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DGAlgebras :: homologyAlgebra

homologyAlgebra -- Compute the homology algebra of a DGAlgebra.

Synopsis

Description

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

o1 = R

o1 : QuotientRing
i2 : A = koszulComplexDGA(R)

o2 = {Ring => R                              }
      Underlying algebra => R[T , T , T , T ]
                               1   2   3   4
      Differential => {a, b, c, d}
      isHomogeneous => true

o2 : DGAlgebra
i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))

o3 = {1, 4, 6, 4, 1}

o3 : List
i4 : HA = homologyAlgebra(A)
Computing generators in degree 1 :      -- used 0.0014292 seconds
Computing generators in degree 2 :      -- used 0.00997588 seconds
Computing generators in degree 3 :      -- used 0.00948602 seconds
Computing generators in degree 4 :      -- used 0.00875903 seconds
Finding easy relations           :      -- used 0.0230406 seconds
Computing relations in degree 1  :      -- used 0.00213878 seconds
Computing relations in degree 2  :      -- used 0.00211746 seconds
Computing relations in degree 3  :      -- used 0.00210353 seconds
Computing relations in degree 4  :      -- used 0.0020963 seconds
Computing relations in degree 5  :      -- used 0.00191361 seconds

o4 = HA

o4 : PolynomialRing

Note that HA is a graded commutative polynomial ring (i.e. an exterior algebra) since R is a complete intersection.

i5 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4,a^3*b^3*c^3*d^3}

o5 = R

o5 : QuotientRing
i6 : A = koszulComplexDGA(R)

o6 = {Ring => R                              }
      Underlying algebra => R[T , T , T , T ]
                               1   2   3   4
      Differential => {a, b, c, d}
      isHomogeneous => true

o6 : DGAlgebra
i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))

o7 = {1, 5, 10, 10, 4}

o7 : List
i8 : HA = homologyAlgebra(A)
Computing generators in degree 1 :      -- used 0.00143835 seconds
Computing generators in degree 2 :      -- used 0.0109316 seconds
Computing generators in degree 3 :      -- used 0.0116391 seconds
Computing generators in degree 4 :      -- used 0.0119316 seconds
Finding easy relations           :      -- used 0.107806 seconds
Computing relations in degree 1  :      -- used 0.00863478 seconds
Computing relations in degree 2  :      -- used 0.00894605 seconds
Computing relations in degree 3  :      -- used 0.00886002 seconds
Computing relations in degree 4  :      -- used 0.0106262 seconds
Computing relations in degree 5  :      -- used 0.0182346 seconds

o8 = HA

o8 : QuotientRing
i9 : numgens HA

o9 = 19
i10 : HA.cache.cycles

        3     3     3     3     2 3 3 3     2 3 3 3       3 2 3 3     
o10 = {a T , b T , c T , d T , a b c d T , a b c d T T , a b c d T T ,
          1     2     3     4           1           1 2           1 2 
      -----------------------------------------------------------------------
       2 3 3 3       2 3 3 3       2 3 3 3         3 2 3 3       
      a b c d T T , a b c d T T , a b c d T T T , a b c d T T T ,
               1 3           1 4           1 2 3           1 2 3 
      -----------------------------------------------------------------------
       3 3 2 3         2 3 3 3         3 2 3 3         2 3 3 3       
      a b c d T T T , a b c d T T T , a b c d T T T , a b c d T T T ,
               1 2 3           1 2 4           1 2 4           1 3 4 
      -----------------------------------------------------------------------
       2 3 3 3           3 2 3 3           3 3 2 3           3 3 3 2
      a b c d T T T T , a b c d T T T T , a b c d T T T T , a b c d T T T T }
               1 2 3 4           1 2 3 4           1 2 3 4           1 2 3 4

o10 : List
i11 : Q = ZZ/101[x,y,z]

o11 = Q

o11 : PolynomialRing
i12 : I = ideal{y^3,z*x^2,y*(z^2+y*x),z^3+2*x*y*z,x*(z^2+y*x),z*y^2,x^3,z*(z^2+2*x*y)}

