Hom bundle¶

This module implements the HomBundle class, for vector bundles constructed as homomorphism sheaves between two vector bundles.

The class inherits from the VectorBundle class, but sections, either local or global, are displayed as matrices.

EXAMPLES

sage: from vector_bundle import VectorBundle, trivial_bundle, savin_bundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x^5 - 1)

We construct a vector bundle of rank 2 and degree 4:

sage: F = VectorBundle(K, 3 * K.places_infinite()[0].divisor())
sage: F1 = VectorBundle(K, 2 * K.places_finite()[0].divisor())
sage: F2 = VectorBundle(K, 2 * K.places_finite()[1].divisor())
sage: V = savin_bundle(K, 2, 4, F, F1, F2); V.h0()
[(1, 0), (2*x, 1)]

We construct the HomBundle from $\mathcal{O}_X^2$ to V. Its global sections should represent linear maps from $k^2$ to $H^0(V)$ , where $k$ is the constant field of $K$ :

sage: domain = trivial_bundle(K).direct_sum_repeat(2)
sage: hom_bundle = domain.hom(V); hom_bundle.h0()
[
[1 0]  [0 1]  [2*x   0]  [  0 2*x]
[0 0], [0 0], [  1   0], [  0   1]
]

class vector_bundle.hom_bundle.EndBundle(bundle)¶

Bases: HomBundle

Vector bundles representing endomorphism sheaves of vector bundles.

EXAMPLES:

sage: from vector_bundle import trivial_bundle, canonical_bundle
sage: F.<x> = FunctionField(GF(7))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^4 - x^-2 - 1)
sage: triv = trivial_bundle(K)
sage: can = canonical_bundle(K)
sage: triv.direct_sum(can).end()
Endomorphism bundle of Vector bundle of rank 2 over Function field in y defined by y^4 + (6*x^2 + 6)/x^2

global_algebra()¶

Return self.h0() as a k-algebra in which computations may be done. Also return maps to and from the algebra.

EXAMPLES:

sage: from vector_bundle import trivial_bundle, canonical_bundle
sage: F.<x> = FunctionField(GF(7))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^4 - x^-2 - 1)
sage: triv = trivial_bundle(K)
sage: can = canonical_bundle(K)
sage: end = triv.direct_sum(can).end()
sage: A, to_A, from_A = end.global_algebra(); A
Finite-dimensional algebra of degree 4 over Finite Field of size 7
sage: h0 = end.h0()

class vector_bundle.hom_bundle.HomBundle(domain, codomain)¶

Bases: VectorBundle

Vector bundles representing homomorphism sheaves of vector bundles.

EXAMPLES

sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 + x + 2)
sage: ideals = [P.prime_ideal() for P in K.places_finite()[:2]]
sage: g_finite = matrix([[1, x], [2, y]])
sage: g_infinite = matrix([[x, 1], [2, y]])
sage: V1 = VectorBundle(K, ideals, g_finite, g_infinite)
sage: V2 = VectorBundle(K, K.places_infinite()[0].divisor())
sage: V = V1.hom(V2); V
Homomorphism bundle from Vector bundle of rank 2 over Function field in y defined by y^2 + x + 2 to Vector bundle of rank 1 over Function field in y defined by y^2 + x + 2

basis_finite()¶

Return basis of the finite lattice of the hom bundle.

OUTPUT:

The basis elements are represented as matrices.

EXAMPLES

sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 + x + 2)
sage: ideals = [P.prime_ideal() for P in K.places_finite()[:2]]
sage: g_finite = matrix([[1, x], [2, y]])
sage: g_infinite = matrix([[x, 1], [2, y]])
sage: V1 = VectorBundle(K, ideals, g_finite, g_infinite)
sage: O = K.maximal_order()
sage: V2 = VectorBundle(K, K.places_finite()[2].prime_ideal(),1,x^2)
sage: V = V1.hom(V2)
sage: V.basis_finite() 
[[(x/(x^2 + x + 2))*y + (x + 2)/(x^2 + x + 2)   (x/(x^2 + x + 2))*y + 2*x^2/(x^2 + x + 2)],
 [(2/(x^2 + x + 2))*y + x/(x^2 + x + 2) (2/(x^2 + x + 2))*y + x/(x^2 + x + 2)]]

basis_infinite()¶

Return basis of the infinite lattice of the hom bundle.

OUTPUT:

The basis elements are represented as matrices.

