Finitely presented graded modules#
Let \(R\) be a connected graded algebra. A finitely presented module over \(R\) is a module isomorphic to the cokernel of an \(R\)-linear homomorphism \(f: F_1 \to F_0\) of finitely generated free modules: the generators of \(F_0\) correspond to the generators of the module, and the generators of \(F_1\) correspond to its relations, via the map \(f\).
This module class takes as input a set of generators and relations
and uses them to construct a presentation, using the class
FreeGradedModuleMorphism
.
It also allows such a morphism as input.
This package was designed with homological algebra in mind, and its API
focuses on maps rather than objects. A good example of this is the kernel
function sage.modules.fp_graded.morphism.FPModuleMorphism.kernel_inclusion()
,
which computes the kernel of a homomorphism \(f: M\to N\). Its return value
is not an instance of the module class, but rather an injective homomorphism
\(i: K\to M\) with the property that \(\operatorname{im}(i) = \ker(f)\).
Note
Some methods here require in addition that \(R\) be an algebra over a field or a PID and that Sage has a description of a basis for \(R\).
AUTHORS:
Robert R. Bruner, Michael J. Catanzaro (2012): Initial version.
Sverre Lunoee–Nielsen and Koen van Woerden (2019-11-29): Updated the original software to Sage version 8.9.
Sverre Lunoee–Nielsen (2020-07-01): Refactored the code and added new documentation and tests.
- class sage.modules.fp_graded.module.FPModule(j, names)#
Bases:
UniqueRepresentation
,IndexedGenerators
,Module
A finitely presented module over a connected graded algebra.
INPUT:
One of the following:
arg0
– a morphism such that the module is the cokernel, or a free graded module, in which case the output is the same module, viewed as finitely presented
Otherwise:
arg0
– the graded connected algebra over which the module is defined; this algebra must be equipped with a graded basisgenerator_degrees
– tuple of integer degreesrelations
– tuple of relations; a relation is a tuple of coefficients \((c_1, \ldots, c_n)\), ordered so that they correspond to the module generators, that is, such a tuple corresponds to the relation\[c_1 g_1 + \ldots + c_n g_n = 0\]if the generators are \((g_1, \ldots, g_n)\)
EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: E.<x,y> = ExteriorAlgebra(QQ) sage: M = FPModule(E, [0, 1], [[x, 1]]) sage: a, b = M.generators() sage: x*a + b == 0 True sage: (x*a + b).normalize() 0 sage: A3 = SteenrodAlgebra(2, profile=(4,3,2,1)) sage: M = FPModule(A3, [0, 1], [[Sq(2), Sq(1)]]) sage: M.generators() (g[0], g[1]) sage: M.relations() (Sq(2)*g[0] + Sq(1)*g[1],) sage: M.is_trivial() False sage: Z = FPModule(A3, []) sage: Z.generators() () sage: Z.relations() () sage: Z.is_trivial() True sage: from sage.modules.fp_graded.free_module import FreeGradedModule sage: F = FreeGradedModule(E, [0, 1]) sage: one = Hom(F, F).identity() sage: Z = FPModule(one) sage: Z.is_trivial() True sage: FPModule(E.free_graded_module([0, 1])) Free graded left module on 2 generators over The exterior algebra of rank 2 over Rational Field
- an_element(n=None)#
An element of this module.
This function chooses deterministically an element, i.e. the output depends only on the module and its input
n
.INPUT:
n
– (optional) the degree of the element to construct
OUTPUT:
A module element of the given degree.
EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A2 = SteenrodAlgebra(2, profile=(3,2,1)) sage: M = FPModule(A2, [0,2,4], [[0, Sq(5), Sq(3)], [Sq(7), 0, Sq(2)*Sq(1)]]) sage: [M.an_element(i) for i in range(10)] [g[0], Sq(1)*g[0], Sq(2)*g[0] + g[2], Sq(0,1)*g[0] + Sq(1)*g[2], Sq(1,1)*g[0] + Sq(2)*g[2] + g[4], Sq(2,1)*g[0] + Sq(0,1)*g[2] + Sq(1)*g[4], Sq(0,2)*g[0] + Sq(1,1)*g[2] + Sq(2)*g[4], Sq(0,0,1)*g[0] + Sq(2,1)*g[2] + Sq(0,1)*g[4], Sq(1,0,1)*g[0] + Sq(6)*g[2] + Sq(1,1)*g[4], Sq(2,0,1)*g[0] + Sq(4,1)*g[2] + Sq(2,1)*g[4]]
- basis_elements(n, verbose=False)#
Return a basis for the free module of degree
n
module elements.Note
Suppose
self
is a module over the graded algebra \(A\) with base ring \(R\). This returns a basis as a free module over \(R\).INPUT:
n
– an integerverbose
– (default:False
) a boolean to control if log messages should be emitted
OUTPUT:
A list of homogeneous module elements of degree
n
which is a basis for the free module of all degreen
module elements.EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A2 = SteenrodAlgebra(2, profile=(3,2,1)) sage: M.<m0,m2> = FPModule(A2, [0,2], [[Sq(4), Sq(2)], [0, Sq(6)]]) sage: M.basis_elements(4) (Sq(1,1)*m0, Sq(4)*m0) sage: M.basis_elements(5) (Sq(2,1)*m0, Sq(5)*m0, Sq(0,1)*m2) sage: M.basis_elements(25) () sage: M.basis_elements(0) (m0,) sage: M.basis_elements(2) (Sq(2)*m0, m2)
- change_ring(algebra)#
Change the base ring of
self
.INPUT:
algebra
– a connected graded algebra
OUTPUT:
The finitely presented module over
algebra
defined with the exact same number of generators of the same degrees and relations asself
.EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A = SteenrodAlgebra(2) sage: A2 = SteenrodAlgebra(2, profile=(3,2,1)) sage: M = FPModule(A, [0,1], [[Sq(2), Sq(1)]]) sage: N = M.change_ring(A2); N Finitely presented left module on 2 generators and 1 relation over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1]
Changing back yields the original module:
sage: N.change_ring(A) is M True
- connectivity()#
The connectivity of
self
.Since a finitely presented module over a connected algebra is in particular bounded below, the connectivity is an integer when the module is non-trivial, and \(+\infty\) when the module is trivial.
EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A = SteenrodAlgebra(2) sage: M = FPModule(A, [0,2,4], [[0, Sq(5), Sq(3)], [Sq(7), 0, Sq(2)*Sq(1)]]) sage: M.connectivity() 0 sage: G = FPModule(A, [0,2], [[1,0]]) sage: G.connectivity() 2
- defining_homomorphism()#
Return the homomorphism defining
self
.self
is a finitely presented module defined as the cokernel of a map \(j: F_0 o F_1\) of free modules, and this returns that map.EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: E.<x,y> = ExteriorAlgebra(QQ) sage: M = FPModule(E, [0, 1], [[x, 1]]) sage: M.defining_homomorphism() Module morphism: From: Free graded left module on 1 generator over The exterior algebra of rank 2 over Rational Field To: Free graded left module on 2 generators over The exterior algebra of rank 2 over Rational Field Defn: g[1] |--> x*g[0] + g[1]
- element_from_coordinates(coordinates, n)#
Return the module element in degree
n
having the given coordinates with respect to the basis returned bybasis_elements()
.This function is inverse to
sage.modules.fp_graded.element.FPElement.vector_presentation()
.INPUT:
coordinates
– a vector of coordinatesn
– the degree of the element to construct
OUTPUT:
A module element of degree
n
having the given coordinates with respect to the basis returned bybasis_elements()
.EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A = SteenrodAlgebra(2) sage: M = FPModule(A, [0], [[Sq(4)], [Sq(7)], [Sq(4)*Sq(9)]]) sage: M.vector_presentation(12).dimension() 3 sage: x = M.element_from_coordinates((0,1,0), 12); x Sq(0,4)*g[0]
Applying the inverse function brings us back to the coordinate form:
sage: x.vector_presentation() (0, 1, 0)
- gen(index)#
Return the module generator with the given index.
EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A4 = SteenrodAlgebra(2, profile=(4,3,2,1)) sage: M = FPModule(A4, [0,2,3]) sage: M.generator(0) g[0] sage: N = FPModule(A4, [0, 1], [[Sq(2), Sq(1)]], names='h') sage: N.generator(1) h[1]
- generator(index)#
Return the module generator with the given index.
EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A4 = SteenrodAlgebra(2, profile=(4,3,2,1)) sage: M = FPModule(A4, [0,2,3]) sage: M.generator(0) g[0] sage: N = FPModule(A4, [0, 1], [[Sq(2), Sq(1)]], names='h') sage: N.generator(1) h[1]
- generator_degrees()#
Return the degrees of the generators for
self
.EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A4 = SteenrodAlgebra(2, profile=(4,3,2,1)) sage: N = FPModule(A4, [0, 1], [[Sq(2), Sq(1)]]) sage: N.generator_degrees() (0, 1)
- generators()#
Return the generators of
self
.EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A4 = SteenrodAlgebra(2, profile=(4,3,2,1)) sage: M = FPModule(A4, [0,0,2,3]) sage: M.generators() (g[0, 0], g[0, 1], g[2, 0], g[3, 0]) sage: N = FPModule(A4, [0, 1], [[Sq(2), Sq(1)]], names='h') sage: N.generators() (h[0], h[1]) sage: Z = FPModule(A4, []) sage: Z.generators() ()
- gens()#
Return the generators of
self
.EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A4 = SteenrodAlgebra(2, profile=(4,3,2,1)) sage: M = FPModule(A4, [0,0,2,3]) sage: M.generators() (g[0, 0], g[0, 1], g[2, 0], g[3, 0]) sage: N = FPModule(A4, [0, 1], [[Sq(2), Sq(1)]], names='h') sage: N.generators() (h[0], h[1]) sage: Z = FPModule(A4, []) sage: Z.generators() ()
- has_relations()#
Return
True
if no relations are defined, andFalse
otherwise.Note
This module is free if this function returns
False
, but a free module can have (redundant) relations.EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A2 = SteenrodAlgebra(2, profile=(3,2,1)) sage: F = FPModule(A2, [1,2]) sage: F.has_relations() False sage: M = FPModule(A2, [1,2], [[Sq(2), Sq(1)]]) sage: M.has_relations() True
A free module constructed with a redundant generator and relation:
sage: N = FPModule(A2, [0, 0], [[0, 1]]) sage: N.has_relations() True sage: # Computing a minimal presentation reveals an ....: # isomorphic module with no relations. sage: N_min = N.minimal_presentation().domain() sage: N_min.has_relations() False
- is_trivial()#
Return
True
ifself
is isomorphic to the trivial module andFalse
otherwise.EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A = SteenrodAlgebra(2) sage: A2 = SteenrodAlgebra(2, profile=(3,2,1)) sage: M = FPModule(A2, []) sage: M.is_trivial() True sage: N = FPModule(A, [1,2]) sage: N.is_trivial() False sage: P = FPModule(A, [1,2], [[1,0], [0,1]]) sage: P.is_trivial() True
- minimal_presentation(top_dim=None, verbose=False)#
Return a minimal presentation of
self
.OUTPUT:
An isomorphism \(M \to S\), where \(M\) has minimal presentation and \(S\) is
self
.EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A2 = SteenrodAlgebra(2, profile=(3,2,1)) sage: M = FPModule(A2, [0,1], [[Sq(2),Sq(1)],[0,Sq(2)],[Sq(3),0]]) sage: i = M.minimal_presentation() sage: M_min = i.domain()
i
is an isomorphism betweenM_min
andM
:sage: i.codomain() is M True sage: i.is_injective() True sage: i.is_surjective() True
There are more relations in
M
than inM_min
:sage: M.relations() (Sq(2)*g[0] + Sq(1)*g[1], Sq(2)*g[1], Sq(3)*g[0]) sage: M_min.relations() (Sq(2)*g[0] + Sq(1)*g[1], Sq(2)*g[1])
- monomial()#
Return the basis element indexed by
i
.INPUT:
i
– an element of the index set
EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: M = FPModule(SteenrodAlgebra(2), [0,1], [[Sq(4), Sq(3)]]) sage: M.monomial(0) g[0] sage: M.monomial(1) g[1]
- relation(index)#
Return the module relation of the given index.
EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A4 = SteenrodAlgebra(2, profile=(4,3,2,1)) sage: N = FPModule(A4, [0, 1], [[Sq(2), Sq(1)]]) sage: N.relation(0) Sq(2)*g[0] + Sq(1)*g[1]
- relations()#
Return the relations of
self
.EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A4 = SteenrodAlgebra(2, profile=(4,3,2,1)) sage: M = FPModule(A4, [0,2,3]) sage: M.relations() () sage: N = FPModule(A4, [0, 1], [[Sq(2), Sq(1)]]) sage: N.relations() (Sq(2)*g[0] + Sq(1)*g[1],) sage: Z = FPModule(A4, []) sage: Z.relations() ()
- resolution(k, top_dim=None, verbose=False)#
Return a free resolution of this module of length
k
.INPUT:
k
– a non-negative integertop_dim
– stop the computation at this degree (optional, defaultNone
, but required if the algebra is not finite-dimensional)verbose
– (default:False
) a boolean to control if log messages should be emitted
OUTPUT:
A list of homomorphisms \([\epsilon, f_1, \ldots, f_k]\) such that
\[f_i: F_i \to F_{i-1} \text{ for } 1 < i \leq k, \qquad \epsilon: F_0 \to M,\]where each \(F_i\) is a finitely generated free module, and the sequence
\[F_k \xrightarrow{\mathit{f_k}} F_{k-1} \xrightarrow{\mathit{f_{k-1}}} \ldots \rightarrow F_0 \xrightarrow{\epsilon} M \rightarrow 0\]is exact. Note that the 0th element in this list is the map \(\epsilon\), and the rest of the maps are between free modules.
EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: E.<x,y> = ExteriorAlgebra(QQ) sage: M = FPModule(E, [0], [[x], [y]]) sage: res = M.resolution(3); res [Module morphism: From: Free graded left module on 1 generator over The exterior algebra of rank 2 over Rational Field To: Finitely presented left module on 1 generator and 2 relations over The exterior algebra of rank 2 over Rational Field Defn: g[0] |--> g[0], Module morphism: From: Free graded left module on 2 generators over The exterior algebra of rank 2 over Rational Field To: Free graded left module on 1 generator over The exterior algebra of rank 2 over Rational Field Defn: g[1, 0] |--> x*g[0] g[1, 1] |--> y*g[0], Module morphism: From: Free graded left module on 3 generators over The exterior algebra of rank 2 over Rational Field To: Free graded left module on 2 generators over The exterior algebra of rank 2 over Rational Field Defn: g[2, 0] |--> x*g[1, 0] g[2, 1] |--> y*g[1, 0] + x*g[1, 1] g[2, 2] |--> y*g[1, 1], Module morphism: From: Free graded left module on 4 generators over The exterior algebra of rank 2 over Rational Field To: Free graded left module on 3 generators over The exterior algebra of rank 2 over Rational Field Defn: g[3, 0] |--> x*g[2, 0] g[3, 1] |--> y*g[2, 0] + x*g[2, 1] g[3, 2] |--> y*g[2, 1] + x*g[2, 2] g[3, 3] |--> y*g[2, 2]] sage: all((res[i] * res[i+1]).is_zero() for i in range(len(res)-1)) True sage: e = SymmetricFunctions(QQ).e() sage: M = FPModule(e, [0], [[e[2]+e[1,1]], [e[1,1]]]) sage: res = M.resolution(3, top_dim=10) sage: all((res[i] * res[i+1]).is_zero() for i in range(2)) True sage: res[-1].domain().is_trivial() True sage: M = FPModule(e, [0,2], [[e[2]+e[1,1], 0], [e[2,1], e[1]], [0, e[1,1]]]) sage: res = M.resolution(3, top_dim=10) sage: all((res[i] * res[i+1]).is_zero() for i in range(2)) True sage: res[-1].domain().is_trivial() True sage: A2 = SteenrodAlgebra(2, profile=(3,2,1)) sage: M = FPModule(A2, [0,1], [[Sq(2), Sq(1)]]) sage: M.resolution(0) [Module morphism: From: Free graded left module on 2 generators over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1] To: Finitely presented left module on 2 generators and 1 relation over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1] Defn: g[0] |--> g[0] g[1] |--> g[1]] sage: res = M.resolution(4, verbose=True) Computing f_1 (1/4) Computing f_2 (2/4) Computing using the profile: (2, 1) Resolving the kernel in the range of dimensions [2, 8]: 2 3 4 5 6 7 8. Computing f_3 (3/4) Computing using the profile: (2, 1) Resolving the kernel in the range of dimensions [8, 14]: 8 9 10 11 12 13 14. Computing f_4 (4/4) Computing using the profile: (2, 1) Resolving the kernel in the range of dimensions [9, 16]: 9 10 11 12 13 14 15 16. sage: len(res) 5 sage: res [Module morphism: From: Free graded left module on 2 generators over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1] To: Finitely presented left module on 2 generators and 1 relation over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1] Defn: g[0] |--> g[0] g[1] |--> g[1], Module morphism: