Singular Manual: Syzygies and resolutions

C.3 Syzygies and resolutions

Syzygies

Let

be a quotient of $\hbox{Loc}_< K[\underline{x}]$ and let

be a submodule of

. Then the module of syzygies (or 1st syzygy module, module of relations) of

, syz(

), is defined to be the kernel of the map $R^s \rightarrow R^r,\; \sum_{i=1}^s w_ie_i \mapsto \sum_{i=1}^s w_ig_i$ .

The k-th syzygy module is defined inductively to be the module of syzygies of the -stsyzygy module.

Note, that the syzygy modules of depend on a choice of generators . But one can show that they depend on 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 $I=(g_1,...,g_s)\subseteq R^r$ and

. A free resolution of is a long exact sequence

$\begin{displaymath}...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F... ...1}\over{\longrightarrow} F_0\longrightarrow M\longrightarrow 0,\end{displaymath}$

where the columns of the matrix

generate

. Note that resolutions need not to be finite (i.e., of finite length). The Hilbert Syzygy Theorem states that for $R = \hbox{Loc}_< K[\underline{x}]$ 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

be a graded ring (e.g., $R = \hbox{Loc}_< K[\underline{x}]$ ) and let $I \subset R^r$ be a graded submodule. Let

$\begin{displaymath} R^r = \bigoplus_a R\cdot e_{a,0} \buildrel A_1 \over \longl... ...ts \longleftarrow \bigoplus_a R\cdot e_{a,n} \longleftarrow 0 \end{displaymath}$

be a minimal free resolution of

considered with homogeneous maps of degree 0. Then the graded Betti number $b_{i,j}$ of

is the minimal number of generators $e_{a,j}$ in degree

of the

-th syzygy module of

(i.e., the

-st syzygy module of

). Note that, by definition, the

-th syzygy module of

and the 1st syzygy module of

The regularity of is the smallest integer such that

$\begin{displaymath} \hbox{deg}(e_{a,j}) \le s+j-1 \quad \hbox{for all $j$.} \end{displaymath}$

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

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