EXAMPLES

sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 + x + 2)
sage: ideals = [P.prime_ideal() for P in K.places_finite()[:2]]
sage: g_finite = matrix([[1, x], [2, y]])
sage: g_infinite = matrix([[x, 1],[2, y]])
sage: V1 = VectorBundle(K, ideals,g_finite,g_infinite)
sage: O = K.maximal_order()
sage: V2 = VectorBundle(K, K.places_finite()[2].prime_ideal(),1,x^2)
sage: V = V1.hom(V2)
sage: V.basis_infinite()
[[(x^2/(x^3 + 2*x^2 + 1))*y + (x^4 + 2*x^3)/(x^3 + 2*x^2 + 1) (x^3/(x^3 + 2*x^2 + 1))*y + 2*x^2/(x^3 + 2*x^2 + 1)],
[(2*x^3/(x^3 + 2*x^2 + 1))*y + x^2/(x^3 + 2*x^2 + 1) (2*x^4/(x^3 + 2*x^2 + 1))*y + x^3/(x^3 + 2*x^2 + 1)]]

basis_local(place)¶

Return basis of the infinite lattice of the hom bundle.

OUTPUT:

The basis elements are represented as matrices.

EXAMPLES

sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 + x + 2)
sage: ideals = [P.prime_ideal() for P in K.places_finite()[:2]]
sage: g_finite = matrix([[1,x],[2,y]])
sage: g_infinite = matrix([[x,1],[2,y]])
sage: V1 = VectorBundle(K, ideals,g_finite,g_infinite)
sage: O = K.maximal_order()
sage: V2 = VectorBundle(K, K.places_finite()[2].prime_ideal(),1,x^2)
sage: V = V1.hom(V2)
sage: place = K.places_finite()[0]
sage: V.basis_local(place)
[[(1/(x^2 + x + 2))*y + (x + 2)/(x^3 + x^2 + 2*x) (1/(x^2 + x + 2))*y + 2*x/(x^2 + x + 2)],
[(2/(x^2 + x + 2))*y + x/(x^2 + x + 2) (2/(x^2 + x + 2))*y + x/(x^2 + x + 2)]]
sage: all([all([(mat * g_finite)[0, j].valuation(place) >= (V2._ideals[0] * V1._ideals[j]**-1).divisor().valuation(place) for j in range(2)]) for mat in V.basis_local(place)])
True

codomain()¶

Return the codomain of self

EXAMPLES

sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: L1 = VectorBundle(F, x.poles()[0].divisor())
sage: L2 = VectorBundle(F, x.zeros()[0].divisor())
sage: V = L1.hom(L2); V.codomain() == L2
True

conorm(K)¶

Return the conorm of a hom bundle.

It is the same thing as the hom bundle of the conorms of its domain and codomain.

EXAMPLES

sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 + x + 2)
sage: ideals = [P.prime_ideal() for P in F.places_finite()[:2]]
sage: g_finite = matrix([[1, x], [2, x]])
sage: g_infinite = matrix([[x, 1], [2, x]])
sage: V1 = VectorBundle(F, ideals, g_finite, g_infinite)
sage: ideals = [P.prime_ideal() for P in F.places_finite()[1:3]]
sage: g_finite = matrix([[0, x], [1, 1/x]])
sage: g_infinite = matrix([[x, 2*x^2], [2, 1]])
sage: V2 = VectorBundle(F, ideals, g_finite, g_infinite)
sage: V1.conorm(K).hom(V2.conorm(K)) == V1.hom(V2).conorm(K)
True

coordinates_in_h0(mat)¶

Return the coordinates in the basis of self.h0() of the matrix mat EXAMPLES:

sage: from vector_bundle import trivial_bundle, canonical_bundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^4 - x^-2 - 1)
sage: triv = trivial_bundle(K)
sage: can = canonical_bundle(K)
sage: V1 = triv.direct_sum(can)
sage: V2 = can.direct_sum(triv)
sage: hom = V1.hom(V2)
sage: hom.coordinates_in_h0(matrix([[0, 1], [1, 0]]))
(1, 1, 0, 0)

domain()¶

Return the domain of self

EXAMPLES

sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: L1 = VectorBundle(F, x.poles()[0].divisor())
sage: L2 = VectorBundle(F, x.zeros()[0].divisor())
sage: V = L1.hom(L2); V.domain() == L1
True

h0()¶

Returns the 0th cohomology group of the hom bundle. The global sections are output in matrix form, they are the global homomorphisms from self._domain to self._codomain.

