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DGAlgebras > The Koszul complex as a DG Algebra

The Koszul complex as a DG Algebra -- an example

The Koszul complex on a sequence of elements $f_1,\dots,f_r$ is a complex of R-modules whose underlying graded R-module is the exterior algebra on R^r generated in homological degree one. This algebra structure also respects the boundary map of the complex in the sense that it satisfies the Liebniz rule. That is, $d(ab) = d(a)b + (-1)^{deg a}ad(b)$. When one speaks of 'the' Koszul complex of a ring, one means the Koszul complex on a minimal set of generators of the maximal ideal of R.

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

o1 = R

o1 : QuotientRing
i2 : KR = koszulComplexDGA R

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

o2 : DGAlgebra

One can specify the name of the variable to easily handle multiple Koszul complexes at once.

i3 : S = ZZ/101[x,y,z]/ideal{x^3,y^3,z^3,x^2*y^2,y^2*z^2}

o3 = S

o3 : QuotientRing
i4 : KS = koszulComplexDGA(S,Variable=>"U")

o4 = {Ring => S                      }
      Underlying algebra => S[U ..U ]
                               1   3
      Differential => {x, y, z}

o4 : DGAlgebra

To obtain the chain complex associated to the Koszul complex, one may use toComplex. One can also obtain this complex directly without using the DGAlgebras package by using the command koszul.

i5 : cxKR = toComplex KR

      1      4      6      4      1
o5 = R  <-- R  <-- R  <-- R  <-- R
                                  
     0      1      2      3      4

o5 : ChainComplex
i6 : prune HH cxKR

o6 = 0 : cokernel | d c b a |                                            

     1 : cokernel {3} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 |
                  {3} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 |
                  {3} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 |
                  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |                

     2 : cokernel {6} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                  {6} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                  {6} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 0 0 0 0 |
                  {6} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 |
                  {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 |
                  {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |

     3 : cokernel {9} | d c b a 0 0 0 0 0 0 0 0 0 0 0 0 |
                  {9} | 0 0 0 0 d c b a 0 0 0 0 0 0 0 0 |
                  {9} | 0 0 0 0 0 0 0 0 d c b a 0 0 0 0 |
                  {9} | 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |                

     4 : cokernel {12} | d c b a |                                       

o6 : GradedModule

Since the Koszul complex is a DG algebra, its homology is itself an algebra. One can obtain this algebra using the command homology, homologyAlgebra, or HH (all commands work). This algebra structure can detect whether or not the ring is a complete intersection or Gorenstein.

i7 : HKR = HH KR
Finding easy relations           :      -- used 0.020997 seconds

o7 = HKR

o7 : PolynomialRing, 4 skew commutative variables
i8 : ideal HKR

o8 = ideal ()

o8 : Ideal of HKR
i9 : R' = ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^3,d^3,a*c,a*d,b*c,b*d,a^2*b^2-c^2*d^2}

o9 = R'

o9 : QuotientRing
i10 : HKR' = HH koszulComplexDGA R'
Finding easy relations           :      -- used 0.796604 seconds

o10 = HKR'

o10 : QuotientRing
i11 : numgens HKR'

o11 = 34
i12 : ann ideal gens HKR'

o12 = ideal(X X  )
             4 34

o12 : Ideal of HKR'

Note that since the socle of HKR' is one dimensional, HKR' has Poincare duality, and hence R' is Gorenstein.

One can also consider the Koszul complex of an ideal, or a sequence of elements.

i13 : Q = ambient R

o13 = Q

o13 : PolynomialRing
i14 : I = ideal {a^3,b^3,c^3,d^3}

              3   3   3   3
o14 = ideal (a , b , c , d )

o14 : Ideal of Q
i15 : KI = koszulComplexDGA I

o15 = {Ring => Q                       }
       Underlying algebra => Q[T ..T ]
                                1   4
                         3   3   3   3
       Differential => {a , b , c , d }

o15 : DGAlgebra
i16 : HKI = HH KI

o16 = HKI

o16 : QuotientRing
i17 : describe HKI

              Q
o17 = ----------------
        3   3   3   3
      (d , c , b , a )
i18 : use Q

o18 = Q

o18 : PolynomialRing
i19 : I' = I + ideal{a^2*b^2*c^2*d^2}

              3   3   3   3   2 2 2 2
o19 = ideal (a , b , c , d , a b c d )

o19 : Ideal of Q
i20 : KI' = koszulComplexDGA I'

o20 = {Ring => Q                                 }
       Underlying algebra => Q[T ..T ]
                                1   5
                         3   3   3   3   2 2 2 2
       Differential => {a , b , c , d , a b c d }

o20 : DGAlgebra
i21 : HKI' = HH KI'

o21 = HKI'

o21 : QuotientRing
i22 : describe HKI'

                                                                       Q
                                                          --------------------------[X ..X ]
                                                            3   3   3   3   2 2 2 2   1   4
                                                          (d , c , b , a , a b c d )
o22 = ------------------------------------------------------------------------------------------------------------------------------------------
        2                  2                               2                                            2
      (a X , a*X  - b*X , b X , b*X  - c*X , a*X  - c*X , c X , c*X  - d*X , b*X  - d*X , a*X  - d*X , d X , X X , X X , X X , X X , X X , X X )
          4     3      4     3     2      3     2      4     2     1      2     1      3     1      4     1   3 4   2 4   1 4   2 3   1 3   1 2
i23 : HKI'.cache.cycles

        2 2 2            2 2 2            2 2 2            2 2 2
o23 = {a b c T  - d*T , a b d T  - c*T , a c d T  - b*T , b c d T  - a*T }
              4      5         3      5         2      5         1      5

o23 : List

Note that since I is a Q-regular sequence, the Koszul complex is acyclic, and that both homology algebras are algebras over the zeroth homology of the Koszul complex.