(Ringed spaces)
A ringed space is a pair \((X, \OO_X)\) of a topological space \(X\) and a sheaf of rings \(\OO_X\) on \(X\), its structure sheaf. An isomorphism of ringed spaces \((X, \OO_X) \to (Y, \OO_Y)\) is a homeomorphism \(f: X \to Y\) together with an isomorphism of sheaves of rings \(\OO_Y \to f_* \OO_X\). A ringed space is a locally ringed space if every stalk \(\OO_{X,p}\) is a local ring.
(Scheme)
A scheme is a locally ringed space \((X, \OO_X)\) admitting an open cover by spaces isomorphic, as locally ringed spaces, to \(\Spec A\) for various rings \(A\). A morphism of schemes is a morphism of locally ringed spaces: a continuous \(f: X \to Y\) together with \(f^\sharp : \OO_Y \to f_* \OO_X\) whose induced maps on stalks are local homomorphisms.
(Scheme over a ring)
Let \(A\) be a ring. A scheme over \(A\) is a scheme \(X\) with a morphism \(X \to \Spec A\). Morphisms \(X \to \Spec A\) correspond to ring homomorphisms \(A \to \Gamma(X, \OO_X)\), so this is the same as making every \(\OO_X(U)\) an \(A\)-algebra with \(A\)-algebra restriction maps. [@Har10a, Exercise II.2.4]
(A morphism of ringed spaces that is not a morphism of schemes)
Let \(R\) be a discrete valuation ring with fraction field \(K\), closed point \(\mfm\) and generic point \(\eta\). Send the one point of \(\Spec K\) to \(\mfm\), and let \(f^\sharp\) be the inclusion \(R \injects K\) on sections over \(\Spec R\) and zero over the open set \(\ts{\eta}\), whose preimage is empty. This is a morphism of ringed spaces. Its map on stalks at the point is the inclusion \(R = \OO_{\Spec R, \mfm} \injects K\), which sends a uniformizer to a unit and so is not local. It is not induced by a ring homomorphism: the only candidate \(R \to K\) is the inclusion, and \(\Spec\) of the inclusion sends the point to \(\eta\).
(Morphisms to affine space)
For an \(S\)-scheme \(X\), an \(S\)-morphism \(X \to \AA^n_S\) is the same as an \(n\)-tuple of global sections \(h_1, \ldots, h_n \in \Gamma(X, \OO_X)\), the pullbacks of the coordinates. [@Har10a]
(The scheme of a variety)
Let \(k\) be algebraically closed. There is a fully faithful functor \(t\) from varieties over \(k\) to schemes over \(k\). The points of \(t(V)\) are the nonempty irreducible closed subsets of \(V\), with closed sets the sets \(t(Y)\) for \(Y \subseteq V\) closed. The closed points of \(t(V)\) form a subspace homeomorphic to \(V\), and restricting \(\OO_{t(V)}\) to it recovers the sheaf of regular functions on \(V\). [@Har10a]
For rings \(A\) and \(B\), every morphism of locally ringed spaces \(\Spec B\to\Spec A\) is induced by a unique ring homomorphism \(A\to B\) [@Har10a, Proposition II.2.3], so \(\Spec\) is an anti-equivalence from the category of rings to the category of affine schemes. The morphism \(\Spec K\to\Spec R\) in the example of a morphism of ringed spaces that is not a morphism of schemes has a non-local map on stalks, and it is induced by no ring homomorphism.
A scheme is glued from affine schemes along open subschemes, as a manifold is glued from open subsets of \(\RR^n\) [@Har10a, Exercise II.2.12].
Dependencies
D-VKR54$\Spec A$ as a locally ringed spaceuses
Backlinks
T-SK599$\Spec$ is an anti-equivalence, and the adjunction that recognises affinesusesPR-SCHUNITCOVERDistinguished opens cover exactly when the functions generate the unit idealusesD-SCHGLUEGluing schemes along open subschemesusesD-SCHSUBOpen and closed subschemes, and closed immersionsusesD-T2J3QSeparated and quasi-separated morphismsusesD-SCHFPRThe fibre product of schemesuses