$\Spec A$ as a locally ringed space (Tag D-VKR54)
\(\Spec A\) is the set of prime ideals of \(A\), with closed sets \(V(J) = \ts{\mfp \st \mfp \supseteq J}\). Its structure sheaf sends \(U\) to the functions \[ \varphi : U \to \coprod_{\mfp \in U} A_\mfp \] with \(\varphi(\mfp) \in A_\mfp\), locally of the form \(f/g\) with \(g \notin \mfp\).
\(\OO_{\Spec A}(D_f) = A_f\), and in particular \(\OO_{\Spec A}(\Spec A) = A\). The stalk at \(\mfp\) is \(A_\mfp\), a local ring.
For a field \(k\), the affine line punctured at \(0\) and \(1\) is the distinguished open set \(D_{x(x-1)} \subseteq \AA^1_k = \Spec k[x]\), whose closed complement is \(V(x(x-1)) = \{(x), (x-1)\}\). It is the affine scheme \(\Spec R\) with \[R = k[x]_{x(x-1)} = k[x]\left[\frac{1}{x(x-1)}\right].\]
For a ring map \(\varphi\colon A\to B\) and a prime \(\mfq\) of \(B\), \(\inverseof{\varphi}(\mfq)\) is a prime of \(A\), so \(\varphi\) induces a continuous map \(\Spec B\to\Spec A\). The preimage of a maximal ideal need not be maximal: for \(\ZZ\subseteq\QQ\), the preimage of the maximal ideal \((0)\) of \(\QQ\) is the prime \((0)\) of \(\ZZ\), which is not maximal.
Morphisms of schemes are morphisms of locally ringed spaces: the induced maps on stalks are local homomorphisms. With this condition, ring maps \(A\to B\) correspond bijectively to morphisms \(\Spec B\to\Spec A\) [@Har10a, Proposition II.2.3]; as morphisms of ringed spaces there are more (D-AN662).
Schemes, and morphisms of schemes (Tag D-AN662)
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].
Why the definition is shaped that way
Points are prime ideals because the preimage of a prime ideal under a ring map \(\varphi\colon A\to B\) is prime, so \(\mathfrak q\mapsto\varphi^{-1}(\mathfrak q)\) is a continuous map \(\Spec B\to\Spec A\); the preimage of a maximal ideal need not be maximal, as \((0)\subset\QQ\) pulls back to \((0)\subset\ZZ\). The structure sheaf is defined by a local condition so that its sections over \(D_f\) come out as \(A_f\). Morphisms are local on stalks so that \(\Hom(\Spec B,\Spec A)=\Hom(A,B)\).
The schemes that the classical picture cannot hold (Tag FE-O12TX)
\(\Spec k[\varepsilon]/(\varepsilon^2)\) has one point and a two-dimensional ring of functions. It is the scheme-theoretic intersection of the parabola \(y = x^2\) with the line \(y = 0\), and it is what distinguishes tangency from transversality: the reduced intersection is a single point either way.
The function \(\varepsilon\) is nonzero, but its value at the unique point, its image in the residue field \(k[\varepsilon]/(\varepsilon) = k\), is \(0\); so \(\varepsilon\) and \(0\) are different functions with the same values at every point.
Its \(k\)-points valued in a \(k\)-algebra compute tangent vectors: a morphism \(\Spec k[\varepsilon]/(\varepsilon^2) \to X\) is a point of \(X\) together with an element of \(T_p X\).
(Two points that are not the same size)
\(\Spec \ZZ\) has a closed point for each prime and a dense generic point \((0)\). Its residue fields are \(\FF_p\) and \(\QQ\), so the “functions” on it take values in different fields at different points.
(Values of a regular function on $D(6) \subseteq \Spec \ZZ$)
The distinguished open set \(D(6) \subseteq \Spec \ZZ\) consists of the generic point \((0)\) and the points \((p)\) for primes \(p \neq 2, 3\), and \(\OO_{\Spec \ZZ}(D(6)) \cong \ZZ[1/6]\). So \(\varphi = 5/6\) is a regular function on \(D(6)\), and its value at a point is its image in the residue field there.
At \((0)\), the value is \(5/6 \in \QQ = \kappa((0))\).
At \((p)\) with \(p \neq 2, 3\), the value is \(\bar{5} \cdot \bar{6}^{-1} \in \FF_p = \kappa((p))\). Hence \(\varphi\) vanishes at \((5)\), and \(\varphi((11)) = \bar{5} \cdot \bar{2} = \bar{10} \in \FF_{11}\), since \(\bar{6} \cdot \bar{2} = \bar{1}\) in \(\FF_{11}\).
(A nonreduced multiple structure)
\(\Spec k[x]/(x^2)\) and \(\Spec k[x]/(x)\) have the same topological space. The first is the fibre of \(\AA^1 \to \AA^1\), \(t \mapsto t^2\), over the branch point, and its length \(2\) is what makes the degree of that map constant.
The three examples are the three things schemes add: nilpotents record multiplicity, generic points record “generically”, and arithmetic rings make the base something other than a field. Any one of them is a sufficient answer to why the classical language was replaced.
