Graded free resolutions¶

Let \(R\) be a commutative ring. A graded free resolution of a graded \(R\)-module \(M\) is a free resolution such that all maps are homogeneous module homomorphisms.

EXAMPLES:

Sage

sage: S.<x,y,z,w> = PolynomialRing(QQ)
sage: I = S.ideal([y*w - z^2, -x*w + y*z, x*z - y^2])
sage: r = I.graded_free_resolution(algorithm='minimal')
sage: r
S(0) <-- S(-2)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3) <-- 0
sage: I.graded_free_resolution(algorithm='shreyer')
S(0) <-- S(-2)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3) <-- 0
sage: I.graded_free_resolution(algorithm='standard')
S(0) <-- S(-2)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3) <-- 0
sage: I.graded_free_resolution(algorithm='heuristic')
S(0) <-- S(-2)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3) <-- 0

Python

>>> from sage.all import *
>>> S = PolynomialRing(QQ, names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = S._first_ngens(4)
>>> I = S.ideal([y*w - z**Integer(2), -x*w + y*z, x*z - y**Integer(2)])
>>> r = I.graded_free_resolution(algorithm='minimal')
>>> r
S(0) <-- S(-2)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3) <-- 0
>>> I.graded_free_resolution(algorithm='shreyer')
S(0) <-- S(-2)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3) <-- 0
>>> I.graded_free_resolution(algorithm='standard')
S(0) <-- S(-2)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3) <-- 0
>>> I.graded_free_resolution(algorithm='heuristic')
S(0) <-- S(-2)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3) <-- 0

Sage Live

S.<x,y,z,w> = PolynomialRing(QQ)
I = S.ideal([y*w - z^2, -x*w + y*z, x*z - y^2])
r = I.graded_free_resolution(algorithm='minimal')
r
I.graded_free_resolution(algorithm='shreyer')
I.graded_free_resolution(algorithm='standard')
I.graded_free_resolution(algorithm='heuristic')

Sage

sage: d = r.differential(2)
sage: d
Free module morphism defined as left-multiplication by the matrix
  [ y  x]
  [-z -y]
  [ w  z]
  Domain:   Ambient free module of rank 2 over the integral domain
            Multivariate Polynomial Ring in x, y, z, w over Rational Field
  Codomain: Ambient free module of rank 3 over the integral domain
            Multivariate Polynomial Ring in x, y, z, w over Rational Field
sage: d.image()
Submodule of Ambient free module of rank 3 over the integral domain
 Multivariate Polynomial Ring in x, y, z, w over Rational Field
Generated by the rows of the matrix:
[ y -z  w]
[ x -y  z]
sage: m = d.image()
sage: m.graded_free_resolution(shifts=(2,2,2))
S(-2)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3) <-- 0

Python

>>> from sage.all import *
>>> d = r.differential(Integer(2))
>>> d
Free module morphism defined as left-multiplication by the matrix
  [ y  x]
  [-z -y]
  [ w  z]
  Domain:   Ambient free module of rank 2 over the integral domain
            Multivariate Polynomial Ring in x, y, z, w over Rational Field
  Codomain: Ambient free module of rank 3 over the integral domain
            Multivariate Polynomial Ring in x, y, z, w over Rational Field
>>> d.image()
Submodule of Ambient free module of rank 3 over the integral domain
 Multivariate Polynomial Ring in x, y, z, w over Rational Field
Generated by the rows of the matrix:
[ y -z  w]
[ x -y  z]
>>> m = d.image()
>>> m.graded_free_resolution(shifts=(Integer(2),Integer(2),Integer(2)))
S(-2)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3) <-- 0

Sage Live

d = r.differential(2)
d
d.image()
m = d.image()
m.graded_free_resolution(shifts=(2,2,2))

An example of multigraded resolution from Example 9.1 of [MilStu2005]:

