ZZdFactorization -- the class of all ZZ/d-graded factorizations

Description

A ZZ/d-graded factorization is a sequence of objects $C_i$, connected by maps $dd^C_i : C_i \rightarrow C_{i-1}$ such that the composition of any d consecutive maps is equal to a fixed scalar multiple of the identity.

In practice, a ZZdFactorization may be visualized as a sequence of R-module maps:

$$F_0 \leftarrow F_1 \leftarrow \cdots \leftarrow F_{d-1}$$

with the caveat that $d^F_0 : F_0 \to F_{d-1}$, since one should count degree modulo d. Any 2-periodic complex may be reinterpretted as a ZZ/2-graded factorization of 0, and likewise a matrix factorization of a ring element f is equivalently a ZZ/2-graded factorization of f. Because of their similarity with complexes, much of the functionality and syntax for this package closely resembles the "Complexes" package, with some key differences that we will highlight below.

i1 : Q = QQ[x_1..x_3];

i2 : F1 = ZZdfactorization {x_1,x_2}

      1      1      1
o2 = Q  <-- Q  <-- Q
                    
     0      1      0

o2 : ZZdFactorization

i3 : F1.dd

          1               1
o3 = 1 : Q  <----------- Q  : 0
               | x_2 |

          1               1
     0 : Q  <----------- Q  : 1
               | x_1 |

o3 : ZZdFactorizationMap

i4 : F2 = ZZdfactorization {x_1,x_2,x_3}

      1      1      1      1
o4 = Q  <-- Q  <-- Q  <-- Q
                           
     0      1      2      0

o4 : ZZdFactorization

i5 : F2.dd

          1               1
o5 = 2 : Q  <----------- Q  : 0
               | x_3 |

          1               1
     0 : Q  <----------- Q  : 1
               | x_1 |

          1               1
     1 : Q  <----------- Q  : 2
               | x_2 |

o5 : ZZdFactorizationMap

Notice that in the above, both the modules and the differentials are displayed modulo the period of the complex. Moreover, if the user tries to access the data of the modules or differentials, this input is also taken modulo the period:

i6 : F1_0

      1
o6 = Q

o6 : Q-module, free

i7 : F1_2

      1
o7 = Q

o7 : Q-module, free

i8 : F1.dd_123

o8 = | x_1 |

             1      1
o8 : Matrix Q  <-- Q

The package is implemented to not actually check for well-definedness of the factorization (that is, it does not check if all of the differentials actually compose to a scalar multiple of the identity). The user can check this by using the isWellDefined and isdFactorization commands:

i9 : isdFactorization F2

o9 = (true, x x x )
             1 2 3

o9 : Sequence

Much of the syntax and functionality of the types ZZdFactorization and ZZdFactorizationMap are based on current functionalities for the analogous objects in the Complexes package, so there should be essentially no learning curve for users already familiar with working with chain complexes.

Functions and methods returning a ZZ/d graded factorization:

adjointFactorization(Matrix) -- see adjointFactorization -- Constructs matrix factorization of a determinant of a matrix
basis(List,ZZdFactorization) (missing documentation)
basis(ZZ,ZZdFactorization) (missing documentation)
branchedToMF(Matrix) -- see branchedToMF -- Converts a maximal Cohen-Macaulay module over a d-fold branched cover into a d-fold factorization
branchedToMF(Matrix,Ring) -- see branchedToMF -- Converts a maximal Cohen-Macaulay module over a d-fold branched cover into a d-fold factorization
branchedToMF(Module) -- see branchedToMF -- Converts a maximal Cohen-Macaulay module over a d-fold branched cover into a d-fold factorization
branchedToMF(Module,Ring) -- see 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
cone(ZZdFactorizationMap) -- make the mapping cone of a morphism of ZZ/d-graded factorizations
ZZdFactorization ++ ZZdFactorization -- see directSum(ZZdFactorization) -- direct sum of ZZ/d-graded factorizations
dual(ZZdFactorization) -- make the dual of a ZZ/d-graded factorization
dual(ZZdFactorization,RingElement) -- see dual(ZZdFactorization) -- make the dual of a ZZ/d-graded factorization
dual(ZZdFactorization,Symbol) -- see dual(ZZdFactorization) -- make the dual of a ZZ/d-graded factorization
eulerMF(RingElement) -- see eulerMF -- Construct a Koszul matrix factorization with respect to the Jacobian ideal
fullCollapse(ZZdFactorization,ZZ,ZZ) -- see 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
Hom(Complex,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization formed by folding the complex, then taking the ZZ/d-graded Hom factorization
Hom(Complex,ZZdFactorization,RingElement) -- see Hom(Complex,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization formed by folding the complex, then taking the ZZ/d-graded Hom factorization
Hom(Complex,ZZdFactorization,Symbol) -- see Hom(Complex,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization formed by folding the complex, then taking the ZZ/d-graded Hom factorization
Hom(ZZdFactorization,Complex) -- see Hom(Complex,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization formed by folding the complex, then taking the ZZ/d-graded Hom factorization
Hom(ZZdFactorization,Complex,RingElement) -- see Hom(Complex,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization formed by folding the complex, then taking the ZZ/d-graded Hom factorization
Hom(ZZdFactorization,Complex,Symbol) -- see Hom(Complex,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization formed by folding the complex, then taking the ZZ/d-graded Hom factorization
Hom(Module,ZZdFactorization) -- see Hom(ZZdFactorization,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization between two ZZ/d-graded factorizations
Hom(ZZdFactorization,Module) -- see Hom(ZZdFactorization,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization between two ZZ/d-graded factorizations
Hom(ZZdFactorization,Ring) -- see Hom(ZZdFactorization,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization between two ZZ/d-graded factorizations
Hom(ZZdFactorization,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization between two ZZ/d-graded factorizations
Hom(ZZdFactorization,Module,RingElement) -- see Hom(ZZdFactorization,ZZdFactorization,Symbol) -- the ZZ/d-graded homomorphism factorization between two ZZ/d-graded factorizations
Hom(ZZdFactorization,Module,Symbol) -- see Hom(ZZdFactorization,ZZdFactorization,Symbol) -- the ZZ/d-graded homomorphism factorization between two ZZ/d-graded factorizations
Hom(ZZdFactorization,ZZdFactorization,RingElement) -- see Hom(ZZdFactorization,ZZdFactorization,Symbol) -- 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
image(ZZdFactorizationMap) -- make the image of a map of ZZ/d-graded factorizations
kernel(ZZdFactorizationMap) -- make the kernel of a map of ZZ/d-graded factorizations
linearMF(RingElement) -- see linearMF -- Constructs a Koszul d-fold factorization of a homogeneous element of degree d
linearMF(RingElement,RingElement) -- see linearMF -- Constructs a Koszul d-fold factorization of a homogeneous element of degree d
linearMF(RingElement,Symbol) -- see linearMF -- Constructs a Koszul d-fold factorization of a homogeneous element of degree d
