
C.3 Syzygies and resolutions
Syzygies
Let be a quotient of
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
.
The kth 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
and .
A free resolution of is a long exact sequence
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
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.,
) and
let be a graded submodule. Let
be a minimal free resolution of considered with homogeneous maps
of degree 0. Then the graded Betti number of is
the minimal number of generators in degree of the th
syzygy module of (i.e., the st syzygy module of
). Note that, by definition, the th syzygy module of is
and the 1st syzygy module of is .
The regularity of
is the smallest integer
such that
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

