Commutative Differential Graded Algebras¶
An algebra is said to be graded commutative if it is endowed with a grading and its multiplication satisfies the Koszul sign convention: \(yx = (-1)^{ij} xy\) if \(x\) and \(y\) are homogeneous of degrees \(i\) and \(j\), respectively. Thus the multiplication is anticommutative for odd degree elements, commutative otherwise. Commutative differential graded algebras are graded commutative algebras endowed with a graded differential of degree 1. These algebras can be graded over the integers or they can be multi-graded (i.e., graded over a finite rank free abelian group \(\ZZ^n\)); if multi-graded, the total degree is used in the Koszul sign convention, and the differential must have total degree 1.
EXAMPLES:
All of these algebras may be constructed with the function
GradedCommutativeAlgebra()
. For most users, that will be the
main function of interest. See its documentation for many more
examples.
We start by constructing some graded commutative algebras. Generators have degree 1 by default:
sage: A.<x,y,z> = GradedCommutativeAlgebra(QQ)
sage: x.degree()
1
sage: x^2
0
sage: y*x
-x*y
sage: B.<a,b> = GradedCommutativeAlgebra(QQ, degrees = (2,3))
sage: a.degree()
2
sage: b.degree()
3
Once we have defined a graded commutative algebra, it is easy to
define a differential on it using the GCAlgebra.cdg_algebra()
method:
sage: A.<x,y,z> = GradedCommutativeAlgebra(QQ, degrees=(1,1,2))
sage: B = A.cdg_algebra({x: x*y, y: -x*y})
sage: B
Commutative Differential Graded Algebra with generators ('x', 'y', 'z') in degrees (1, 1, 2) over Rational Field with differential:
x --> x*y
y --> -x*y
z --> 0
sage: B.cohomology(3)
Free module generated by {[x*z + y*z]} over Rational Field
sage: B.cohomology(4)
Free module generated by {[z^2]} over Rational Field
We can also compute algebra generators for the cohomology in a range of degrees, and in this case we compute up to degree 10:
sage: B.cohomology_generators(10)
{1: [x + y], 2: [z]}
AUTHORS:
- Miguel Marco, John Palmieri (2014-07): initial version
-
class
sage.algebras.commutative_dga.
CohomologyClass
(x)¶ Bases:
sage.structure.sage_object.SageObject
A class for representing cohomology classes.
This just has
_repr_
and_latex_
methods which put brackets around the object’s name.EXAMPLES:
sage: from sage.algebras.commutative_dga import CohomologyClass sage: CohomologyClass(3) [3] sage: A.<x,y,z,t> = GradedCommutativeAlgebra(QQ, degrees = (2,3,3,1)) sage: CohomologyClass(x^2+2*y*z) [2*y*z + x^2]
-
representative
()¶ Return the representative of
self
.EXAMPLES:
sage: from sage.algebras.commutative_dga import CohomologyClass sage: x = CohomologyClass(sin) sage: x.representative() == sin True
-
-
sage.algebras.commutative_dga.
Differential
¶ Differential of a commutative graded algebra.
INPUT:
A
– algebra where the differential is definedim_gens
– tuple containing the image of each generator
EXAMPLES:
sage: A.<x,y,z,t> = GradedCommutativeAlgebra(QQ, degrees=(1,1,2,3)) sage: B = A.cdg_algebra({x: x*y, y: -x*y , z: t}) sage: B Commutative Differential Graded Algebra with generators ('x', 'y', 'z', 't') in degrees (1, 1, 2, 3) over Rational Field with differential: x --> x*y y --> -x*y z --> t t --> 0 sage: B.differential()(x) x*y
-
sage.algebras.commutative_dga.
DifferentialGCAlgebra
¶ A commutative differential graded algebra.
INPUT:
A
– a graded commutative algebra; that is, an instance ofGCAlgebra
differential
– a differential
As described in the module-level documentation, these are graded algebras for which oddly graded elements anticommute and evenly graded elements commute, and on which there is a graded differential of degree 1.