minimalPresentation(ZZdFactorization) -- minimal presentation of all terms in a ZZ/d-graded factorization
prune(ZZdFactorization) -- see minimalPresentation(ZZdFactorization) -- minimal presentation of all terms in a ZZ/d-graded factorization
randomTailMF(RingElement) -- see randomTailMF -- Construct a matrix factorization from a high syzygy of a random module over a hypersurface
randomTailMF(RingElement,ZZ,ZZ) -- see randomTailMF -- Construct a matrix factorization from a high syzygy of a random module over a hypersurface
randomTailMF(RingElement,ZZ,ZZ,ZZ) -- see randomTailMF -- Construct a matrix factorization from a high syzygy of a random module over a hypersurface
ring(ZZdFactorizationMap) -- see ring(ZZdFactorization) -- access the ring of a ZZ/d-graded factorization or a factorization map
Ring ** ZZdFactorization -- see RingMap ** ZZdFactorization -- tensor a ZZ/d-graded factorization along a ring map
RingMap ** ZZdFactorization -- tensor a ZZ/d-graded factorization along a ring map
tensor(RingMap,ZZdFactorization) -- see RingMap ** ZZdFactorization -- tensor a ZZ/d-graded factorization along a ring map
tensor(ZZdFactorization,RingMap) -- see RingMap ** ZZdFactorization -- tensor a ZZ/d-graded factorization along a ring map
ZZdFactorization ** Ring -- see RingMap ** ZZdFactorization -- tensor a ZZ/d-graded factorization along a ring map
ZZdFactorization ** RingMap -- see RingMap ** ZZdFactorization -- tensor a ZZ/d-graded factorization along a ring map
RingMap ZZdFactorization -- apply a ring map
rk1MCM2gen(List) -- see rk1MCM2gen -- Construct every possible rank 1, 2-generated maximal Cohen-Macaulay module over the cubic Fermat hypersurface in 4 variables
rk1MCM2gen(List,ZZ) -- see 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(ZZ,ZZdFactorization) -- see suspension -- Constructs the suspension of a d-fold factorization
suspension(ZZdFactorization) -- see suspension -- Constructs the suspension of a d-fold factorization
tailMF(Ideal) -- see tailMF -- Generate a ZZdFactorization from a module over a hypersurface ring
tailMF(Module) -- see tailMF -- Generate a ZZdFactorization from a module over a hypersurface ring
target(ZZdFactorizationMap) -- get the target of a map of ZZ/d-graded factorizations
tensor(ZZdFactorization,ZZdFactorization,RingElement) (missing documentation)
tensor(ZZdFactorization,ZZdFactorization,Symbol) (missing documentation)
trivialMF(Module,RingElement) -- see trivialMF -- Constructs the trivial matrix factorization of an element f
trivialMF(ZZ,Module,RingElement) -- see trivialMF -- Constructs the trivial matrix factorization of an element f
trivialMF(ZZ,ZZ,Module,RingElement) -- see trivialMF -- Constructs the trivial matrix factorization of an element f
trivialMF(ZZ,ZZ,Module,ZZ) -- see trivialMF -- Constructs the trivial matrix factorization of an element f
truncate(List,ZZdFactorization) (missing documentation)
truncate(ZZ,ZZdFactorization) (missing documentation)
ZZdfactorization(List) -- see ZZdfactorization -- make a ZZ/d-graded factorization
Complex ** ZZdFactorization -- see ZZdFactorization ** ZZdFactorization -- tensor product of ZZ/d-graded factorizations
Module ** ZZdFactorization -- see ZZdFactorization ** ZZdFactorization -- tensor product of ZZ/d-graded factorizations
tensor(ZZdFactorization,ZZdFactorization) -- see ZZdFactorization ** ZZdFactorization -- tensor product of ZZ/d-graded factorizations
ZZdFactorization ** Complex -- see ZZdFactorization ** ZZdFactorization -- tensor product of ZZ/d-graded factorizations
ZZdFactorization ** Module -- see ZZdFactorization ** ZZdFactorization -- tensor product of ZZ/d-graded factorizations
ZZdFactorization ** ZZdFactorization -- tensor product of ZZ/d-graded factorizations
ZZdFactorization Array -- shift a ZZ/d-graded factorization or map of ZZ/d-graded factorizations
ZZdfactorization(HashTable) -- make a ZZ/d-graded factorization
ZZdfactorization(Ideal,ZZ) -- see ZZdfactorization(Module,ZZ) -- convert a module/ring/ideal into a ZZ/d-graded factorization for specified period d
ZZdfactorization(Module,ZZ) -- convert a module/ring/ideal into a ZZ/d-graded factorization for specified period d
ZZdfactorization(Ring,ZZ) -- see 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