EXAMPLES

sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(3))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 + x + 2)
sage: ideals = [P.prime_ideal() for P in K.places_finite()[:2]]
sage: g_finite = matrix([[1, x], [2, y]])
sage: g_infinite = matrix([[x, 1], [2, y]])
sage: V1 = VectorBundle(K, ideals, g_finite, g_infinite)
sage: O = K.maximal_order()
sage: V2 = VectorBundle(K, K.places_finite()[2].prime_ideal(),1,x^2)
sage: V = V1.hom(V2)
sage: h0 = V.h0(); len(h0) == V.degree() + (1-K.genus())*V.rank()
True
sage: all([all([(mat * g_finite)[0, j] in V2._ideals[0] * V1._ideals[j]**-1 for j in range(2)]) for mat in h0])
True
sage: O_infinity = K.maximal_order_infinite()
sage: all([all([a in O_infinity for a in (x**-2 * mat * g_infinite).list()]) for mat in h0])
True

hom(other)¶

Return the Hom bundle from self to other.

If other is a vector bundle, this is the hom bundle from self._codomain to self._domain.tensor_product(other). If other is also a hom bundle, this is the hom bundle from self.codomain().tensor_product(other.domain()) to self.domain().tensor_product(other.codomain()

EXAMPLES:

sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(7))
sage: L1 = VectorBundle(F, 2*x.zeros()[0].divisor())
sage: L2 = VectorBundle(F, -3*x.poles()[0].divisor())
sage: hom = L1.hom(L2)
sage: T = L1.tensor_product(L2)
sage: hom.hom(hom) == T.end()
True

image(f)¶

Return an image of global homomorphism f which is an element of $H^0(\mathrm{self})$ .

That is, a vector bundle $V$ together with an injective morphism of $V$ into self.codomain such that the image of $V$ in self.codomain is also the image of f.

EXAMPLES

sage: from vector_bundle import trivial_bundle, canonical_bundle
sage: F.<x> = FunctionField(GF(7))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x^3 - x)
sage: triv = trivial_bundle(K)
sage: can = canonical_bundle(K)
sage: V1 = triv.direct_sum(can)
sage: V2 = can.direct_sum(triv)
sage: hom = V1.hom(V2)
sage: image, map = hom.image(matrix(K, [[0, 1], [0, 0]]))
sage: image.isomorphism_to(can) is not None
True
sage: image.hom(V2).coordinates_in_h0(map)
(1, 0)

is_isomorphism(f)¶

Check if f is an isomorphism from self.domain() to self.codomain()

EXAMPLES:

sage: from vector_bundle import atiyah_bundle, canonical_bundle
sage: F.<x> = FunctionField(GF(7))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x^3 - x)
sage: V = atiyah_bundle(K, 2, 0)
sage: W = atiyah_bundle(K, 2, 0, canonical_bundle(K))
sage: isom = x^2/(x^2 + 3) * identity_matrix(K, 2)
sage: V.hom(W).is_isomorphism(isom)
True

kernel(f)¶

Return a kernel of global homomorphism f which is an element of $H^0(\mathrm{self})$ .

That is, a vector bundle $V$ together with an injective morphism of $V$ into self.domain such that the image of $V$ in self.domain is the kernel of f.

EXAMPLES

sage: from vector_bundle import trivial_bundle, canonical_bundle
sage: F.<x> = FunctionField(GF(7))
sage: R.<y> = F[]
sage: K.<y> = F.extension(y^2 - x^3 - x)
sage: triv = trivial_bundle(K)
sage: can = canonical_bundle(K)
sage: V1 = triv.direct_sum(can)
sage: V2 = can.direct_sum(triv)
sage: hom = V1.hom(V2)
sage: kernel, map = hom.kernel(matrix(K, [[0, 1], [0, 0]]))
sage: kernel.isomorphism_to(triv) is not None
True
sage: kernel.hom(V1).coordinates_in_h0(map)
(1, 0)

tensor_product(other)¶

Return the tensor product of a hom bundle and a vector bundle. This is the same thing as self._domain.hom(self._codomain.tensor_product(other))

EXAMPLES:

sage: from vector_bundle import VectorBundle
sage: F.<x> = FunctionField(GF(7))
sage: L1 = VectorBundle(F, 2*x.zeros()[0].divisor())
sage: L2 = VectorBundle(F, -3*x.poles()[0].divisor())
sage: hom = L1.hom(L2)
sage: hom2 = L1.hom(L2.tensor_product(L1))
sage: hom.tensor_product(L1) == hom2
True

Hom bundle¶

Table of Contents

Previous topic

Next topic

This Page