These algebras should be graded over the integers; multi-graded algebras should be constructed using
DifferentialGCAlgebra_multigraded
instead.Note that a natural way to construct these is to use the
GradedCommutativeAlgebra()
function and theGCAlgebra.cdg_algebra()
method.EXAMPLES:
sage: A.<x,y,z,t> = GradedCommutativeAlgebra(QQ, degrees=(3, 2, 2, 3)) sage: A.cdg_algebra({x: y*z}) Commutative Differential Graded Algebra with generators ('x', 'y', 'z', 't') in degrees (3, 2, 2, 3) over Rational Field with differential: x --> y*z y --> 0 z --> 0 t --> 0
Alternatively, starting with
GradedCommutativeAlgebra()
:sage: A.<x,y,z,t> = GradedCommutativeAlgebra(QQ, degrees=(3, 2, 2, 3)) sage: A.cdg_algebra(differential={x: y*z}) Commutative Differential Graded Algebra with generators ('x', 'y', 'z', 't') in degrees (3, 2, 2, 3) over Rational Field with differential: x --> y*z y --> 0 z --> 0 t --> 0
See the function
GradedCommutativeAlgebra()
for more examples.
-
sage.algebras.commutative_dga.
DifferentialGCAlgebra_multigraded
¶ A commutative differential multi-graded algebras.
INPUT:
A
– a commutative multi-graded algebradifferential
– a differential
EXAMPLES:
sage: A.<a,b,c> = GradedCommutativeAlgebra(QQ, degrees=((1,0), (0, 1), (0,2))) sage: B = A.cdg_algebra(differential={a: c}) sage: B.basis((1,0)) [a] sage: B.basis(1, total=True) [b, a] sage: B.cohomology((1, 0)) Free module generated by {} over Rational Field sage: B.cohomology(1, total=True) Free module generated by {[b]} over Rational Field
-
sage.algebras.commutative_dga.
Differential_multigraded
¶ Differential of a commutative multi-graded algebra.
-
sage.algebras.commutative_dga.
GCAlgebra
¶ A graded commutative algebra.
INPUT:
base
– the base fieldnames
– (optional) names of the generators: a list of strings or a single string with the names separated by commas. If not specified, the generators are named “x0”, “x1”, …degrees
– (optional) a tuple or list specifying the degrees of the generators; if omitted, each generator is given degree 1, and if bothnames
anddegrees
are omitted, an error is raised.R
(optional, default None) – the ring over which the algebra is defined: if this is specified, the algebra is defined to beR/I
.I
(optional, default None) – an ideal inR
. It is should include, among other relations, the squares of the generators of odd degree
As described in the module-level documentation, these are graded algebras for which oddly graded elements anticommute and evenly graded elements commute.
The arguments
R
andI
are primarily for use by thequotient()
method.These algebras should be graded over the integers; multi-graded algebras should be constructed using
GCAlgebra_multigraded
instead.EXAMPLES:
sage: A.<a,b> = GradedCommutativeAlgebra(QQ, degrees = (2,3)) sage: a.degree() 2 sage: B = A.quotient(A.ideal(a**2*b)) sage: B Graded Commutative Algebra with generators ('a', 'b') in degrees (2, 3) with relations [a^2*b] over Rational Field sage: A.basis(7) [a^2*b] sage: B.basis(7) []
Note that the function
GradedCommutativeAlgebra()
can also be used to construct these algebras.
-
class
sage.algebras.commutative_dga.
GCAlgebraHomset
(R, S, category=None)¶ Bases:
sage.rings.homset.RingHomset_generic
Set of morphisms between two graded commutative algebras.
Note
Homsets (and thus morphisms) have only been implemented when the base fields are the same for the domain and codomain.
