General Rings

1. The first six sections are rather elementary and deal with more general rings than local rings.

Defining

1. A very interesting way, due to Zariski, of defining a nonsingular variety intrinsically in terms of local rings is given.

Discrete Topology

1. Then the induced morphism on local rings is formally étale with respect to the discrete topology.

Algebras

1. Here, a formal scheme is admissible if it is coverable by formal spectra of topologically finitely presented R algebras whose local rings are R -flat.

Field

1. Given a field K, which may or may not be a function field, we may look for local rings in it.

Fields

1. An important class of local rings are discrete valuation rings, which are local principal ideal domains that are not fields.

Sheaves

1. Sheaves of local rings A similar, but more technical, approach to defining differentiable manifolds can be formulated using the notion of a ringed space.

Prime Ideals

1. No silly topological spaces, no prime ideals, no axiom of choice, no local rings, no fields.

Algebraic Geometry

1. This approach is strongly influenced by the theory of schemes in algebraic geometry, but uses local rings of the germs of differentiable functions.

Rarr

1. Moreover for every prime P the homomorphism f descends to homomorphisms O f -1(P) → O P, of local rings.

Completions

1. For function fields, the local fields are completions of the local rings at all points of the curve for function fields.
2. We present the modern theory of local rings: completions, dimension theory, homological methods.

Local

1. If (R, m) and (S, n) are local rings, then a local ring homomorphism from R to S is a ring homomorphism f: R → S with the property f(m)⊆ n.

Localization

1. The localization of f # is an isomorphism of local rings f # p: O f(p) → O M, p.

Rings

1. The text starts with the theory of rings and modules and standard bases with emphasis on local rings and localization.
2. Non-commutative local rings arise naturally as endomorphism rings in the study of direct sum decompositions of modules over some other rings.

Quotients

1. Localization: multiplicative sets; rings of quotients and quotient fields; local rings.

Maximal Ideal

1. All fields (and skew fields) are local rings, since {0} is the only maximal ideal in these rings.

Stalks

1. A locally ringed space is a ringed space (X, O X) such that all stalks of O X are local rings (i.e.
2. Ringed spaces are sheaves of commutative rings; especially important are the locally ringed spaces where all stalks (see below) are local rings.

Sheaf

1. It is the stalks of the sheaf that are local rings, not the collections of sections (which are rings, but in general are not close to being local).
2. In applications to geometry, for example complex manifolds and algebraic geometry, the idea of a sheaf of local rings is central.
3. One must be rather more careful in defining sheaf of local rings than with sheaf of groups or other algebraic structure.

Local Rings

1. Rather, it emerges as a sheaf of functions as a consequence of the construction (via the quotients of local rings by their maximal ideals).
2. The pair (R n, O) is an example of a locally ringed space: it is a topological space equipped with a sheaf whose stalks are each local rings.
3. A locally ringed space is a topological space X, together with a sheaf F of commutative rings on X, such that all stalks of F are local rings.

Categories

1. Sheaf
2. Stalks
3. Maximal Ideals
4. Quotients
5. Localization
6. Books about "Local Rings" in Amazon.com
   

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