• Ext^ZZ(Matrix,Module) -- map between Ext modules
• Ext^ZZ(Module,Matrix) -- map between Ext modules
• Ext^ZZ(Module,Module) -- Ext module