EXAMPLES:
sage: A.<x,y> = GradedCommutativeAlgebra(QQ, degrees=(1,2)) sage: H = Hom(A,A) sage: H([x,y]) == H.identity() True sage: H([x,x]) == H.identity() False sage: A.<w,x> = GradedCommutativeAlgebra(QQ, degrees=(1,2)) sage: B.<y,z> = GradedCommutativeAlgebra(QQ, degrees=(1,1)) sage: H = Hom(A,B) sage: H([y,0]) Graded Commutative Algebra morphism: From: Graded Commutative Algebra with generators ('w', 'x') in degrees (1, 2) over Rational Field To: Graded Commutative Algebra with generators ('y', 'z') in degrees (1, 1) over Rational Field Defn: (w, x) --> (y, 0) sage: H([y,y*z]) Graded Commutative Algebra morphism: From: Graded Commutative Algebra with generators ('w', 'x') in degrees (1, 2) over Rational Field To: Graded Commutative Algebra with generators ('y', 'z') in degrees (1, 1) over Rational Field Defn: (w, x) --> (y, y*z)
-
identity
()¶ Construct the identity morphism of this homset.
EXAMPLES:
sage: A.<x,y> = GradedCommutativeAlgebra(QQ, degrees=(1,2)) sage: H = Hom(A,A) sage: H([x,y]) == H.identity() True sage: H([x,x]) == H.identity() False
-
zero
()¶ Construct the “zero” morphism of this homset: the map sending each generator to zero.
EXAMPLES:
sage: A.<x,y> = GradedCommutativeAlgebra(QQ, degrees=(1,2)) sage: B.<a,b,c> = GradedCommutativeAlgebra(QQ, degrees=(1,1,1)) sage: zero = Hom(A,B).zero() sage: zero(x) == zero(y) == 0 True
-
-
class
sage.algebras.commutative_dga.
GCAlgebraMorphism
(parent, im_gens, check=True)¶ Bases:
sage.rings.morphism.RingHomomorphism_im_gens
Create a morphism between two
graded commutative algebras
.INPUT:
parent
– the parent homsetim_gens
– the images, in the codomain, of the generators of the domaincheck
– boolean (default:True
); check whether the proposed map is actually an algebra map; if the domain and codomain have differentials, also check that the map respects those.
EXAMPLES:
sage: A.<x,y> = GradedCommutativeAlgebra(QQ) sage: H = Hom(A,A) sage: f = H([y,x]) sage: f Graded Commutative Algebra endomorphism of Graded Commutative Algebra with generators ('x', 'y') in degrees (1, 1) over Rational Field Defn: (x, y) --> (y, x) sage: f(x*y) -x*y
-
is_graded
(total=False)¶ Return
True
if this morphism is graded.That is, return
True
if \(f(x)\) is zero, or if \(f(x)\) is homogeneous and has the same degree as \(x\), for each generator \(x\).INPUT:
total
(optional, defaultFalse
) – ifTrue
, use the total degree to determine whether the morphism is graded (relevant only in the multigraded case)
EXAMPLES:
sage: C.<a,b,c> = GradedCommutativeAlgebra(QQ, degrees=(1,1,2)) sage: H = Hom(C,C) sage: H([a, b, a*b + 2*a]).is_graded() False sage: H([a, b, a*b]).is_graded() True sage: A.<w,x> = GradedCommutativeAlgebra(QQ, degrees=((1,0), (1,0))) sage: B.<y,z> = GradedCommutativeAlgebra(QQ, degrees=((1,0), (0,1))) sage: H = Hom(A,B) sage: H([y,0]).is_graded() True sage: H([z,z]).is_graded() False sage: H([z,z]).is_graded(total=True) True
-
sage.algebras.commutative_dga.
GCAlgebra_multigraded
¶ A multi-graded commutative algebra.