Methods that use a ZZ/d graded factorization:

adjoinRoot(ZZdFactorization,RingElement) -- see adjoinRoot -- Adjoin a root of unity to the underlying object input
adjoinRoot(ZZdFactorization,Symbol) -- see adjoinRoot -- Adjoin a root of unity to the underlying object input
collapseMF(ZZdFactorization,ZZ) -- see collapseMF -- Converts a d-fold factorization into a (d-1)-fold factorization by composing the kth differential with the (k+1)th differential
components(ZZdFactorization) -- list the components of a direct sum
Symbol ^ ZZdFactorization -- see 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
euler(ZZdFactorization) -- Calculates the Euler characteristic of a ZZdFactorization
extend(ZZdFactorization,ZZdFactorization,Matrix) (missing documentation)
extend(ZZdFactorization,ZZdFactorization,Matrix,Sequence) (missing documentation)
HH_ZZ ZZdFactorization -- homology or cohomology module of a ZZ/d-graded factorization
HH_ZZ(ZZ,ZZdFactorization) -- see HH_ZZ ZZdFactorization -- homology or cohomology module of a ZZ/d-graded factorization
Hom(Ring,ZZdFactorization) -- see Hom(ZZdFactorization,ZZdFactorization) -- the ZZ/d-graded homomorphism factorization between two ZZ/d-graded factorizations
Hom(Matrix,ZZdFactorization) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(ZZdFactorization,Matrix) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(ZZdFactorization,ZZdFactorizationMap) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,ZZdFactorization) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap) -- the map of factorizations between Hom complexes
Hom(ZZdFactorization,Matrix,RingElement) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorization,Matrix,Symbol) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorization,ZZdFactorizationMap,RingElement) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorization,ZZdFactorizationMap,Symbol) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,ZZdFactorization,RingElement) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
Hom(ZZdFactorizationMap,ZZdFactorization,Symbol) -- see Hom(ZZdFactorizationMap,ZZdFactorizationMap,Symbol) -- the map of factorizations between Hom complexes
homomorphism(ZZ,Matrix,ZZdFactorization) -- get the homomorphism from an element of Hom
inducedMap(ZZdFactorization,ZZdFactorization) -- make the map of ZZ/d-graded factorizations induced at each term by the identity map
inducedMap(ZZdFactorization,ZZdFactorization,ZZdFactorizationMap) (missing documentation)
injectiveCover(ZZdFactorization) -- see injectiveCover -- Constructs the injective hull of a d-fold matrix factorization
isdFactorization(ZZdFactorization) -- see isdFactorization -- Checks if the differentials of a factorization compose to a scalar multiple of the identity
isFree(ZZdFactorization) -- whether a ZZ/d-graded factorization consists of free modules
isWellDefined(ZZdFactorization) -- whether a ZZ/d-graded factorization is well-defined
isZZdComplex(ZZdFactorization) -- see isZZdComplex -- Checks if the differentials of a factorization compose to 0
map(ZZdFactorization,ZZdFactorization,Function) (missing documentation)
map(ZZdFactorization,ZZdFactorization,HashTable) -- make a map of ZZ/d-graded factorizations
map(ZZdFactorization,ZZdFactorization,List) (missing documentation)
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)
net(ZZdFactorization) (missing documentation)
period(ZZdFactorization) -- see period -- the period of a ZZ/d-graded factorization or map
potential(ZZdFactorization) -- see potential -- Outputs the polynomial $f$ such that the dth power of the differentials is equal to $ f \cdot \text{id}$
projectiveCover(ZZdFactorization) -- see projectiveCover -- Constructs the projective cover of a d-fold matrix factorization
randomFactorizationMap(ZZdFactorization,ZZdFactorization) -- see randomFactorizationMap -- a random map of ZZ/d-graded factorizations
ring(ZZdFactorization) -- access the ring of a ZZ/d-graded factorization or a factorization map
support(ZZdFactorization) (missing documentation)
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(ZZdFactorization,RingElement) -- see toBranchedCover -- Converts a d-fold matrix factorization into a maximal Cohen-Macaulay module over a d-fold branched cover of the potential
toBranchedCover(ZZdFactorization,Symbol) -- see toBranchedCover -- Converts a d-fold matrix factorization into a maximal Cohen-Macaulay module over a d-fold branched cover of the potential
zeroOutDegrees(ZZdFactorization) -- see toBranchedCover -- Converts a d-fold matrix factorization into a maximal Cohen-Macaulay module over a d-fold branched cover of the potential
unfold(ZZdFactorization,List) -- see unfold -- Converts a factorization into a complex
unfold(ZZdFactorization,ZZ) -- see unfold -- Converts a factorization into a complex
Matrix ** ZZdFactorization -- see ZZdFactorization ** Matrix -- create the tensor product of a ZZ/d-graded factorization and a map of modules
ZZdFactorization ** Matrix -- create the tensor product of a ZZ/d-graded factorization and a map of modules
ZZ == ZZdFactorization -- see ZZdFactorization == ZZdFactorization -- whether two ZZ/d-graded factorizations are equal
ZZdFactorization == ZZ -- see ZZdFactorization == ZZdFactorization -- whether two ZZ/d-graded factorizations are equal
ZZdFactorization == ZZdFactorization -- whether two ZZ/d-graded factorizations are equal
ZZdFactorization ^ Array -- see ZZdFactorization _ Array -- the canonical inclusion or projection map of a direct sum
ZZdFactorization _ Array -- the canonical inclusion or projection map of a direct sum
ZZdFactorization ^ ZZ -- see ZZdFactorization _ ZZ -- access individual object in a ZZ/d-graded factorization
ZZdFactorization _ ZZ -- access individual object in a ZZ/d-graded factorization
ZZdFactorization ** ZZdFactorizationMap -- see ZZdFactorizationMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations
ZZdFactorizationMap ** ZZdFactorization -- see ZZdFactorizationMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations

For the programmer

The object ZZdFactorization is a type, with ancestor classes MutableHashTable < HashTable < Thing.

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:400:0.

ZZdFactorization -- the class of all ZZ/d-graded factorizations

Description

See also

Functions and methods returning a ZZ/d graded factorization:

Methods that use a ZZ/d graded factorization:

For the programmer