MatrixFactorizations : Table of Contents

• MatrixFactorizations -- A package for creating and computing objects in the category of ZZ/d-graded factorizations, such as matrix factorizations
• adjoinRoot -- Adjoin a root of unity to the underlying object input
• adjointFactorization -- Constructs matrix factorization of a determinant of a matrix
• arithmetic with ZZ/d-graded factorization maps -- perform arithmetic operations on ZZ/d-graded factorization maps
• Basic invariants and properties -- information about accessing basic features
• branchedToMF -- Converts a maximal Cohen-Macaulay module over a d-fold branched cover into a d-fold factorization
• coimage(ZZdFactorizationMap) -- make the coimage of a map of ZZ/d-graded factorizations
• cokernel(ZZdFactorizationMap) -- make the cokernel of a map of ZZ/d-graded factorizations
• collapseMF -- Converts a d-fold factorization into a (d-1)-fold factorization by composing the kth differential with the (k+1)th differential
• ComplexMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations, obtained by first folding the relevant complex map
• components(ZZdFactorization) -- list the components of a direct sum
• components(ZZdFactorizationMap) -- list the components of a direct sum
• cone(ZZdFactorizationMap) -- make the mapping cone of a morphism of ZZ/d-graded factorizations
• degree(ZZdFactorizationMap) -- get the degree of a map of ZZ/d-graded factorizations
• differential of a ZZ/d-graded factorization -- get the maps between the terms in a ZZ/d-graded factorization
• directSum(ZZdFactorization) -- direct sum of ZZ/d-graded factorizations
• dual(ZZdFactorization) -- make the dual of a ZZ/d-graded factorization
• dual(ZZdFactorizationMap) -- the dual of a map of ZZ/d-graded factorizations
• dual(ZZdFactorizationMap,RingElement) -- the dual of a map of ZZ/d-graded factorizations
• euler(ZZdFactorization) -- Calculates the Euler characteristic of a ZZdFactorization
• eulerMF -- Construct a Koszul matrix factorization with respect to the Jacobian ideal
• Fold -- Convert any complex or complex map into a ZZ/d-graded factorization (or map) for a fixed integer d
• fullCollapse -- Converts a d-fold factorization into a 2-fold factorization
• gradedModule(ZZdFactorization) -- a new ZZ/d-graded factorization in which the differential is zero
• HH ZZdFactorization -- homology of a ZZ/d-graded factorization
• HH ZZdFactorizationMap -- induced map on homology or cohomology
• HH_ZZ ZZdFactorization -- homology or cohomology module of a ZZ/d-graded factorization
• higherHomotopyFactorization -- Constructs the matrix factorization associated with a system of higher homotopies
• Hom(Complex,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization formed by folding the complex, then taking the ZZ/d-graded Hom factorization
• Hom(ComplexMap,ZZdFactorizationMap) -- the map of factorizations between Hom factorizations formed by first folding the relevant complex map
• Hom(ZZdFactorization,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization between two ZZ/d-graded factorizations
• Hom(ZZdFactorization,ZZdFactorization,Symbol) -- the ZZ/d-graded homomorphism factorization between two ZZ/d-graded factorizations
• Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
• Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
• homomorphism'(ZZdFactorizationMap) -- get the element of Hom from a map of factorizations
• homomorphism(ZZ,Matrix,ZZdFactorization) -- get the homomorphism from an element of Hom
• homomorphism(ZZdFactorizationMap) -- get the homomorphism from an element of Hom
• id _ ZZdFactorization -- the identity map of a ZZ/d-graded factorization
• image(ZZdFactorizationMap) -- make the image of a map of ZZ/d-graded factorizations
• inducedMap(ZZdFactorization,ZZdFactorization) -- make the map of ZZ/d-graded factorizations induced at each term by the identity map
• injectiveCover -- Constructs the injective hull of a d-fold matrix factorization
• isCommutative(ZZdFactorizationMap) -- whether a ZZ/d-graded factorization map commutes with the differentials
• isdFactorization -- Checks if the differentials of a factorization compose to a scalar multiple of the identity
• isFactorizationMorphism(ZZdFactorizationMap) -- whether a ZZdFactorizationMap is a morphism of factorizations
• isFree(ZZdFactorization) -- whether a ZZ/d-graded factorization consists of free modules
• isHomogeneous(ZZdFactorizationMap) -- whether a map of ZZ/d-graded factorizations is homogeneous
• isNullHomotopic(ZZdFactorizationMap) -- whether a map of ZZ/d-graded factorizations is null-homotopic
• isNullHomotopyOf(ZZdFactorizationMap,ZZdFactorizationMap) -- whether the first map of ZZ/d-graded factorizations is a null homotopy for the second
• isQuasiIsomorphism(ZZdFactorizationMap) -- whether a map of ZZ/d-graded factorizations is a quasi-isomorphism
• isShortExactSequence(ZZdFactorizationMap,ZZdFactorizationMap) -- whether a pair of ZZ/d-graded factorization maps forms a short exact sequence
• isWellDefined(ZZdFactorization) -- whether a ZZ/d-graded factorization is well-defined
• isWellDefined(ZZdFactorizationMap) -- whether a map of ZZ/d-graded factorizations is well-defined
