A.3.4 Free resolution

In SINGULAR a free resolution of a module or ideal has its own type: resolution. It is a structure that stores all information related to free resolutions. This allows partial computations of resolutions via the command res. After applying res, only a pre-format of the resolution is computed which allows to determine invariants like Betti-numbers or homological dimension. To see the differentials of the complex, a resolution must be converted into the type list which yields a list of modules: the k-th module in this list is the first syzygy-module (module of relations) of the (k-1)st module. There are the following commands to compute a resolution:

res

res
computes a free resolution of an ideal or module using a heuristically chosen method. This is the preferred method to compute free resolutions of ideals or modules.

fres

fres
improved version of sres, computes a free resolution of an ideal or module using Schreyer's method. The input has to be a standard basis.

lres

lres
computes a free resolution of an ideal or module with LaScala's method. The input needs to be homogeneous.

mres

mres
computes a minimal free resolution of an ideal or module with the syzygy method.

sres

sres
computes a free resolution of an ideal or module with Schreyer's method. The input has to be a standard basis.

nres

nres
computes a free resolution of an ideal or module with the standard basis method.

minres

minres
minimizes a free resolution of an ideal or module.

syz

syz
computes the first syzygy module.

Now let us take a look at an example which uses resolutions: The Hilbert-Burch theorem says that the ideal i of a reduced curve in has a free resolution of length 2 and that i is given by the 2x2 minors of the 2nd matrix in the resolution. We test this for two transversal cusps in .Afterwards we compute the resolution of the ideal j of the tangent developable of the rational normal curve in from above. Finally we demonstrate the use of the type resolution in connection with the lres command.

// Two transversal cusps in (k^3,0): ring r2 =0,(x,y,z),ds; ideal i =z2-1y3+x3y,xz,-1xy2+x4,x3z; resolution rs=mres(i,0); // computes a minimal resolution rs; // the standard representation of complexes ==> 1 3 2 ==> r2 <-- r2 <-- r2 ==> ==> 0 1 2 ==> list resi=rs; // conversion to a list print(resi[1]); // the 1st module is i minimized ==> xz, ==> z2-y3+x3y, ==> xy2-x4 print(resi[2]); // the 1st syzygy module of i ==> -z,-y2+x3, ==> x, 0, ==> y, z resi[3]; // the 2nd syzygy module of i ==> _[1]=0 ideal j=minor(resi[2],2); reduce(j,std(i)); // check whether j is contained in i ==> _[1]=0 ==> _[2]=0 ==> _[3]=0 size(reduce(i,std(j))); // check whether i is contained in j ==> 0 // size(<ideal>) counts the non-zero generators // --------------------------------------------- // The tangent developable of the rational normal curve in P^4: ring P = 0,(a,b,c,d,e),dp; ideal j= 3c2-4bd+ae, -2bcd+3ad2+3b2e-4ace, 8b2d2-9acd2-9b2ce+9ac2e+2abde-1a2e2; resolution rs=mres(j,0); rs; ==> 1 2 1 ==> P <-- P <-- P ==> ==> 0 1 2 ==> list L=rs; print(L[2]); ==> 2bcd-3ad2-3b2e+4ace, ==> -3c2+4bd-ae // create an intmat with graded Betti numbers intmat B=betti(rs); // this gives a nice output of Betti numbers print(B,"betti"); ==> 0 1 2 ==> ------------------------ ==> 0: 1 - - ==> 1: - 1 - ==> 2: - 1 - ==> 3: - - 1 ==> ------------------------ ==> total: 1 2 1 ==> // the user has access to all Betti numbers // the 2-nd column of B: B[1..4,2]; ==> 0 1 1 0 ring cyc5=32003,(a,b,c,d,e,h),dp; ideal i= a+b+c+d+e, ab+bc+cd+de+ea, abc+bcd+cde+dea+eab, abcd+bcde+cdea+deab+eabc, h5-abcde; resolution rs=lres(i,0); //computes the resolution according LaScala rs; //the shape of the minimal resolution ==> 1 5 10 10 5 1 ==> cyc5 <-- cyc5 <-- cyc5 <-- cyc5 <-- cyc5 <-- cyc5 ==> ==> 0 1 2 3 4 5 ==> print(betti(rs),"betti"); //shows the Betti-numbers of cyclic 5 ==> 0 1 2 3 4 5 ==> ------------------------------------------ ==> 0: 1 1 - - - - ==> 1: - 1 1 - - - ==> 2: - 1 1 - - - ==> 3: - 1 2 1 - - ==> 4: - 1 2 1 - - ==> 5: - - 2 2 - - ==> 6: - - 1 2 1 - ==> 7: - - 1 2 1 - ==> 8: - - - 1 1 - ==> 9: - - - 1 1 - ==> 10: - - - - 1 1 ==> ------------------------------------------ ==> total: 1 5 10 10 5 1 ==> dim(rs); //the homological dimension ==> 4 size(list(rs)); //gets the full (non-reduced) resolution ==> 6 minres(rs); //minimizes the resolution ==> 1 5 10 10 5 1 ==> cyc5 <-- cyc5 <-- cyc5 <-- cyc5 <-- cyc5 <-- cyc5 ==> ==> 0 1 2 3 4 5 ==> size(list(rs)); //gets the minimized resolution ==> 6