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.
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.
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.
An affine scheme is a locally ringed space that is isomorphic to the spectrum Spec(R) of a commutative ring R. General schemes are constructed by gluing together open sets of this simple kind, called affine schemes.
Conditions: The object is an affine scheme.; It is a locally ringed space.; It is isomorphic to the spectrum of a commutative ring.
An affine scheme is a locally ringed space that is isomorphic to the spectrum Spec(R) of a commutative ring R. General schemes are constructed by gluing together open sets of this simple kind, called affine schemes.
Conditions: The object is an affine scheme.; It is a locally ringed space.; It is isomorphic to the spectrum of a commutative ring.