acyclicClosure(...,EndDegree=>...) -- Option to specify the degree to stop computing the acyclic closure

Usage:

A = acyclicClosure(B,EndDegree=>n)
Inputs:
- B, an instance of the type DGAlgebra,
- n, an integer,

Description

All generators of the acyclic closure will be found up to homological degree n. Note that B can be an ordinary ring, a factor ring of a polynomial ring, treated as a DGAlgebra generated in homological degree 0, as in the following example:

The default value of EndDegree is 3.

i1 : R = ZZ/101[x,y,z,w]/(ideal"x3,y3,z3,x2yz")

o1 = R

o1 : QuotientRing

i2 : acyclicClosure R

o2 = {Ring => R                                                                                                                                                                 }
      Underlying algebra => R[T   ..T   , T   ..T   , T   ..T   ]
                               1,1   2,4   3,1   3,3   4,1   4,6
                                    2       2       2                             2       2       2       2        2                                             2        2
      Differential => {x, y, z, w, x T   , y T   , z T   , x*y*z*T   , y*z*T   , z T   , y T   , z T   , y T   , -x T   T   , -x*y*z*T   T   , -x*y*z*T   T   , y T    - z T   }
                                      1,1     1,2     1,3         1,1       2,1     2,4     2,4     3,1     3,1      1,1 2,4          1,1 2,3          1,1 2,2     3,2      3,3

o2 : DGAlgebra

i3 : acyclicClosure(R,EndDegree => 3)

o3 = {Ring => R                                                                                                                                                                 }
      Underlying algebra => R[T   ..T   , T   ..T   , T   ..T   ]
                               1,1   2,4   3,1   3,3   4,1   4,6
                                    2       2       2                             2       2       2       2        2                                             2        2
      Differential => {x, y, z, w, x T   , y T   , z T   , x*y*z*T   , y*z*T   , z T   , y T   , z T   , y T   , -x T   T   , -x*y*z*T   T   , -x*y*z*T   T   , y T    - z T   }
                                      1,1     1,2     1,3         1,1       2,1     2,4     2,4     3,1     3,1      1,1 2,4          1,1 2,3          1,1 2,2     3,2      3,3

o3 : DGAlgebra

Functions with optional argument named EndDegree:

acyclicClosure(...,EndDegree=>...) -- Option to specify the degree to stop computing the acyclic closure
isAcyclic(...,EndDegree=>...) -- Option to specify the degree to finish checking acyclicity
killCycles(...,EndDegree=>...) -- Last homological degree processed by killCycles
liftToDGMap(...,EndDegree=>...) -- see liftToDGMap -- Lift a ring homomorphism in degree zero to a DG algebra morphism
liftToDGModuleMap(...,EndDegree=>...) -- see liftToDGModuleMap -- Lift an image of hom-degree-0 generators to a full DGModuleMap
minimalModel(...,EndDegree=>...) -- see minimalModel -- Build the minimal DG algebra resolution of a quotient ring
minimalSemifreeResolution(...,EndDegree=>...) -- Option to specify the last degree at which minimalSemifreeResolution kills cycles
semifreeResolution(...,EndDegree=>...) -- Option to specify the last degree at which semifreeResolution kills cycles
toComplexMap(...,EndDegree=>...) -- see toComplexMap -- Construct the ComplexMap associated to a DGAlgebraMap

Further information

Default value: 3
Function: acyclicClosure -- Tate's construction of a semifree acyclic DG algebra extension
Option key: EndDegree -- Option to specify the last homological degree processed by an iterative DG construction

The source of this document is in DGAlgebras/doc.m2:9470:0.