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When S is finite, its completion is also finite, and has the smallest number of elements among all finite complete lattices containing S. [ 12 ] The partially ordered set S is join-dense and meet-dense in the Dedekind–MacNeille completion; that is, every element of the completion is a join of some set of elements of S , and is also the meet ...
A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers. [1] A simple example is , the set of natural numbers.
A Dedekind domain can also be characterized in terms of homological algebra: an integral domain is a Dedekind domain if and only if it is a hereditary ring; that is, every submodule of a projective module over it is projective. Similarly, an integral domain is a Dedekind domain if and only if every divisible module over it is injective. [3]
However, over a Dedekind domain the ideal class group is the only obstruction, and the structure theorem generalizes to finitely generated modules over a Dedekind domain with minor modifications. There is still a unique torsion part, with a torsionfree complement (unique up to isomorphism), but a torsionfree module over a Dedekind domain is no ...
Some of the most important applications of the conductor arise when B is a Dedekind domain and B /A is finite. For example, B can be the ring of integers of a number field and A a non-maximal order. Or, B can be the affine coordinate ring of a smooth projective curve over a finite field and A the affine
In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations. Linear algebra is the branch of mathematics concerning linear equations such as:
Kollár (2007, example 3.4.4, page 121) gives the following example showing that one cannot expect a sufficiently good resolution procedure to commute with products. If f : A → B is the blowup of the origin of a quadric cone B in affine 3-space, then f × f : A × A → B × B cannot be produced by an étale local resolution procedure ...
In algebraic geometry, the Néron model (or Néron minimal model, or minimal model) for an abelian variety A K defined over the field of fractions K of a Dedekind domain R is the "push-forward" of A K from Spec(K) to Spec(R), in other words the "best possible" group scheme A R defined over R corresponding to A K.