Schemes¶
Job: glue affines into schemes; fiber products; standard morphism classes; \(\operatorname{Proj}\) for projective geometry.
Core¶
Affine schemes and gluing¶
An affine scheme is a ringed space isomorphic to \((\operatorname{Spec} A,\mathcal{O}_{\operatorname{Spec} A})\). A scheme is a locally ringed space \(X\) admitting an open cover \(\{U_i\}\) with each \(U_i\) affine.
Gluing data. Opens \(U_i\), isomorphisms \(\varphi_{ij}:U_i\cap U_j\to U_j\cap U_i\) (as locally ringed spaces) with cocycle \(\varphi_{ik}=\varphi_{jk}\circ\varphi_{ij}\) on triple overlaps \(\Rightarrow\) unique scheme \(X\) with \(U_i\hookrightarrow X\).
| Example | Gluing |
|---|---|
| \(\mathbb{A}^1_k\) | single \(\operatorname{Spec} k[t]\) |
| \(\mathbb{P}^1_k\) | two \(\mathbb{A}^1\): \(k[t]\), \(k[u]\), \(t\leftrightarrow u^{-1}\) on \(D(t)\cong D(u)\) |
| Line with double origin | two \(\mathbb{A}^1\) glued on \(\mathbb{A}^1\setminus\{0\}\) by identity — not separated |
Locally ringed spaces¶
A ringed space \((X,\mathcal{O}_X)\): topological space + sheaf of rings. Locally ringed: each stalk \(\mathcal{O}_{X,x}\) is a local ring; morphisms require local stalk maps (residue fields \(\kappa(x)=\mathcal{O}_{X,x}/\mathfrak{m}_x\) behave well).
Schemes form a full subcategory of locally ringed spaces; not every locally ringed space is a scheme.
Fiber products¶
Theorem. Fiber products exist in \(\mathbf{Sch}\): for \(X\to S\leftarrow Y\) there is \(X\times_S Y\) with the usual universal property (as schemes, not merely as topological spaces).
Affine case: $$ \operatorname{Spec} B\times_{\operatorname{Spec} A}\operatorname{Spec} C \cong \operatorname{Spec}(B\otimes_A C). $$ General case: glue these on affine covers of \(X,Y,S\). Topology of underlying space of \(X\times_S Y\) is not the topological fiber product in general.
Fibers. For \(x\in X\) over \(s=f(x)\), the scheme-theoretic fiber \(X_s=X\times_S\operatorname{Spec}\kappa(s)\) carries the geometric fiber after base change to \(\overline{\kappa(s)}\).
Morphisms: one-liners that matter¶
Assume \(f:X\to Y\) of schemes (often locally of finite type / finite presentation as needed).
| Property | Rough meaning | Typical use |
|---|---|---|
| Separated | diagonal \(\Delta:X\to X\times_Y X\) is a closed immersion | uniqueness of limits of maps; no double origins |
| Proper | separated + universally closed + finite type | valuative criterion; pushforwards of coherent sheaves stay coherent (under hyp.) |
| Finite | affine; \(\mathcal{O}_Y\to f_*\mathcal{O}_X\) finite as modules | branched covers; normalization (under excellence) |
| Flat | \(\mathcal{O}_{X,x}\) flat over \(\mathcal{O}_{Y,f(x)}\) all \(x\) | fibers vary continuously; \(\chi\) constant in flat families |
| Smooth | flat + geometrically regular fibers (equiv. form. smooth of rel. dim \(d\)) | relative differentials locally free rank \(d\); étale = smooth rel. dim \(0\) |
| Étale | flat + unramified (or smooth of relative dimension 0) | algebraic analogue of local iso / covering space |
| Open/closed immersion | iso onto open / closed subscheme | subschemes, local study |
Valuative criteria (separated / proper)¶
Let \(R\) be a valuation ring with fraction field \(K\) (often a DVR in practice), and \(f:X\to S\) a morphism of schemes locally of finite type (Stacks/Hartshorne hypotheses; Noetherian bases simplify).