Sage

sage: R.<s,t> = QQ[]
sage: S.<a,b,c,d> = QQ[]
sage: phi = S.hom([s, s*t, s*t^2, s*t^3])
sage: I = phi.kernel(); I                                                           # needs sage.rings.function_field
Ideal (c^2 - b*d, b*c - a*d, b^2 - a*c) of
 Multivariate Polynomial Ring in a, b, c, d over Rational Field
sage: P3 = ProjectiveSpace(S)
sage: C = P3.subscheme(I)  # twisted cubic curve
sage: r = I.graded_free_resolution(degrees=[(1,0), (1,1), (1,2), (1,3)])
sage: r
S((0, 0)) <-- S((-2, -4))⊕S((-2, -3))⊕S((-2, -2)) <-- S((-3, -5))⊕S((-3, -4)) <-- 0
sage: r.K_polynomial(names='s,t')
s^3*t^5 + s^3*t^4 - s^2*t^4 - s^2*t^3 - s^2*t^2 + 1

Python

>>> from sage.all import *
>>> R = QQ['s, t']; (s, t,) = R._first_ngens(2)
>>> S = QQ['a, b, c, d']; (a, b, c, d,) = S._first_ngens(4)
>>> phi = S.hom([s, s*t, s*t**Integer(2), s*t**Integer(3)])
>>> I = phi.kernel(); I                                                           # needs sage.rings.function_field
Ideal (c^2 - b*d, b*c - a*d, b^2 - a*c) of
 Multivariate Polynomial Ring in a, b, c, d over Rational Field
>>> P3 = ProjectiveSpace(S)
>>> C = P3.subscheme(I)  # twisted cubic curve
>>> r = I.graded_free_resolution(degrees=[(Integer(1),Integer(0)), (Integer(1),Integer(1)), (Integer(1),Integer(2)), (Integer(1),Integer(3))])
>>> r
S((0, 0)) <-- S((-2, -4))⊕S((-2, -3))⊕S((-2, -2)) <-- S((-3, -5))⊕S((-3, -4)) <-- 0
>>> r.K_polynomial(names='s,t')
s^3*t^5 + s^3*t^4 - s^2*t^4 - s^2*t^3 - s^2*t^2 + 1

Sage Live

R.<s,t> = QQ[]
S.<a,b,c,d> = QQ[]
phi = S.hom([s, s*t, s*t^2, s*t^3])
I = phi.kernel(); I                                                           # needs sage.rings.function_field
P3 = ProjectiveSpace(S)
C = P3.subscheme(I)  # twisted cubic curve
r = I.graded_free_resolution(degrees=[(1,0), (1,1), (1,2), (1,3)])
r
r.K_polynomial(names='s,t')

AUTHORS:

Kwankyu Lee (2022-05): initial version
Travis Scrimshaw (2022-08-23): refactored for free module inputs

class sage.homology.graded_resolution.GradedFiniteFreeResolution(module, degrees=None, shifts=None, name='S', **kwds)[source]¶

Bases: FiniteFreeResolution

Graded finite free resolutions.

INPUT:

module – a homogeneous submodule of a free module \(M\) of rank \(n\) over \(S\) or a homogeneous ideal of a multivariate polynomial ring \(S\)
degrees – (default: a list with all entries \(1\)) a list of integers or integer vectors giving degrees of variables of \(S\)
shifts – list of integers or integer vectors giving shifts of degrees of \(n\) summands of the free module \(M\); this is a list of zero degrees of length \(n\) by default
name – string; name of the base ring

K_polynomial(names=None)[source]¶

Return the K-polynomial of this resolution.

INPUT:

names – (optional) a string of names of the variables of the K-polynomial

EXAMPLES:

Sage

sage: S.<x,y,z,w> = PolynomialRing(QQ)
sage: I = S.ideal([y*w - z^2, -x*w + y*z, x*z - y^2])
sage: r = I.graded_free_resolution()
sage: r.K_polynomial()
2*t^3 - 3*t^2 + 1

Python

>>> from sage.all import *
>>> S = PolynomialRing(QQ, names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = S._first_ngens(4)
>>> I = S.ideal([y*w - z**Integer(2), -x*w + y*z, x*z - y**Integer(2)])
>>> r = I.graded_free_resolution()
>>> r.K_polynomial()
2*t^3 - 3*t^2 + 1

Sage Live

S.<x,y,z,w> = PolynomialRing(QQ)
I = S.ideal([y*w - z^2, -x*w + y*z, x*z - y^2])
r = I.graded_free_resolution()
r.K_polynomial()

betti(i, a=None)[source]¶

Return the \(i\)-th Betti number in degree \(a\).