INPUT:
base
– the base fielddegrees
– a tuple or list specifying the degrees of the generatorsnames
– (optional) names of the generators: a list of strings or a single string with the names separated by commas; if not specified, the generators are namedx0
,x1
, …R
– (optional) the ring over which the algebra is definedI
– (optional) an ideal inR
; it should include, among other relations, the squares of the generators of odd degree
When defining such an algebra, each entry of
degrees
should be a list, tuple, or element of an additive (free) abelian group. Regardless of how the user specifies the degrees, Sage converts them to group elements.The arguments
R
andI
are primarily for use by theGCAlgebra.quotient()
method.EXAMPLES:
sage: A.<a,b,c> = GradedCommutativeAlgebra(QQ, degrees=((1,0), (0,1), (1,1))) sage: A Graded Commutative Algebra with generators ('a', 'b', 'c') in degrees ((1, 0), (0, 1), (1, 1)) over Rational Field sage: a**2 0 sage: c.degree(total=True) 2 sage: c**2 c^2 sage: c.degree() (1, 1)
Although the degree of
c
was defined using a Python tuple, it is returned as an element of an additive abelian group, and so it can be manipulated via arithmetic operations:sage: type(c.degree()) <class 'sage.groups.additive_abelian.additive_abelian_group.AdditiveAbelianGroup_fixed_gens_with_category.element_class'> sage: 2 * c.degree() (2, 2) sage: (a*b).degree() == a.degree() + b.degree() True
The
basis()
method and theElement.degree()
method both accept the boolean keywordtotal
. IfTrue
, use the total degree:sage: A.basis(2, total=True) [a*b, c] sage: c.degree(total=True) 2
-
sage.algebras.commutative_dga.
GradedCommutativeAlgebra
(ring, names=None, degrees=None, relations=None)¶ A graded commutative algebra.
INPUT:
There are two ways to call this. The first way defines a free graded commutative algebra:
ring
– the base field over which to worknames
– names of the generators. You may also use Sage’sA.<x,y,...> = ...
syntax to define the names. If no names are specified, the generators are namedx0
,x1
, …degrees
– degrees of the generators; if this is omitted, the degree of each generator is 1, and if bothnames
anddegrees
are omitted, an error is raised
Once such an algebra has been defined, one can use its associated methods to take a quotient, impose a differential, etc. See the examples below.
The second way takes a graded commutative algebra and imposes relations:
ring
– a graded commutative algebrarelations
– a list or tuple of elements ofring
EXAMPLES:
Defining a graded commutative algebra:
sage: GradedCommutativeAlgebra(QQ, 'x, y, z') Graded Commutative Algebra with generators ('x', 'y', 'z') in degrees (1, 1, 1) over Rational Field sage: GradedCommutativeAlgebra(QQ, degrees=(2, 3, 4)) Graded Commutative Algebra with generators ('x0', 'x1', 'x2') in degrees (2, 3, 4) over Rational Field
As usual in Sage, the
A.<...>
notation defines both the algebra and the generator names:sage: A.<x,y,z> = GradedCommutativeAlgebra(QQ, degrees=(1, 2, 1)) sage: x^2 0 sage: z*x # Odd classes anticommute. -x*z sage: z*y # y is central since it is in degree 2. y*z sage: (x*y**3*z).degree() 8 sage: A.basis(3) # basis of homogeneous degree 3 elements [y*z, x*y]
Defining a quotient:
sage: I = A.ideal(x*y) sage: AQ = A.quotient(I) sage: AQ Graded Commutative Algebra with generators ('x', 'y', 'z') in degrees (1, 2, 1) with relations [x*y] over Rational Field sage: AQ.basis(3) [y*z]
Note that
AQ
has no specified differential. This is reflected in its print representation:AQ
is described as a “graded commutative algebra” – the word “differential” is missing. Also, it has no defaultdifferential
:sage: AQ.differential() # py2 Traceback (most recent call last): ... TypeError: differential() takes exactly 2 arguments (1 given) sage: AQ.differential() # py3 Traceback (most recent call last): ... TypeError: differential() missing 1 required positional argument: 'diff'
Now we add a differential to
AQ
:sage: B = AQ.cdg_algebra({y:y*z}) sage: B Commutative Differential Graded Algebra with generators ('x', 'y', 'z') in degrees (1, 2, 1) with relations [x*y] over Rational Field with differential: x --> 0 y --> y*z z --> 0 sage: B.differential() Differential of Commutative Differential Graded Algebra with generators ('x', 'y', 'z') in degrees (1, 2, 1) with relations [x*y] over Rational Field Defn: x --> 0 y --> y*z z --> 0 sage: B.cohomology(1) Free module generated by {[z], [x]} over Rational Field sage: B.cohomology(2) Free module generated by {[x*z]} over Rational Field
We compute algebra generators for cohomology in a range of degrees. This cohomology algebra appears to be finitely generated:
sage: B.cohomology_generators(15) {1: [z, x]}
We can construct multi-graded rings as well. We work in characteristic 2 for a change, so the algebras here are honestly commutative:
sage: C.<a,b,c,d> = GradedCommutativeAlgebra(GF(2), degrees=((1,0), (1,1), (0,2), (0,3))) sage: D = C.cdg_algebra(differential={a:c, b:d}) sage: D Commutative Differential Graded Algebra with generators ('a', 'b', 'c', 'd') in degrees ((1, 0), (1, 1), (0, 2), (0, 3)) over Finite Field of size 2 with differential: a --> c b --> d c --> 0 d --> 0
We can examine
D
using both total degrees and multidegrees. Use tuples, lists, vectors, or elements of additive abelian groups to specify degrees:sage: D.basis(3) # basis in total degree 3 [d, a*c, a*b, a^3] sage: D.basis((1,2)) # basis in degree (1,2) [a*c] sage: D.basis([1,2]) [a*c] sage: D.basis(vector([1,2])) [a*c] sage: G = AdditiveAbelianGroup([0,0]); G Additive abelian group isomorphic to Z + Z sage: D.basis(G(vector([1,2]))) [a*c]
At this point,
a
, for example, is an element ofC
. We can redefine it so that it is instead an element ofD
in several ways, for instance usinggens()
method:sage: a, b, c, d = D.gens() sage: a.differential() c
Or the
inject_variables()
method:sage: D.inject_variables() Defining a, b, c, d sage: (a*b).differential() b*c + a*d sage: (a*b*c**2).degree() (2, 5)
Degrees are returned as elements of additive abelian groups:
sage: (a*b*c**2).degree() in G True sage: (a*b*c**2).degree(total=True) # total degree 7 sage: D.cohomology(4) Free module generated by {[b^2], [a^4]} over Finite Field of size 2 sage: D.cohomology((2,2)) Free module generated by {[b^2]} over Finite Field of size 2
-
sage.algebras.commutative_dga.
exterior_algebra_basis
(n, degrees)¶ Basis of an exterior algebra in degree
n
, where the generators are in degreesdegrees
.INPUT:
n
- integerdegrees
- iterable of integers
Return list of lists, each list representing exponents for the corresponding generators. (So each list consists of 0’s and 1’s.)
EXAMPLES:
sage: from sage.algebras.commutative_dga import exterior_algebra_basis sage: exterior_algebra_basis(1, (1,3,1)) [[0, 0, 1], [1, 0, 0]] sage: exterior_algebra_basis(4, (1,3,1)) [[0, 1, 1], [1, 1, 0]] sage: exterior_algebra_basis(10, (1,5,1,1)) []
-
sage.algebras.commutative_dga.
total_degree
(deg)¶ Total degree of
deg
.INPUT:
deg
- an element of a free abelian group.
In fact,
deg
could be an integer, a Python int, a list, a tuple, a vector, etc. This function returns the sum of the components ofdeg
.EXAMPLES:
sage: from sage.algebras.commutative_dga import total_degree sage: total_degree(12) 12 sage: total_degree(range(5)) 10 sage: total_degree(vector(range(5))) 10 sage: G = AdditiveAbelianGroup((0,0)) sage: x = G.gen(0); y = G.gen(1) sage: 3*x+4*y (3, 4) sage: total_degree(3*x+4*y) 7