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C.3 Syzygies and resolutions
Syzygies
Let 53#53 be a quotient of
657#657 and let
658#658 be a submodule of 622#622.
Then the module of syzygies (or 1st syzygy module, module of relations) of 251#251, syz(251#251), is defined to be the kernel of the map
659#659.
The k-th syzygy module is defined inductively to be the module
of syzygies of the
660#660-stsyzygy module.
Note, that the syzygy modules of 251#251 depend on a choice of generators 661#661.
But one can show that they depend on 251#251 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
662#662 and 663#663.
A free resolution of 13#13 is a long exact sequence
664#664
where the columns of the matrix
166#166generate
251#251. Note that resolutions need not to be finite (i.e., of
finite length). The Hilbert Syzygy Theorem states that for
621#621there 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.,
621#621) and
let 665#665 be a graded submodule. Let
666#666
be a minimal free resolution of 667#667 considered with homogeneous maps
of degree 0. Then the graded Betti number 668#668 of 667#667 is
the minimal number of generators 669#669 in degree 172#172 of the 55#55-th
syzygy module of 667#667 (i.e., the 670#670-st syzygy module of
251#251). Note that, by definition, the 2#2-th syzygy module of 667#667 is 622#622
and the 1st syzygy module of 667#667 is 251#251.
The regularity of
251#251 is the smallest integer
177#177 such that
671#671
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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