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A.3 Syzygies and resolutions

Syzygies

Let $K$ be a field and $>$ a well ordering on $A^r$. A left syzygy between $k$ elements $f_1,\dots,f_k \in A^r =\oplus_{i=1}^{r}Ae_i $ is $k$-tuple $(g_1,\dots ,g_k)\in A^k$ satisfying

\begin{displaymath}\sum_{i=1}^{k}g_i f_i =0 \end{displaymath}

The set of all left (resp.right) syzygies between $f_1,...,f_k$ is left (resp. right) submodule $S$ of $A^k$.


Remark: With respect to the definitions of ideal and module (see PLURAL conventions ) PLURAL works with left syzygies only (by syz we understand a left syzygy). If S is a matrix of a left syzygy module of left submodule given by matrix M, then transpose(S)*transpose(M) = 0.

Note, that the syzygy modules of $I$ depend on a choice of generators $g_1, \dots , g_s$. But one can show that they depend on $I$ uniquely up to direct summands.

Free resolutions

Let $I=\langle g_1,\dots ,g_s\rangle \subseteq A^r$ and $M= A^r/I$. A free resolution of $M$ is a long exact sequence

\begin{displaymath}\dots \longrightarrow F_2 \buildrel{B_2}\over{\longrightarrow...
...1}\over{\longrightarrow} F_0\longrightarrow M\longrightarrow
0,\end{displaymath}


where the columns of the matrix $B_1$generate $I$. Note, that resolutions over factor-algebras need not to be finite (i.e., of finite length). The Generalized Hilbert Syzygy Theorem states, that for G-algebra $A$, generated by n variables, there exists a resolution of length smaller or equal than n.

Example:
 
LIB "poly.lib";
ring R=0,(x,y,z),dp;
matrix d[3][3];
d[1,2]=-z;  d[1,3]=2x;  d[2,3]=-2y;
ncalgebra(1,d); // this is U(sl_2)
ideal I=x3,y3,z3-z;
I=std(I);
I;
==> I[1]=z3-z
==> I[2]=y3
==> I[3]=x3
==> I[4]=y2z2-y2z
==> I[5]=x2z2+x2z
==> I[6]=x2y2z-2xyz2-2xyz+2z2+2z
resolution resI = mres(I,0); 
resI;
==>  1      5      7      3      
==> R <--  R <--  R <--  R
==> 
==> 0      1      2      3      
==> 
// The matrix A_1 is given by
print(matrix(resI[1]));
==> z3-z,y3,x3,y2z2-y2z,x2z2+x2z
// We see that the columns of A_1 generate I.
// The matrix A_2 is given by
print(matrix(resI[2]));
==> 0,        0,        y2,  x2,  6yz,        -36xy+18z+24,-6xz,       
==> z2+11z+30,0,        0,   0,   2x2z+12x2,  2x3,         0,          
==> 0,        z2-11z+30,0,   0,   0,          -2y3,        2y2z-12y2,  
==> -y,       0,        -z-5,0,   x2y-6xz-30x,9x2,         x3,         
==> 0,        -x,       0,   -z+5,-y3,        -9y2,        -xy2-4yz+28y
ideal tst; // now let us show that the resolution is exact
matrix TST;
TST = transpose(resI[3])*transpose(resI[2]); // 2nd term
tst = std(flatten(TST));
tst;
==> tst[1]=0
TST = transpose(resI[2])*transpose(resI[1]); // 1st term
tst = std(flatten(TST));
tst;
==> tst[1]=0


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