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A.11 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.-
lres lres
computes a free resolution of an ideal or module with La Scala’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.
res(i,r), lres(i,r), sres(i,r), mres(i,r),
nres(i,r) compute the first r modules of the resolution
of i, resp. the full resolution if r=0 and the basering is not a qring.
See the manual for a precise description of these commands.
Note: The command betti does not require a minimal
resolution for the minimal betti numbers.
Now let’s 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; // convertion 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 La Scala
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
→ resolution not minimized yet
→
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
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