• isZZdComplex -- Checks if the differentials of a factorization compose to 0
• kernel(ZZdFactorizationMap) -- make the kernel of a map of ZZ/d-graded factorizations
• koszulMF -- Construct a Koszul matrix factorization from an ideal and a polynomial
• linearMF -- Constructs a Koszul d-fold factorization of a homogeneous element of degree d
• Making maps between factorizations -- information about the basic constructors
• Making maps between ZZ/d-graded factorizations -- information about the basic constructors
• Making ZZdFactorizations -- Information about the basic constructors
• map(ZZdFactorization,ZZdFactorization,HashTable) -- make a map of ZZ/d-graded factorizations
• map(ZZdFactorization,ZZdFactorization,ZZ) -- make the zero map or identity between ZZ/d-graded factorizations
• map(ZZdFactorization,ZZdFactorization,ZZdFactorizationMap) -- make a new map of ZZ/d-graded factorizations from an existing one (for inducing maps)
• minimalPresentation(ZZdFactorization) -- minimal presentation of all terms in a ZZ/d-graded factorization
• monomialMF -- Construct a trivial d-fold factorization of a monomial
• mooreMF -- Constructs the matrix factorization of the Moore matrix and its adjoint
• nullHomotopy(ZZdFactorizationMap) -- a map which is a candidate for being a null homotopy
• period -- the period of a ZZ/d-graded factorization or map
• potential -- Outputs the polynomial $f$ such that the dth power of the differentials is equal to $ f \cdot \text{id}$
• Preprogrammed examples and operations -- Information about the basic constructors
• projectiveCover -- Constructs the projective cover of a d-fold matrix factorization
• randomFactorizationMap -- a random map of ZZ/d-graded factorizations
• randomLinearMF -- Generates a linear d-fold factorization of a random degree d polynomial
• randomTailMF -- Construct a matrix factorization from a high syzygy of a random module over a hypersurface
• ring(ZZdFactorization) -- access the ring of a ZZ/d-graded factorization or a factorization map
• RingMap ** ZZdFactorization -- tensor a ZZ/d-graded factorization along a ring map
• RingMap ** ZZdFactorizationMap -- tensor a map of ZZ/d-graded factorizations along a ring map
• RingMap ZZdFactorization -- apply a ring map
• RingMap ZZdFactorizationMap -- apply a ring map to a map of ZZ/d-graded factorizations
• rk1MCM2gen -- Construct every possible rank 1, 2-generated maximal Cohen-Macaulay module over the cubic Fermat hypersurface in 4 variables
• source(ZZdFactorizationMap) -- get the source of a map of ZZ/d-graded factorizations
• suspension -- Constructs the suspension of a d-fold factorization
• tailMF -- Generate a ZZdFactorization from a module over a hypersurface ring
• target(ZZdFactorizationMap) -- get the target of a map of ZZ/d-graded factorizations
• tensorAssociativity(ZZdFactorization,ZZdFactorization,ZZdFactorization) -- make the canonical isomorphism arising from associativity
• tensorCommutativity(ZZdFactorization,ZZdFactorization) -- make the canonical isomorphism arising from commutativity of the tensor product operation
• toBranchedCover -- Converts a d-fold matrix factorization into a maximal Cohen-Macaulay module over a d-fold branched cover of the potential
• trivialMF -- Constructs the trivial matrix factorization of an element f
• unfold -- Converts a factorization into a complex
• WellDefined (missing documentation)
• ZZdFactorization -- the class of all ZZ/d-graded factorizations
• ZZdfactorization -- make a ZZ/d-graded factorization
• ZZdFactorization ** Matrix -- create the tensor product of a ZZ/d-graded factorization and a map of modules
• ZZdFactorization ** ZZdFactorization -- tensor product of ZZ/d-graded factorizations
• ZZdFactorization == ZZdFactorization -- whether two ZZ/d-graded factorizations are equal
• ZZdFactorization _ Array -- the canonical inclusion or projection map of a direct sum
• ZZdFactorization _ ZZ -- access individual object in a ZZ/d-graded factorization
• ZZdFactorization Array -- shift a ZZ/d-graded factorization or map of ZZ/d-graded factorizations
• ZZdfactorization(HashTable) -- make a ZZ/d-graded factorization
• ZZdfactorization(Module,ZZ) -- convert a module/ring/ideal into a ZZ/d-graded factorization for specified period d
• ZZdfactorization(ZZdFactorization) -- make a ZZ/d-graded factorization with shifted base; used for permuting the modules without adding signs
• ZZdFactorizationMap -- the class of all maps between ZZ/d-graded factorizations
• ZZdFactorizationMap * ZZdFactorizationMap -- composition of homomorphisms of ZZ/d-graded factorizations
• ZZdFactorizationMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations
• ZZdFactorizationMap ++ ZZdFactorizationMap -- direct sum of ZZ/d-graded factorization maps
• ZZdFactorizationMap == ZZdFactorizationMap -- whether two ZZ/d-graded factorization maps are equal
• ZZdFactorizationMap ^ Array -- the composition with the canonical inclusion or projection map
• ZZdFactorizationMap ^ ZZ -- the n-fold composition
• ZZdFactorizationMap _ ZZ -- access individual matrices in a ZZ/d-graded factorization map
• ZZdFactorizationMap | ZZdFactorizationMap -- join or concatenate maps horizontally
• ZZdFactorizationMap || ZZdFactorizationMap -- join or concatenate maps vertically