How is a scheme defined in terms of gluing spectra and as a ringed 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.
Reasoning, step by step
- Identify the spectra of commutative rings as the basic building blocks.
- Glue these spectra together along their open subsets to form a topological space.
- Assign commutative rings to all open sets of this space.
- Alternatively, define the scheme as a ringed space that is locally a spectrum of a commutative ring.
- Conclude that this is equivalent to being locally isomorphic to an affine space.
Example
The video states: 'A scheme is a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra (spaces of prime ideals) of commutative rings along their open subsets. Equivalently, a scheme is a ringed space which is locally a spectrum of a commutative ring. In other words, a scheme is a ringed space that is locally isomorphic to an affine space.'
Common misconceptions
- Thinking that a scheme is just a topological space without the structure sheaf of rings.
- Assuming that schemes are globally isomorphic to affine spaces, rather than locally.
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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.
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