Top
Back: C.2 Hilbert function
Forward: C.4 Characteristic sets
FastBack: Appendix C Mathematical background
FastForward: Appendix D SINGULAR libraries
Up: Appendix C Mathematical background
Top: 1 Preface
Contents: Table of Contents
Index: F Index
About: About This Document

C.3 Syzygies and resolutions

Syzygies

Let R be a quotient of Loc<K[x] and let I = (g1,...,gs) be a submodule of Rr. Then the module of syzygies (or 1st syzygy module, module of relations) of I, syz(I), is defined to be the kernel of the map Rs Rr, i=1swiei↦→ i=1swigi.

The k-th syzygy module is defined inductively to be the module of syzygies of the (k 1)-st syzygy module.

Note, that the syzygy modules of I depend on a choice of generators g1,...,gs. But one can show that they depend on I uniquely up to direct summands.
Example:
  ring R= 0,(u,v,x,y,z),dp;
  ideal i=ux, vx, uy, vy;
  print(syz(i));
→ -y,0, -v,0, 
→ 0, -y,u, 0, 
→ x, 0, 0, -v,
→ 0, x, 0, u  

Free resolutions

Let I = (g1,...,gs) Rr and M = Rr∕I. A free resolution of M is a long exact sequence
        A2     A1
...−→ F2 −→ F1 −→  F0 −→ M − → 0,

where the columns of the matrix A1 generate I . Note, that resolutions need not to be finite (i.e., of finite length). The Hilbert Syzygy Theorem states, that for R = Loc<K[x] there exists a ("minimal") resolution of length not exceeding the number of variables.

Example:
  ring R= 0,(u,v,x,y,z),dp;
  ideal I = ux, vx, uy, vy;
  resolution resI = mres(I,0); resI;
→  1      4      4      1      
→ R <--  R <--  R <--  R
→ 
→ 0      1      2      3      
→ 
  // The matrix A_1 is given by
  print(matrix(resI[1]));
→ vy,uy,vx,ux
  // We see that the columns of A_1 generate I.
  // The matrix A_2 is given by
  print(matrix(resI[3]));
→ u, 
→ -v,
→ -x,
→ y  

Betti numbers and regularity

Let R be a graded ring (e.g., R = Loc<K[x]) and let I Rr be a graded submodule. Let
 r  ⊕          A1 ⊕                  ⊕
R =     R ⋅ea,0← −    R ⋅ea,1 ← − ... ←−   R ⋅ea,n ←− 0
     a             a                  a
be a minimal free resolution of Rn∕I considered with homogeneous maps of degree 0. Then the graded Betti number bi,j of Rr∕I is the minimal number of generators ea,j in degree i + j of the j-th syzygy module of Rr∕I (i.e., the (j 1)-st syzygy module of I). Note, that by definition the 0-th syzygy module of Rr∕I is Rr and the 1st syzygy module of Rr∕I is I.

The regularity of I is the smallest integer s

such that

deg(ea,j) ≤ s + j − 1 for all j.

Example:
  ring R= 0,(u,v,x,y,z),dp;
  ideal I = ux, vx, uy, vy;
  resolution resI = mres(I,0); resI;
→  1      4      4      1      
→ R <--  R <--  R <--  R
→ 
→ 0      1      2      3      
→ 
  // the betti number:
  print(betti(resI), "betti");
→            0     1     2     3
→ ------------------------------
→     0:     1     -     -     -
→     1:     -     4     4     1
→ ------------------------------
→ total:     1     4     4     1
  // the regularity:
  regularity(resI);
→ 2

Top Back: C.2 Hilbert function Forward: C.4 Characteristic sets FastBack: Appendix C Mathematical background FastForward: Appendix D SINGULAR libraries Up: Appendix C Mathematical background Top: 1 Preface Contents: Table of Contents Index: F Index About: About This Document
            User manual for Singular version 2-0-4, October 2002, generated by texinfo.