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: 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

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

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

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

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: 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

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: 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

shifts(i)[source]¶

Return the shifts of self.

EXAMPLES:

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)
[]

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: 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

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: 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