From: Free graded left module on 1 generator over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1] To: Free graded left module on 2 generators over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1] Defn: g[2] |--> Sq(2)*g[0] + Sq(1)*g[1], Module morphism: From: Free graded left module on 1 generator over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1] To: Free graded left module on 1 generator over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1] Defn: g[8] |--> Sq(3,1)*g[2], Module morphism: From: Free graded left module on 2 generators over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1] To: Free graded left module on 1 generator over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1] Defn: g[9] |--> Sq(1)*g[8] g[10] |--> Sq(2)*g[8], Module morphism: From: Free graded left module on 2 generators over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1] To: Free graded left module on 2 generators over sub-Hopf algebra of mod 2 Steenrod algebra, milnor basis, profile function [3, 2, 1] Defn: g[10] |--> Sq(1)*g[9] g[12] |--> Sq(0,1)*g[9] + Sq(2)*g[10]] sage: for i in range(len(res)-1): ....: assert (res[i] * res[i+1]).is_zero(), 'the result is not a complex'
We construct \(\GF{3}\) as a \(\ZZ\)-module (with trivial grading concentrated in degree 0) and compute its resolution:
sage: E = ExteriorAlgebra(ZZ, 0) sage: M = FPModule(E, [0], [[3]]) sage: res = M.resolution(3) sage: res [Module morphism: From: Free graded left module on 1 generator over The exterior algebra of rank 0 over Integer Ring To: Finitely presented left module on 1 generator and 1 relation over The exterior algebra of rank 0 over Integer Ring Defn: g[0] |--> g[0], Module endomorphism of Free graded left module on 1 generator over The exterior algebra of rank 0 over Integer Ring Defn: g[0] |--> 3*g[0], Module morphism: From: Free graded left module on 0 generators over The exterior algebra of rank 0 over Integer Ring To: Free graded left module on 1 generator over The exterior algebra of rank 0 over Integer Ring, Module endomorphism of Free graded left module on 0 generators over The exterior algebra of rank 0 over Integer Ring]
- submodule_inclusion(spanning_elements)#
Return the inclusion morphism of the submodule of
self
spanned by the given elements.INPUT:
spanning_elements
– an iterable of elements
OUTPUT:
The inclusion of the submodule into this module.
Because a submodule of a finitely presented module need not be finitely presented, this method will only work if the underlying algebra is finite-dimensional. Indeed, the current implementation only works if the algebra has a
top_class
method, which gets used insage.modules.fp_graded.morphism._resolve_kernel()
.EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A2 = SteenrodAlgebra(2, profile=(3,2,1)) sage: M = FPModule(A2, [0,1], [[Sq(2),Sq(1)]]) sage: i = M.submodule_inclusion([M.generator(0)]) sage: i.codomain() is M True sage: i.is_injective() True sage: i.domain().generator_degrees() (0,) sage: i.domain().relations() (Sq(3)*g[0],)
- suspension(t)#
Return the suspension of
self
by degreet
.INPUT:
t
– an integer degree by which the module is suspended
OUTPUT:
A module which is identical to this module by a shift of degrees by the integer
t
.EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A = SteenrodAlgebra(2) sage: A2 = SteenrodAlgebra(2, profile=(3,2,1)) sage: Y = FPModule(A2, [0], [[Sq(1)]]) sage: X = Y.suspension(4) sage: X.generator_degrees() (4,) sage: X.relations() (Sq(1)*g[4],) sage: M = FPModule(A, [2,3], [[Sq(2), Sq(1)], [0, Sq(2)]]) sage: Q = M.suspension(1) sage: Q.generator_degrees() (3, 4) sage: Q.relations() (Sq(2)*g[3] + Sq(1)*g[4], Sq(2)*g[4]) sage: Q = M.suspension(-3) sage: Q.generator_degrees() (-1, 0) sage: Q = M.suspension(0) sage: Q.generator_degrees() (2, 3)
- vector_presentation(n, verbose=False)#
Return a free module isomorphic to the free module of module elements of degree
n
.INPUT:
n
– the degree of the presentation
OUTPUT:
A vector space.
See also
EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: A = SteenrodAlgebra(2) sage: M = FPModule(A, [0,2,4], [[Sq(4),Sq(2),0]]) sage: V = M.vector_presentation(4) sage: V.dimension() 3 sage: len(M.basis_elements(4)) 3
- zero()#
Return the zero element.
EXAMPLES:
sage: from sage.modules.fp_graded.module import FPModule sage: M = FPModule(SteenrodAlgebra(2), [0,1], [[Sq(4), Sq(3)]]) sage: M.zero() 0