Consider solid commutative diagrams $$ \begin{CD} \operatorname{Spec} K @>>> X \ @VVV @VV f V \ \operatorname{Spec} R @>>> S \end{CD} $$ i.e. an \(S\)-valued \(R\)-point of the base and a lift of the generic point to \(X\).
| Criterion | Condition on lifts \(\operatorname{Spec} R\to X\) | Conclusion |
|---|---|---|
| Separated | at most one extension of \(\operatorname{Spec} K\to X\) to \(\operatorname{Spec} R\to X\) | \(f\) separated |
| Proper | at least one extension (and \(\le 1\) if already separated) | \(f\) universally closed + separated + finite type, i.e. proper |
| Universally closed | existence of extensions after arbitrary base change \(S'\to S\) | (with finite type) part of properness |
Intuition. \(\operatorname{Spec} R\) is a “trait” (generic point \(\eta\) + closed point \(s\)). A map \(\operatorname{Spec} K\to X\) is a point with values in \(K\); extending across \(R\) means the point has a limit specialization in \(X\) compatible with the map to \(S\). Uniqueness of limits \(\Leftrightarrow\) separated (like Hausdorff); existence of limits \(\Leftrightarrow\) universally closed / proper (like compactness in the relative setting).
DVR form (most used). For DVRs, the closed point maps to a specialization of the image of the generic point; checking on complete DVRs or \(k[[t]]\) often suffices in geometric situations (approximation).
| Example | Verdict |
|---|---|
| \(\mathbb{A}^1_k\to\operatorname{Spec} k\) | not proper: \(k((t))\to\mathbb{A}^1\setminus\{0\}\) via \(t\) does not extend to \(k[[t]]\) into \(\mathbb{A}^1\setminus\{0\}\) |
| \(\mathbb{P}^n_k\to\operatorname{Spec} k\) | proper: homogeneous coords can be scaled to land in \(R\) not all in \(\mathfrak{m}\) |
| Line with two origins \(\to\operatorname{Spec} k\) | not separated: two extensions when the limit wants to be the origin |
| Open immersion \(U\hookrightarrow X\) | separated; proper iff also closed (hence clopen under mild hyp.) |
Open diagonal \(\Rightarrow\) not separated (line with two origins: diagonal misses the pair of distinct origins). Finite \(\Rightarrow\) proper (over Noetherian bases, standard hyp.); projective \(\Rightarrow\) proper; proper + quasi-finite \(\Rightarrow\) finite (Zariski’s main theorem package).
Relative Spec of valuation rings also tests the valuative criterion of properness after base change, which is why properness is stable under base change and composition (with standard finite-type caveats).
Proj¶
Let \(S=\bigoplus_{d\ge 0}S_d\) be a graded ring (\(S_0\) typically Noetherian, \(S\) f.g. as \(S_0\)-algebra for geometric cases). $$ \operatorname{Proj} S = {\mathfrak{p}\text{ homogeneous prime}: S_+\not\subset\mathfrak{p}}, $$ \(S_+=\bigoplus_{d>0}S_d\), with Zariski closed sets from homogeneous ideals and structure sheaf via \(S_{(f)}\) (degree-0 localization) on \(D_+(f)\).
| Fact | Statement |
|---|---|
| Cover | \(D_+(f)\) for \(f\in S_d\), \(d>0\); each \(D_+(f)\cong\operatorname{Spec} S_{(f)}\) |
| \(\mathbb{P}^n_A\) | \(\operatorname{Proj} A[x_0,\ldots,x_n]\) standard grading |
| Twists | \(\mathcal{O}(n)\) from graded shifts; \(\Gamma(\mathbb{P}^n_k,\mathcal{O}(n))=k[x]_n\) for \(n\ge 0\) |
| Closed subschemes | \(\operatorname{Proj} S/I\) for homogeneous \(I\) |
Serre’s theorem (pointer). On \(\mathbb{P}^n_A\), coherent sheaves are quotients of \(\bigoplus \mathcal{O}(n_i)^{\oplus}\); cohomology vanishes in high degrees after twisting — cohomology.
