# Singular

### Syzygies

Let be a GR-algebra. A left (resp. right) syzygy between elements is a -tuple satisfying

The set of all left (resp. right) syzygies between is a left (resp. right) submodule of .

Remark: With respect to the definitions of ideal and module (see PLURAL), 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 (but, in general, ).

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.

### Free resolutions

Let and . A free resolution of is a long exact sequence

with

and where the columns of the matrix generate . Note, that resolutions over factor-algebras need not to be of finite length.

### Generalized Hilbert Syzygy Theorem

For a -algebra , generated by variables, there exists a free resolution of length smaller or equal than .

Example:
 ring R=0,(x,y,z),dp; matrix d[3][3]; d[1,2]=-z; d[1,3]=2x; d[2,3]=-2y; def U=nc_algebra(1,d); // this algebra is U(sl_2) setring U; option(redSB); option(redTail); 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 ==> U <-- U <-- U <-- U ==> ==> 0 1 2 3 ==> list l = resI; // The matrix A_1 is given by print(matrix(l[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(l[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(l[3])*transpose(l[2]); // 2nd term size(ideal(TST)); ==> 0 TST = transpose(l[2])*transpose(l[1]); // 1st term size(ideal(TST)); ==> 0