INPUT:

i – nonnegative integer
a – a degree; if None, return Betti numbers in all degrees

EXAMPLES:

Sage

sage: S.<x,y,z,w> = PolynomialRing(QQ)
sage: I = S.ideal([y*w - z^2, -x*w + y*z, x*z - y^2])
sage: r = I.graded_free_resolution()
sage: r.betti(0)
{0: 1}
sage: r.betti(1)
{2: 3}
sage: r.betti(2)
{3: 2}
sage: r.betti(1, 0)
0
sage: r.betti(1, 1)
0
sage: r.betti(1, 2)
3

Python

>>> from sage.all import *
>>> S = PolynomialRing(QQ, names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = S._first_ngens(4)
>>> I = S.ideal([y*w - z**Integer(2), -x*w + y*z, x*z - y**Integer(2)])
>>> r = I.graded_free_resolution()
>>> r.betti(Integer(0))
{0: 1}
>>> r.betti(Integer(1))
{2: 3}
>>> r.betti(Integer(2))
{3: 2}
>>> r.betti(Integer(1), Integer(0))
0
>>> r.betti(Integer(1), Integer(1))
0
>>> r.betti(Integer(1), Integer(2))
3

Sage Live

S.<x,y,z,w> = PolynomialRing(QQ)
I = S.ideal([y*w - z^2, -x*w + y*z, x*z - y^2])
r = I.graded_free_resolution()
r.betti(0)
r.betti(1)
r.betti(2)
r.betti(1, 0)
r.betti(1, 1)
r.betti(1, 2)

shifts(i)[source]¶

Return the shifts of self.

EXAMPLES:

Sage

sage: S.<x,y,z,w> = PolynomialRing(QQ)
sage: I = S.ideal([y*w - z^2, -x*w + y*z, x*z - y^2])
sage: r = I.graded_free_resolution()
sage: r.shifts(0)
[0]
sage: r.shifts(1)
[2, 2, 2]
sage: r.shifts(2)
[3, 3]
sage: r.shifts(3)
[]

Python

>>> from sage.all import *
>>> S = PolynomialRing(QQ, names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = S._first_ngens(4)
>>> I = S.ideal([y*w - z**Integer(2), -x*w + y*z, x*z - y**Integer(2)])
>>> r = I.graded_free_resolution()
>>> r.shifts(Integer(0))
[0]
>>> r.shifts(Integer(1))
[2, 2, 2]
>>> r.shifts(Integer(2))
[3, 3]
>>> r.shifts(Integer(3))
[]

Sage Live

S.<x,y,z,w> = PolynomialRing(QQ)
I = S.ideal([y*w - z^2, -x*w + y*z, x*z - y^2])
r = I.graded_free_resolution()
r.shifts(0)
r.shifts(1)
r.shifts(2)
r.shifts(3)

class sage.homology.graded_resolution.GradedFiniteFreeResolution_free_module(module, degrees=None, *args, **kwds)[source]¶

Bases: GradedFiniteFreeResolution, FiniteFreeResolution_free_module

Graded free resolution of free modules.