Relative Spec / Proj¶
For quasi-coherent \(\mathcal{A}\) of \(\mathcal{O}_S\)-algebras: \(\mathbf{Spec}_S\mathcal{A}\to S\) affine morphism. Graded version \(\mathbf{Proj}_S\mathcal{S}\) for projective morphisms. Vector bundles \(\leftrightarrow\) locally free sheaves via \(\mathbf{Spec}\) of symmetric algebras (dual conventions vary).
See it¶
Elliptic curve as proper smooth genus-1. \(E\subset\mathbb{P}^2_{\mathbb{Z}[a,b,\Delta^{-1}]}\) cut by a Weierstrass equation with \(\Delta\neq 0\) is smooth proper of relative dimension 1 over the base; group law is a morphism \(E\times_S E\to E\). Reduction mod \(p\) is the fiber over \((p)\); multiplicative/additive reduction when the special fiber is singular — scheme-theoretic fibers encode Kodaira types after base change.
Hilbert scheme (existence pointer). Flat proper families of closed subschemes of \(\mathbb{P}^n\) with fixed Hilbert polynomial are represented by a scheme \(\operatorname{Hilb}\); construction uses Grassmannians of subspaces of \(H^0(\mathcal{O}(m))\) for large \(m\) and flattening stratifications — pure scheme technology (fiber products, Proj, coherent cohomology finiteness).
Check¶
Show \(\mathbb{A}^1\setminus\{0\}\hookrightarrow\mathbb{A}^1\) is not proper using the valuative criterion or an explicit map.
Map \(\operatorname{Spec} k((t))\to\mathbb{A}^1\setminus\{0\}\) via \(u\mapsto t\) does not extend to \(\operatorname{Spec} k[[t]]\to\mathbb{A}^1\setminus\{0\}\) (extends to \(\mathbb{A}^1\) sending closed point to \(0\), which is missing). Alternatively: image of closed unit disk under \(z\mapsto z\) is not closed in classical topology after analytification — scheme version uses universal closedness failure.
Why is \(X\times_S Y\) needed instead of set-theoretic fiber products of points?
Scheme structure on fibers (nilpotents, non-reduced) and universal property for morphisms: a map into the product is a pair of maps agreeing on \(S\), which on affines is exactly the tensor product, not a cartesian product of prime sets alone.
Pitfalls¶
- Separatedness is relative (\(X\to Y\)); absolute separated usually means over \(\operatorname{Spec}\mathbb{Z}\) (or over the base field).
- Flat \(\neq\) smooth: \(y^2=x^2(x+1)\) family can be flat with singular fibers; smooth requires regular geometric fibers.
- \(\operatorname{Proj} S\) ignores \(S_+\) in the irrelevant ideal — primes containing \(S_+\) are deleted; \(\operatorname{Proj} k[x,y]/(x,y)\) is empty.
- Topological space of a fiber product can be subtle (e.g. \(\operatorname{Spec}\mathbb{C}\times_{\operatorname{Spec}\mathbb{R}}\operatorname{Spec}\mathbb{C}\cong\operatorname{Spec}(\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C})\cong\operatorname{Spec}(\mathbb{C}\times\mathbb{C})\) two points).
Refs¶
- Hartshorne, Algebraic Geometry (II)
- The Stacks Project, Schemes / Morphisms of Schemes
- EGA I–II
- Vakil, Foundations of Algebraic Geometry (The Rising Sea)
Next¶
Foundations · Cohomology · AG index · Commutative algebra · Homological algebra · Modules · Category theory