              3   2      2      2            3   2       2   2    3          
o12 = ideal (y , x z, x*y  + y*z , 2x*y*z + z , x y + x*z , y z, x , 2x*y*z +
      -----------------------------------------------------------------------
       3
      z )

o12 : Ideal of Q
i13 : R = Q/I

o13 = R

o13 : QuotientRing
i14 : A = koszulComplexDGA(R)

o14 = {Ring => R                          }
       Underlying algebra => R[T , T , T ]
                                1   2   3
       Differential => {x, y, z}
       isHomogeneous => true

o14 : DGAlgebra
i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))

o15 = {1, 7, 7, 1}

o15 : List
i16 : HA = homologyAlgebra(A)
Computing generators in degree 1 :      -- used 0.0013749 seconds
Computing generators in degree 2 :      -- used 0.0114449 seconds
Computing generators in degree 3 :      -- used 0.0114284 seconds
Finding easy relations           :      -- used 0.0604133 seconds
Computing relations in degree 1  :      -- used 0.00723283 seconds
Computing relations in degree 2  :      -- used 0.0193523 seconds
Computing relations in degree 3  :      -- used 0.00681008 seconds
Computing relations in degree 4  :      -- used 0.00682101 seconds

o16 = HA

o16 : QuotientRing

One can check that HA has Poincare duality since R is Gorenstein.

If your DGAlgebra has generators in even degrees, then one must specify the options GenDegreeLimit and RelDegreeLimit.

i17 : R = ZZ/101[a,b,c,d]

o17 = R

o17 : PolynomialRing
i18 : S = R/ideal{a^4,b^4,c^4,d^4}

o18 = S

o18 : QuotientRing
i19 : A = acyclicClosure(R,EndDegree=>3)

o19 = {Ring => R                              }
       Underlying algebra => R[T , T , T , T ]
                                1   2   3   4
       Differential => {a, b, c, d}
       isHomogeneous => true

o19 : DGAlgebra
i20 : B = A ** S

o20 = {Ring => S                              }
       Underlying algebra => S[T , T , T , T ]
                                1   2   3   4
       Differential => {a, b, c, d}
       isHomogeneous => true

o20 : DGAlgebra
i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
Computing generators in degree 1 :      -- used 0.00699921 seconds
Computing generators in degree 2 :      -- used 0.0161563 seconds
Computing generators in degree 3 :      -- used 0.014772 seconds
Computing generators in degree 4 :      -- used 0.00876924 seconds
Computing generators in degree 5 :      -- used 0.00110159 seconds
Computing generators in degree 6 :      -- used 0.00109086 seconds
Computing generators in degree 7 :      -- used 0.00109914 seconds
Finding easy relations           :      -- used 0.016577 seconds
Computing relations in degree 1  :      -- used 0.0021343 seconds
Computing relations in degree 2  :      -- used 0.0021726 seconds
Computing relations in degree 3  :      -- used 0.00215339 seconds
Computing relations in degree 4  :      -- used 0.00211286 seconds
Computing relations in degree 5  :      -- used 0.00950257 seconds
Computing relations in degree 6  :      -- used 0.00186482 seconds
Computing relations in degree 7  :      -- used 0.00180963 seconds
Computing relations in degree 8  :      -- used 0.00185437 seconds
Computing relations in degree 9  :      -- used 0.00180308 seconds
Computing relations in degree 10  :      -- used 0.00178714 seconds
Computing relations in degree 11  :      -- used 0.00180104 seconds
Computing relations in degree 12  :      -- used 0.00179108 seconds
Computing relations in degree 13  :      -- used 0.00179247 seconds
Computing relations in degree 14  :      -- used 0.00193477 seconds

o21 = HB

o21 : PolynomialRing

Ways to use homologyAlgebra :