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Revision as of 07:52, 26 August 2013 by Richard Pinch (talk | contribs) (Starr article: Manin obstruction)
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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 = ZN 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.

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Descendant subgroup

A subgroup of a group for which there is an descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predecessor.

The series may be infinite. If the series is finite, then the subgroup is subnormal.

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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

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.

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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

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:

References

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

References

How to Cite This Entry:
Richard Pinch/sandbox-WP. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-WP&oldid=30250