Algebraic sets, radicals and coordinate groups over a group #
Let H be a group receiving the coefficients G through ι : G →* H. A system of equations
over G in the variables X is a subset S ⊆ G[X], and its solutions in H form the
algebraic set V_H(S) ⊆ H^X. Conversely the equations vanishing on a set of points
Y ⊆ H^X form a normal subgroup I_H(Y) of G[X], the ideal of Y. These two operations are
the upper and lower polars of the relation eval ι v s = 1, so they form an antitone Galois
connection. The radical Rad_H(S) = I_H(V_H(S)) of S is the largest
system with the same solutions as S; it contains the normal closure of S, generally strictly.
A G-homomorphism from a quotient G[X] ⧸ N to H is the same as a point of H^X at which
every equation of N vanishes. Quotienting G[X] by the radical gives the coordinate group
G_{R(S)} of S, while quotienting by the normal closure gives the group G_S presented by the
system itself; there is a surjection G_S →* G_{R(S)} compatible with the coefficients. When ι
is injective and S has a solution in H, both coefficient maps are injective, so G_{R(S)} is a
G-group.
The radical genuinely depends on the group H in which the solutions are taken, and differs from
the normal closure: over a torsion-free H with injective coefficients, the radical of the
one-variable system x ^ n = 1 (n ≠ 0) is the normal closure of x, which does not lie in the
normal closure of x ^ n when n ≠ 1.
Main definitions #
TauCeti.GroupAlgGeom.V ι S: the algebraic setV_H(S)of a systemS ⊆ G[X].TauCeti.GroupAlgGeom.I ι Y: the idealI_H(Y), a normal subgroup ofG[X].TauCeti.GroupAlgGeom.Rad ι S: the radicalRad_H(S) = I_H(V_H(S)).TauCeti.GroupAlgGeom.CoordGroup ι S: the coordinate groupG_{R(S)} = G[X] ⧸ Rad_H(S).TauCeti.GroupAlgGeom.SystemGroup G S: the groupG_S = G[X] ⧸ ncl(S)of the systemS.TauCeti.GroupAlgGeom.SystemGroup.toCoordGroup ι S: the surjectionG_S →* G_{R(S)}.
Main results #
TauCeti.GroupAlgGeom.galoisConnection_V_I:VandIform an antitone Galois connection.TauCeti.GroupAlgGeom.V_normalClosure,TauCeti.GroupAlgGeom.normalClosure_le_Rad: a system and its normal closure have the same solutions, soncl(S) ≤ Rad_H(S).TauCeti.GroupAlgGeom.V_I_V:V(I(V(S))) = V(S).TauCeti.GroupAlgGeom.preimage_wordMap_V: inverse images of algebraic sets under word maps are algebraic.TauCeti.GroupAlgGeom.exists_finite_V_eq_of_finite,TauCeti.GroupAlgGeom.exists_subsingleton_V_eq: with finitely many points, every system has a finite subsystem with the same solutions; with at most one point, for instance with no variables, a single equation suffices.TauCeti.GroupAlgGeom.I_singleton_comp: the ideal of a point with coordinatesι (g x)is the normal closure of the equationsx (g x)⁻¹, for injectiveι.TauCeti.GroupAlgGeom.GX.quotientHomEquiv:G-homomorphisms fromG[X] ⧸ NtoHcorrespond to the points ofV_H(N).TauCeti.GroupAlgGeom.CoordGroup.homEquiv,TauCeti.GroupAlgGeom.SystemGroup.homEquiv:G-homomorphisms fromG_{R(S)}or fromG_StoHcorrespond to the solutions ofSinH.TauCeti.GroupAlgGeom.CoordGroup.coeff_injective:G_{R(S)}is aG-group whenShas a solution in aG-group.TauCeti.GroupAlgGeom.Rad_pow_eq_normalClosure,TauCeti.GroupAlgGeom.normalClosure_pow_ne_normalClosure: over a torsion-freeHwith injective coefficients, the radical ofx ^ n(n ≠ 0) is the normal closure ofx, which strictly contains that ofx ^ nwhenn ≠ 1.
References #
- G. Baumslag, A. Myasnikov, V. Remeslennikov, Algebraic geometry over groups I. Algebraic sets and ideal theory, J. Algebra 219 (1999), §1 and §§3.1–3.4.
- O. Kharlampovich, A. Myasnikov, Implicit function theorem over free groups, J. Algebra 290 (2005), §2.4.
Algebraic sets and ideals #
Algebraic sets and ideals form an antitone Galois connection between systems of equations in
G[X] and sets of points in H^X.
The inverse image of a solution set under the tuple word map is the solution set of the image system.
A word map sends solutions to solutions of the inverse-image system.
A tuple solves S exactly when its word map solves the inverse-image system.
The singleton of a tuple of coefficients is the solution set of the equations specifying all its coordinates.
Finite subsystems #
A homomorphism m : G[X] →* G'[Y] together with a map of points θ compatible with
evaluation carries finite subsystems of m(S) with the same solutions back to finite subsystems
of S with the same solutions.
If there are only finitely many points X → H, every system is equivalent to a finite
subsystem: one equation failing at each non-solution suffices.
If there is at most one point X → H, for instance when there are no variables, a single
equation suffices: a system with a solution is equivalent to the empty subsystem, and a system
without one is equivalent to any of its equations that fails at the point.
Radicals #
For injective coefficients, the ideal of a point ι ∘ g with coordinates in the image of G
is the normal closure of the equations x (g x)⁻¹, x ∈ X.
Quotients of G[X] #
A G-homomorphism G[X] ⧸ N → H, composed with the quotient map, is evaluation at its values
on the variables.
Evaluation at a point v at which every equation of N vanishes, as a G-homomorphism
G[X] ⧸ N → H.
Instances For
The universal property of a quotient G[X] ⧸ N: G-homomorphisms G[X] ⧸ N → H correspond
to the points of H^X at which every equation of N vanishes, by restriction to the variables and
by evaluation.
Instances For
Evaluation at the point corresponding to a G-homomorphism φ : G[X] ⧸ N → H is φ
composed with the quotient map.
A G-homomorphism out of G[X] ⧸ N is determined by its values on the variables.
If ι is injective and the equations of N have a common zero in H, the coefficients
embed in G[X] ⧸ N.
A surjective G-homomorphism π : G[X] →* K induces a G-isomorphism G[X] ⧸ N → K from
the quotient by its kernel N.
Coordinate groups and the group of a system #
The universal property of the coordinate group: G-homomorphisms G_{R(S)} → H correspond
to the solutions of S in H, by restriction to the variables and by evaluation.
Instances For
A G-homomorphism out of G_{R(S)} is determined by its values on the variables.
If ι is injective and the system S has a solution in H, the coefficients embed in the
coordinate group G_{R(S)}.
The coordinate group G_{R(S)} of a system with a solution in a G-group H, as a
G-group.
Instances For
A tuple of words maps solutions of S to solutions of T precisely when its word map
does so at every solution.
A word map between algebraic sets pulls equations vanishing on the target back to equations vanishing on the source.
A word map between algebraic sets induces a coefficient-preserving homomorphism of coordinate groups in the opposite direction.
Instances For
Evaluating the induced coordinate-group homomorphism at a point is the original word map of that point.
A coefficient-preserving word map induces a homomorphism between coordinate groups.
Instances For
Precomposition with the coordinate-group word map corresponds to the word map on tuples under the point–homomorphism equivalence.
The word homomorphism σ maps the radical of the inverse-image system into the
radical of the original system.
The universal property of G_S: G-homomorphisms G_S → H correspond to the solutions of
S in H, by restriction to the variables and by evaluation.
Instances For
A G-homomorphism out of G_S is determined by its values on the variables.
If ι is injective and the system S has a solution in H, the coefficients embed in
G_S.
Restricting a G-homomorphism G_{R(S)} → H along G_S → G_{R(S)} gives the same solution
of S.
The radical is not the normal closure #
Over a torsion-free group H with injective coefficients, the radical of the one-variable
equation x ^ n = 1 (n ≠ 0) is the normal closure of x.
Over G = F(a, b) with one variable x, the radical of x² is the normal closure of x,
which is not the normal closure of x².
The radical depends on the group in which solutions are taken: over G = F(a, b), the
variable x lies in the radical of x² over G, but not over G × ℤ/2, where (1, 1) is a
nontrivial solution of x² = 1.