killCycles -- Adjoin variables to kill non-bounding cycles in the lowest positive degree

Usage:

B = killCycles A
Inputs:
- A, an instance of the type DGAlgebra,
Optional inputs:
- EndDegree => ..., default value -1, Last homological degree processed by killCycles
- StartDegree => ..., default value 1, First homological degree from which killCycles begins searching for cycles
- Variable => ..., default value null, Option to name the DG algebra generators adjoined by killCycles
Outputs:
- B, an instance of the type DGAlgebra, a new DG algebra with extra generators adjoined to make every cycle in the first positive degree where HH_i A is nonzero into a boundary

Description

This is the basic building block of Tate's acyclic closure construction. Given a DG algebra A, killCycles locates the smallest positive homological degree i in which HH_i A is nonzero, picks a generating set of cycles, and adjoins new DG algebra generators in degree i+1 mapping to those cycles. The result is a DG algebra whose homology agrees with A below degree i and is killed in degree i. Iterating this procedure produces the acyclic closure.

i1 : R = ZZ/101[a,b,c,d]/ideal{a^3, b^3, c^3 - d^4}

o1 = R

o1 : QuotientRing

i2 : A = koszulComplexDGA R

o2 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,4
      Differential => {a, b, c, d}

o2 : DGAlgebra

i3 : A.diff

o3 = map (R[T   ..T   ], R[T   ..T   ], {a, b, c, d, a, b, c, d})
             1,1   1,4      1,1   1,4

o3 : RingMap R[T   ..T   ] <-- R[T   ..T   ]
                1,1   1,4         1,1   1,4

i4 : B = killCycles A

o4 = {Ring => R                                                      }
      Underlying algebra => R[T   ..T   , T   ..T   ]
                               1,1   1,4   2,1   2,3
                                    2       2         2        3
      Differential => {a, b, c, d, a T   , b T   , - c T    + d T   }
                                      1,1     1,2       1,3      1,4

o4 : DGAlgebra

i5 : B.diff

                                                                              2       2         2        3
o5 = map (R[T   ..T   , T   ..T   ], R[T   ..T   , T   ..T   ], {a, b, c, d, a T   , b T   , - c T    + d T   , a, b, c, d})
             1,1   1,4   2,1   2,3      1,1   1,4   2,1   2,3                   1,1     1,2       1,3      1,4

o5 : RingMap R[T   ..T   , T   ..T   ] <-- R[T   ..T   , T   ..T   ]
                1,1   1,4   2,1   2,3         1,1   1,4   2,1   2,3

The new generator T_(2,1) in B has differential equal to a chosen nonzero cycle representative in homological degree 1 of A.

Ways to use killCycles:

killCycles(DGAlgebra)
killCycles(DGModule) -- Adjoin free generators to kill the lowest nonvanishing homology of a DG module

For the programmer

The object killCycles is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:8001:0.

killCycles -- Adjoin variables to kill non-bounding cycles in the lowest positive degree

Description

See also

Ways to use killCycles:

For the programmer