EXAMPLES:

Sage

sage: from sage.homology.free_resolution import FreeResolution
sage: R.<x> = QQ[]
sage: M = matrix([[x^3, 3*x^3, 5*x^3],
....:             [0, x, 2*x]])
sage: res = FreeResolution(M, graded=True); res
S(0)⊕S(0)⊕S(0) <-- S(-3)⊕S(-1) <-- 0

Python

>>> from sage.all import *
>>> from sage.homology.free_resolution import FreeResolution
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> M = matrix([[x**Integer(3), Integer(3)*x**Integer(3), Integer(5)*x**Integer(3)],
...             [Integer(0), x, Integer(2)*x]])
>>> res = FreeResolution(M, graded=True); res
S(0)⊕S(0)⊕S(0) <-- S(-3)⊕S(-1) <-- 0

Sage Live

from sage.homology.free_resolution import FreeResolution
R.<x> = QQ[]
M = matrix([[x^3, 3*x^3, 5*x^3],
            [0, x, 2*x]])
res = FreeResolution(M, graded=True); res

class sage.homology.graded_resolution.GradedFiniteFreeResolution_singular(module, degrees=None, shifts=None, name='S', algorithm='heuristic', **kwds)[source]¶

Bases: GradedFiniteFreeResolution, FiniteFreeResolution_singular

Graded free resolutions of submodules and ideals of multivariate polynomial rings implemented using Singular.

INPUT:

module – a homogeneous submodule of a free module \(M\) of rank \(n\) over \(S\) or a homogeneous ideal of a multivariate polynomial ring \(S\)
degrees – (default: a list with all entries \(1\)) a list of integers or integer vectors giving degrees of variables of \(S\)
shifts – list of integers or integer vectors giving shifts of degrees of \(n\) summands of the free module \(M\); this is a list of zero degrees of length \(n\) by default
name – string; name of the base ring
algorithm – Singular algorithm to compute a resolution of ideal

OUTPUT: a graded minimal free resolution of ideal

If module is an ideal of \(S\), it is considered as a submodule of a free module of rank \(1\) over \(S\).

The degrees given to the variables of \(S\) are integers or integer vectors of the same length. In the latter case, \(S\) is said to be multigraded, and the resolution is a multigraded free resolution. The standard grading where all variables have degree \(1\) is used if the degrees are not specified.

A summand of the graded free module \(M\) is a shifted (or twisted) module of rank one over \(S\), denoted \(S(-d)\) with shift \(d\).

The computation of the resolution is done by using libSingular. Different Singular algorithms can be chosen for best performance. The available algorithms and the corresponding Singular commands are shown below:

algorithm	Singular commands
`minimal`	`mres(ideal)`
`shreyer`	`minres(sres(std(ideal)))`
`standard`	`minres(nres(std(ideal)))`
`heuristic`	`minres(res(std(ideal)))`

EXAMPLES:

Sage

sage: S.<x,y,z,w> = PolynomialRing(QQ)
sage: I = S.ideal([y*w - z^2, -x*w + y*z, x*z - y^2])
sage: r = I.graded_free_resolution(); r
S(0) <-- S(-2)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3) <-- 0
sage: len(r)
2

sage: I = S.ideal([z^2 - y*w, y*z - x*w, y - x])
sage: I.is_homogeneous()
True
sage: r = I.graded_free_resolution(); r
S(0) <-- S(-1)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3)⊕S(-4) <-- S(-5) <-- 0

Python

>>> from sage.all import *
>>> S = PolynomialRing(QQ, names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = S._first_ngens(4)
>>> I = S.ideal([y*w - z**Integer(2), -x*w + y*z, x*z - y**Integer(2)])
>>> r = I.graded_free_resolution(); r
S(0) <-- S(-2)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3) <-- 0
>>> len(r)
2

>>> I = S.ideal([z**Integer(2) - y*w, y*z - x*w, y - x])
>>> I.is_homogeneous()
True
>>> r = I.graded_free_resolution(); r
S(0) <-- S(-1)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3)⊕S(-4) <-- S(-5) <-- 0

Sage Live

S.<x,y,z,w> = PolynomialRing(QQ)
I = S.ideal([y*w - z^2, -x*w + y*z, x*z - y^2])
r = I.graded_free_resolution(); r
len(r)
I = S.ideal([z^2 - y*w, y*z - x*w, y - x])
I.is_homogeneous()
r = I.graded_free_resolution(); r