What is the precise definition of an affine scheme in terms of ring theory?
Conditions
- The object is an affine scheme.
- It is a locally ringed space.
- It is isomorphic to the spectrum of a commutative ring.
Reasoning, step by step
- Identify a commutative ring R.
- Form the spectrum Spec(R), which is the set of all prime ideals of R.
- Equip this spectrum with a sheaf of rings to make it a locally ringed space.
- Define the affine scheme as any locally ringed space isomorphic to this construction.
Example
The video states: 'An affine scheme is a locally ringed space isomorphic to the spectrum Spec(R) of a commutative ring R. Schemes are made by gluing together open sets of a simple kind, called affine schemes.'
Common misconceptions
- Thinking that an affine scheme is just the set of prime ideals without the sheaf structure.
- Confusing affine schemes with general schemes, which are formed by gluing affine schemes together.
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Connected concepts
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Related questions
A scheme is defined as a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra of commutative rings along their open subsets. Equivalently, a scheme is a ringed space that is locally a spectrum of a commutative ring, or locally isomorphic to an affine space.
Conditions: It involves a topological space.; It involves commutative rings assigned to open sets.; It is constructed by gluing spectra of commutative rings.; It is a ringed space locally isomorphic to an affine space.
In commutative algebra, the spectrum of a ring R is the set of all prime ideals of R, and is usually denoted by Spec R. In algebraic geometry, it is simultaneously a topological space equipped with the sheaf of rings O.
Conditions: R is a commutative ring.; The context includes both commutative algebra and algebraic geometry.
Scheme theory enlarges the notion of an algebraic variety by taking into account multiplicities and allowing varieties to be defined over any commutative ring, not just algebraically closed fields. For example, the equations and define the same algebraic variety but different schemes.
Conditions: The context is the motivation for scheme theory over classical algebraic geometry.; Schemes account for multiplicities.; Schemes can be defined over any commutative ring.
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