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C.3 Syzygies and resolutions
Syzygies
Let 53#53 be a quotient of
659#659 and let
660#660 be a submodule of 624#624.
Then the module of syzygies (or 1st syzygy module, module of relations) of 253#253, syz(253#253), is defined to be the kernel of the map
661#661.
The k-th syzygy module is defined inductively to be the module
of syzygies of the
662#662-stsyzygy module.
Note, that the syzygy modules of 253#253 depend on a choice of generators 663#663.
But one can show that they depend on 253#253 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
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Free resolutions
Let
664#664 and 665#665.
A free resolution of 13#13 is a long exact sequence
666#666
where the columns of the matrix
166#166generate
253#253. Note that resolutions need not to be finite (i.e., of
finite length). The Hilbert Syzygy Theorem states that for
623#623there 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
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Betti numbers and regularity
Let 53#53 be a graded ring (e.g.,
623#623) and
let 667#667 be a graded submodule. Let
668#668
be a minimal free resolution of 669#669 considered with homogeneous maps
of degree 0. Then the graded Betti number 670#670 of 669#669 is
the minimal number of generators 671#671 in degree 172#172 of the 55#55-th
syzygy module of 669#669 (i.e., the 672#672-st syzygy module of
253#253). Note that, by definition, the 2#2-th syzygy module of 669#669 is 624#624
and the 1st syzygy module of 669#669 is 253#253.
The regularity of
253#253 is the smallest integer
178#178 such that
673#673
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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