koszulComplex(Matrix) -- makes the Koszul complex

Synopsis

Function: koszulComplex
Usage:

K = koszulComplex f
Inputs:
- f, a matrix, having one row, or a a list of ring elements
Optional inputs:
- Concentration => a sequence, default value null, a pair (lo, hi) which limits the non-zero terms in the output
Outputs:
- a complex, the Koszul complex (or a subcomplex)

Description

Let $R$ be a commutative ring and let $E$ be a free $R$-module of finite rank $r$. Given a linear map $f \colon E \to R$, the Koszul complex associated to $f$ is the chain complex of $R$-modules

$\phantom{WWWW} 0 \leftarrow R \leftarrow \bigwedge^1 E \leftarrow \bigwedge^2 E \leftarrow \dotsb \leftarrow \bigwedge^r E \leftarrow 0, $

where the differential is given by

$\phantom{WWWW} dd_k(e_1 \wedge e_2 \wedge \dotsb \wedge e_k) = \sum_{i=1}^k (-1)^{i+1} f(e_i) \, e_1 \wedge e_2 \wedge \dotsb \wedge \widehat{e_i} \wedge \dotsb \wedge e_k, $

and the superscript hat means the term is omitted. For this method, the linear map $f$ is given as either a matrix with one row, or a list of ring elements.

i1 : S = QQ[a..d]

o1 = S

o1 : PolynomialRing

i2 : koszulComplex {a}

      1      1
o2 = S  <-- S
             
     0      1

o2 : Complex

i3 : C = koszulComplex {a^2+b^2,c^3}

      1      2      1
o3 = S  <-- S  <-- S
                    
     0      1      2

o3 : Complex

i4 : dd^C

          1                    2
o4 = 0 : S  <---------------- S  : 1
               | a2+b2 c3 |

          2                     1
     1 : S  <----------------- S  : 2
               {2} | -c3   |
               {3} | a2+b2 |

o4 : ComplexMap

i5 : K4 = koszulComplex vars S

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

o5 : Complex

i6 : dd^K4

          1                   4
o6 = 0 : S  <--------------- S  : 1
               | a b c d |

          4                                 6
     1 : S  <----------------------------- S  : 2
               {1} | -b -c 0  -d 0  0  |
               {1} | a  0  -c 0  -d 0  |
               {1} | 0  a  b  0  0  -d |
               {1} | 0  0  0  a  b  c  |

          6                           4
     2 : S  <----------------------- S  : 3
               {2} | c  d  0  0  |
               {2} | -b 0  d  0  |
               {2} | a  0  0  d  |
               {2} | 0  -b -c 0  |
               {2} | 0  a  0  -c |
               {2} | 0  0  a  b  |

          4                  1
     3 : S  <-------------- S  : 4
               {3} | -d |
               {3} | c  |
               {3} | -b |
               {3} | a  |

o6 : ComplexMap

i7 : assert isWellDefined K4

To obtain natural subcomplexes, use the Concentration option.

i8 : koszulComplex(vars S, Concentration => (2,3))

      6      4
o8 = S  <-- S
             
     2      3

o8 : Complex

i9 : koszulComplex(vars S, Concentration => (-1,5))

      1      4      6      4      1
o9 = S  <-- S  <-- S  <-- S  <-- S
                                  
     0      1      2      3      4

o9 : Complex

The koszul complex can be constructed as an iterated tensor product. The maps are identical, except that the even indexed differentials have the opposite sign.

i10 : C = koszulComplex {d} ** (koszulComplex {c} ** (koszulComplex {b} ** koszulComplex {a}))

       1      4      6      4      1
o10 = S  <-- S  <-- S  <-- S  <-- S
                                   
      0      1      2      3      4

o10 : Complex

i11 : K = koszulComplex {a,b,c,d}

       1      4      6      4      1
o11 = S  <-- S  <-- S  <-- S  <-- S
                                   
      0      1      2      3      4

o11 : Complex

i12 : netList {{dd^C, dd^K}}

      +--------------------------------------------+--------------------------------------------+
      |     1                   4                  |     1                   4                  |
o12 = |0 : S  <--------------- S  : 1              |0 : S  <--------------- S  : 1              |
      |          | a b c d |                       |          | a b c d |                       |
      |                                            |                                            |
      |     4                                 6    |     4                                 6    |
      |1 : S  <----------------------------- S  : 2|1 : S  <----------------------------- S  : 2|
      |          {1} | b  c  0  d  0  0  |         |          {1} | -b -c 0  -d 0  0  |         |
      |          {1} | -a 0  c  0  d  0  |         |          {1} | a  0  -c 0  -d 0  |         |
      |          {1} | 0  -a -b 0  0  d  |         |          {1} | 0  a  b  0  0  -d |         |
      |          {1} | 0  0  0  -a -b -c |         |          {1} | 0  0  0  a  b  c  |         |
      |                                            |                                            |
      |     6                           4          |     6                           4          |
      |2 : S  <----------------------- S  : 3      |2 : S  <----------------------- S  : 3      |
      |          {2} | c  d  0  0  |               |          {2} | c  d  0  0  |               |
      |          {2} | -b 0  d  0  |               |          {2} | -b 0  d  0  |               |
      |          {2} | a  0  0  d  |               |          {2} | a  0  0  d  |               |
      |          {2} | 0  -b -c 0  |               |          {2} | 0  -b -c 0  |               |
      |          {2} | 0  a  0  -c |               |          {2} | 0  a  0  -c |               |
      |          {2} | 0  0  a  b  |               |          {2} | 0  0  a  b  |               |
      |                                            |                                            |
      |     4                  1                   |     4                  1                   |
      |3 : S  <-------------- S  : 4               |3 : S  <-------------- S  : 4               |
      |          {3} | d  |                        |          {3} | -d |                        |
      |          {3} | -c |                        |          {3} | c  |                        |
      |          {3} | b  |                        |          {3} | -b |                        |
      |          {3} | -a |                        |          {3} | a  |                        |
      +--------------------------------------------+--------------------------------------------+

Ways to use this method: