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Hilbert functions and free resolutions

In this section, we give examples of common operations involving modules. Throughout this section, we suppose that the base ring R is graded, with each variable having degree one, and that M is a graded R-module. If the ring is not graded, or is multi-graded, or if M is not graded, some of these functions still work, but care must be taken in interpreting the output. Here, we just consider the standard grading case.

Sections:

  • checking homogeniety
  • codimension, degree, and sectional arithmetic genera
  • the Hilbert series
  • free resolutions
  • betti numbers

    • checking homogeniety

    Let's start by making a module over a ring with 18 variables

    i1 : R = ZZ/32003[vars(0..17)];
    i2 : M = coker genericMatrix(R,a,3,6)

    o2 = cokernel | a d g j m p |
                  | b e h k n q |
                  | c f i l o r |

                                3
    o2 : R-module, quotient of R

    Use isHomogeneous to check whether a given module is graded.

    i3 : isHomogeneous M

    o3 = true

    codimension, degree, and sectional arithmetic genera

    Use codim Module, degree Module, and genera Module for some basic numeric information about a module.

    i4 : codim M

    o4 = 4
    i5 : degree M

    o5 = 15
    i6 : genera M

    o6 = {-2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, 4, 14}

    o6 : List

    The last number in the list of genera is the degree minus one. The second to last number is the genus of the generic linear section curve, ..., and the first number is the arithmetic genus

    the Hilbert series

    The Hilbert series (hilbertSeries Module) of M is by definition the formal power series H(t) = sum(d in ZZ) dim(M_d) t^d. This is a rational function with denominator (1-t)^n, where n is the number of variables in the polynomial ring. The numerator of this rational function is called the poincare polynomial, and is obtained by the poincare Module function.

    i7 : poincare M

                       4       5      6
    o7 = 3 - 6$T + 15$T  - 18$T  + 6$T

    o7 : ZZ[ZZ^1]

    i8 : hilbertSeries M

                       4       5      6
         3 - 6$T + 15$T  - 18$T  + 6$T
    o8 = ------------------------------
                           18
                   (1 - $T)

    o8 : Divide

    Notice that the variable is written as $T. This indicates that the variable cannot be typed in directly.

    It is often useful to divide the poincare polynomial by (1-t) as many times as possible. This can be done by the following function:

    i9 : poincare' = (M) -> (
            H := poincare M;
            t := (ring H)_0;  -- The variable t above
            while H % (1-t) == 0 do H = H // (1-t);
            H)

    o9 = poincare'

    o9 : Function
    i10 : poincare' M

                       2
    o10 = 3 + 6$T + 6$T

    o10 : ZZ[ZZ^1]

    free resolutions

    The minimal free resolution C is computed using resolution Module. The specific matrices are obtained by indexing C.dd.

    i11 : C = resolution M

           3      6      15      18      6
    o11 = R  <-- R  <-- R   <-- R   <-- R  <-- 0
                                                
          0      1      2       3       4      5

    o11 : ChainComplex
    i12 : C.dd_3

    o12 = {4} | m  -n o  p  -q r  0  0  0  0  0  0  0  0  0  0  0  0  |
          {4} | -j k  -l 0  0  0  p  0  0  0  -q r  0  0  0  0  0  0  |
          {4} | g  -h i  0  0  0  0  p  0  0  0  0  -q 0  0  r  0  0  |
          {4} | -d e  -f 0  0  0  0  0  p  0  0  0  0  -q 0  0  r  0  |
          {4} | a  -b c  0  0  0  0  0  0  p  0  0  0  0  -q 0  0  r  |
          {4} | 0  0  0  -j k  -l -m 0  0  0  n  -o 0  0  0  0  0  0  |
          {4} | 0  0  0  g  -h i  0  -m 0  0  0  0  n  0  0  -o 0  0  |
          {4} | 0  0  0  -d e  -f 0  0  -m 0  0  0  0  n  0  0  -o 0  |
          {4} | 0  0  0  a  -b c  0  0  0  -m 0  0  0  0  n  0  0  -o |
          {4} | 0  0  0  0  0  0  g  j  0  0  -h i  -k 0  0  l  0  0  |
          {4} | 0  0  0  0  0  0  -d 0  j  0  e  -f 0  -k 0  0  l  0  |
          {4} | 0  0  0  0  0  0  a  0  0  j  -b c  0  0  -k 0  0  l  |
          {4} | 0  0  0  0  0  0  0  -d -g 0  0  0  e  h  0  -f -i 0  |
          {4} | 0  0  0  0  0  0  0  a  0  -g 0  0  -b 0  h  c  0  -i |
          {4} | 0  0  0  0  0  0  0  0  a  d  0  0  0  -b -e 0  c  f  |

                  15       18
    o12 : Matrix R   <--- R

    For more information about chain complexes and resolutions, see chain complexes and computing resolutions.

    betti numbers

    Use betti ChainComplex to display the graded betti numbers of M.

    i13 : betti C

    o13 = total: 3 6 15 18 6
              0: 3 6  .  . .
              1: . .  .  . .
              2: . . 15 18 6

    This table should be interpreted as follows: the number in the ith row and jth column (indices starting at 0), is the number of jth syzygies in degree i+j. In the above example, there are 15 second syzygies of degree 4, and the entries of the maps CC.d_1, CC.d_3, CC.d_4 are all linear.

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