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Baer–Specker group

An example of an infinite Abelian group which is a building block in the structure theory of such groups.

Definition

The Baer-Specker group is the group B = Z^N of all integer sequences with componentwise addition, that is, the direct product of countably many copies of Z.

Properties

Reinhold Baer proved in 1937 that this group is not free abelian; Specker proved in 1950 that every countable subgroup of B is free abelian.

See also

• Slender group

References

• Template:Cite book

Conductor of an abelian variety

In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is. It is connected to the ramification in the field generated by the division points.

For an abelian variety A defined over a field F with ring of integers R, consider the Néron model of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme over

Spec(R)

(cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism

Spec(F) → Spec(R)

gives back A. Let A⁰ denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a residue field k, A⁰_k is a group variety over k, hence an extension of an abelian variety by a linear group. This linear group is an extension of a torus by a unipotent group. Let u be the dimension of the unipotent group and t the dimension of the torus. The order of the conductor is

\[ f = 2u + t + \delta , \, \]

where δ is a measure of wild ramification.

Properties

• If A has good reduction then f = u = t = δ = 0.
• If A has semistable reduction or, more generally, acquires semistable reduction over a Galois extension of F of degree prime to p, the residue characteristic, then δ = 0.
• If p > 2d + 1, where d is the dimension of A, then δ = 0.

References

• Template:Cite book
• Template:Cite journal

Essential subgroup

A subgroup that determines much of the structure of its containing group. The concept may be generalized to essential submodules.

Definition

A subgroup \(S\) of a (typically abelian) group \(G\) is said to be essential if whenever H is a non-trivial subgroup of G, the intersection of S and H is non-trivial: here "non-trivial" means "containing an element other than the identity".

References

• Template:Cite book

Hall–Littlewood polynomials

In mathematics, the Hall–Littlewood polynomials encode combinatorial data relating to the representations of the general linear group. They are named for Philip Hall and Dudley E. Littlewood.

See also

• Littlewood polynomial

References

• Template:Cite book
• Template:Cite journal

External links

• Template:MathWorld

Manin obstruction

In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction is attached to a geometric object X which measures the failure of the Hasse principle for X: that is, if the value of the obstruction is non-trivial, then X may have points over all local fields but not over a global field.

For abelian varieties the Manin obstruction is just the Tate-Shafarevich group and fully accounts for the failure of the local-to-global principle. There are however examples, due to Skorobogatov, of varieties with trivial Manin obstruction which have points everywhere locally and yet no global points.

References

• Serge Lang. "Survey of Diophantine geometry". (Springer-Verlag, 1997) ISBN 3-540-61223-8. Zbl 0869.11051. pp.250–258.
• Alexei Skorobogatov (1999). "Beyond the Manin obstruction" (with Appendix A by S. Siksek: 4-descent). Inventiones Mathematicae 135 no.2 (1999) 399–424. DOI 10.1007/s002220050291. Zbl 0951.14013.
• Alexei Skorobogatov (2001). "Torsors and rational points". Cambridge Tracts in Mathematics 144 (Cambridge: Cambridge University Press, 2001). ISBN 0-521-80237-7. Zbl 0972.14015. pp.1–7,112.

Pinch point

A pinch point or cuspidal point is a type of singular point on an algebraic surface. It is one of the three types of ordinary singularity of a surface.

The equation for the surface near a pinch point may be put in the form

\[ f(u,v,w) = u^2 - vw^2 + [4] \, \]

where [4] denotes terms of degree 4 or more.

References

• Template:Cite book

Residual property

In the mathematical field of group theory, a group is residually X (where X is some property of groups) if it "can be recovered from groups with property X".

Formally, a group G is residually X if for every non-trivial element g there is a homomorphism h from G to a group with property X such that \(h(g)\neq e\).

More categorically, a group is residually X if it embeds into its pro-X completion (see profinite group, pro-p group), that is, the inverse limit of \(\phi\colon G \to H\) where H is a group with property X.

Examples

Important examples include:

• Residually finite
• Residually nilpotent
• Residually solvable
• Residually free

References

• Template:Cite book

Stably free module

A module which is close to being free.

Definition

A module M over a ring R is stably free if there exist free modules F and G over R such that

\[ M \oplus F = G . \, \]

Properties

• A projective module is stably free if and only if it possesses a finite free resolution.

See also

• Free object

References

• Template:Cite book