Recognising an affine scheme
$\Spec$ is an anti-equivalence, and the adjunction that recognises affines (Tag T-SK599)
\(A \mapsto \Spec A\) is an anti-equivalence from rings to affine schemes. More generally, for any scheme \(X\) and any ring \(A\), \[ \Hom_{\mathsf{Sch}}(X, \Spec A) \cong \Hom_{\mathsf{Ring}}(A, \OO_X(X)) . \]
(Affine varieties)
Over an algebraically closed field \(k\), \(X \mapsto A(X)\) is an anti-equivalence from affine varieties over \(k\) to finitely generated \(k\)-algebras that are domains. [@Har10a]
A scheme \(X\) is affine exactly when the canonical morphism \(X \to \Spec \OO_X(X)\) is an isomorphism.
The adjunction is the practical test, and it is why the question “how can you tell if a scheme is affine” has a first answer that needs no cohomology: compute the global sections, and compare \(X\) with the spectrum of that ring.
Doing this on \(\PP^1_k\) gives \(\OO(\PP^1) = k\) and \(\Spec k\) a point, so the comparison map collapses the line and \(\PP^1\) is not affine. Doing it on \(\AA^2 \sm \ts{0}\) gives \(\OO(\AA^2 \sm 0) = k[x,y]\) by Hartogs, and the comparison map is the inclusion into \(\AA^2\), which is not an isomorphism: the standard example of a non-affine scheme whose ring of functions is nevertheless that of an affine.
The cohomological criterion — \(H^1\) vanishing against all quasicoherent sheaves — is the version that is a criterion, in that it can be checked without already knowing the answer.
By Serre's criterion, a noetherian scheme \(X\) is affine exactly when \(H^1(X,\mci)=0\) for every coherent ideal sheaf \(\mci\subseteq\OO_X\).
Covers of an affine scheme
Distinguished opens cover exactly when the functions generate the unit ideal (Tag PR-SCHUNITCOVER)
Let \(R\) be a ring and \(\{f_i\}_{i \in I}\) elements of \(R\). Then \(\bigcup_i D(f_i) = \Spec R\) if and only if the \(f_i\) generate the unit ideal. In that case finitely many of the \(f_i\) already generate it, so \(\Spec R\) is quasicompact.
Formal schemes
Formal schemes, and the formal disc (Tag D-SCHFORMAL)
Let \(A\) be a noetherian ring, complete and separated in the \(I\)-adic topology for an ideal \(I\). The formal spectrum \(\operatorname{Spf} A\) is the topological space \(\Spec(A/I)\) with the sheaf of topological rings \(\varprojlim_{n\ge1}\OO_{\Spec(A/I^n)}\). A locally noetherian formal scheme is a topologically ringed space locally isomorphic to such formal spectra. It is a noetherian formal scheme if it is also quasi-compact, equivalently if finitely many of these affine formal opens cover it [@Har10a, Chapter II, §9]. Its underlying space is then noetherian: each affine formal open has the topology of a noetherian affine scheme, and a finite union of noetherian open subspaces is noetherian.
(Completion along a closed subscheme)
Let \(X\) be a Noetherian scheme and \(Z \subseteq X\) a closed subscheme with ideal sheaf \(\mathcal{I}\). The formal completion of \(X\) along \(Z\) is \(\hat{X} = (Z, \varprojlim_n \OO_X/\mathcal{I}^n)\), a Noetherian formal scheme with underlying space \(Z\). For \(X = \Spec A\) and \(Z = V(I)\), \(\hat{X} = \operatorname{Spf} \hat{A}\) with \(\hat{A} = \varprojlim_n A/I^n\).
(The formal disc)
\(\Spec k[[t]]\) has two points: the closed point \((t)\) and the generic point \((0)\), whose residue field is \(k((t))\). It plays the role of a small disc \(\mathbb{D}\) around \(0\) in the \(t\)-line, and its open subscheme \(\Spec k((t))\), the generic point, plays the role of the punctured disc \(\mathbb{D} \setminus \{0\}\). The formal completion of \(\AA^1_k\) at the origin is \(\operatorname{Spf} k[[t]]\), whose underlying space is the single point \(0\) and which remembers every infinitesimal neighbourhood \(\Spec k[t]/(t^n)\).
Checking a property on one affine cover
The affine communication lemma (Tag T-AFFCOMM)
Let \(X\) be a scheme and \(P\) a property of affine open subschemes of \(X\) such that
if \(\Spec A \subseteq X\) has \(P\) and \(f \in A\), then \(\Spec A_f \subseteq X\) has \(P\), and
if \(\Spec A \subseteq X\) is an affine open, \(f_1, \ldots, f_n \in A\) generate the unit ideal, and every \(\Spec A_{f_i}\) has \(P\), then \(\Spec A\) has \(P\).
If \(X\) has a cover by affine opens \(\Spec A_i\) with \(P\), then every affine open of \(X\) has \(P\) [@Vak25, Lemma 5.3.2].
Being Noetherian, and for a scheme over a ring \(R\) being of finite type over \(R\), satisfy both hypotheses, so they can be checked on one affine cover; so do properties defined on stalks, such as being reduced.
Integrality does not satisfy hypothesis 2. For nonzero integral domains \(A\) and \(B\), the elements \(e_1 = (1, 0)\) and \(e_2 = (0, 1)\) of \(A \times B\) generate the unit ideal, and \((A \times B)_{e_1} \cong A\) and \((A \times B)_{e_2} \cong B\) are integral domains, but \(A \times B\) is not, since \(e_1 e_2 = 0\). Geometrically, \(\Spec(A \times B) = \Spec A \sqcup \Spec B\) is covered by integral affine opens and is not irreducible.