Difference between revisions of "User:Richard Pinch/sandbox-WP"
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==References== | ==References== | ||
* {{cite book | author=Phillip A. Griffith | title=Infinite Abelian group theory | series=Chicago Lectures in Mathematics | publisher=University of Chicago Press | year=1970 | isbn=0-226-30870-7 | page=19}} | * {{cite book | author=Phillip A. Griffith | title=Infinite Abelian group theory | series=Chicago Lectures in Mathematics | publisher=University of Chicago Press | year=1970 | isbn=0-226-30870-7 | page=19}} | ||
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| + | =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 field]]s but not over a [[global field]]. | ||
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| + | For [[abelian variety|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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| + | ==References== | ||
| + | * {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | zbl=0869.11051 | pages=250–258 }} | ||
| + | * {{cite journal | author=Alexei N. Skorobogatov | title=Beyond the Manin obstruction | journal=[[Inventiones Mathematicae]] | others=Appendix A by S. Siksek: 4-descent | volume=135 | issue=2 | pages=399–424 | year=1999 | doi=10.1007/s002220050291 | zbl=0951.14013 }} | ||
| + | * {{cite book | title=Torsors and rational points | author=Alexei Skorobogatov | series=Cambridge Tracts in Mathematics | volume=144 | year=2001 | isbn=0-521-80237-7 | pages=1–7,112 | publisher=[[Cambridge University Press]] | location=Cambridge | zbl=0972.14015 }} | ||
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=Pinch point= | =Pinch point= | ||
Revision as of 07:52, 26 August 2013
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.
See also
References
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.
See also
References
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.
References
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:
- Residually finite
- Residually nilpotent
- Residually solvable
- Residually free
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
Richard Pinch/sandbox-WP. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-WP&oldid=30250