Algebraic Geometry: Foundations¶

Job: bridge classical varieties to scheme language — $\operatorname{Spec}$, Zariski topology, morphisms, and the structure sheaf $\mathcal{O}_X$.

Core¶

Varieties vs schemes (intuition)¶

Classical variety (e.g. over $\overline{k}$)	Scheme
Zero set $V(I)\subset\mathbb{A}^n$ or $\mathbb{P}^n$	Locally $\operatorname{Spec} A$ for rings $A$
Points = closed points (max ideals, Nullstellensatz)	All primes; generic points, non-closed points
Reduced, irreducible components as sets	Allows nilpotents; structure sheaf carries multiplicity
Morphisms = polynomial maps	Morphisms of (locally) ringed spaces
Base field fixed	Relative geometry over any base $S$

Nullstellensatz (Hilbert). $k$ alg. closed: max ideals of $k[x_1,\ldots,x_n]$ are $(x_1-a_1,\ldots,x_n-a_n)$; radical ideals $\leftrightarrow$ algebraic sets. This identifies closed points of $\mathbb{A}^n_k$ with $k^n$.

Moral: varieties are schemes that are reduced, of finite type over an alg. closed field, and often separated/integral; schemes add arithmetic, families, and infinitesimal structure.

Spec as a space¶

For a commutative ring $A$: $$ \operatorname{Spec} A = {\mathfrak{p}\subset A \text{ prime}}, $$ with closed sets $V(I)=\{\mathfrak{p}:I\subset\mathfrak{p}\}$ (Zariski topology). Basis of opens: $D(f)=\{\mathfrak{p}:f\notin\mathfrak{p}\}$.

Feature	Meaning
Closed points	often max ideals (always if Jacobson, e.g. finite type over fields)
Generic point of $V$	prime corresponding to the ideal of an irreducible closed $V$
Specialization $\mathfrak{p}\rightsquigarrow\mathfrak{q}$	$\mathfrak{p}\subset\mathfrak{q}$ (in closure)
Irreducible components	minimal primes; $V(\mathfrak{p})$ for $\mathfrak{p}$ min

Functoriality. Ring map $\varphi:A\to B$ induces continuous $\operatorname{Spec}\varphi:\operatorname{Spec} B\to\operatorname{Spec} A$, $\mathfrak{q}\mapsto\varphi^{-1}(\mathfrak{q})$. Contrariant equivalence (with sheaf data): affine schemes $\simeq$ rings$^{\mathrm{op}}$ (up to standard caveats of locally ringed maps).

Structure sheaf $\mathcal{O}$ (sketch)¶

On $D(f)\subset\operatorname{Spec} A$, set $\mathcal{O}(D(f))=A_f$ (localization). This extends uniquely to a sheaf of rings on $\operatorname{Spec} A$ with: $$ \mathcal{O}{\operatorname{Spec} A,\mathfrak{p}} = A)=A $$ (when }},\qquad \Gamma(\operatorname{Spec} A,\mathcal{O$A$ is what it is — always for the standard construction). Stalks are local rings. A scheme is a ringed space locally isomorphic to $(\operatorname{Spec} A,\mathcal{O})$.

Quasi-coherent modules. An $A$-module $M$ yields $\widetilde{M}$ on $\operatorname{Spec} A$ with $\widetilde{M}(D(f))=M_f$. On affines, $\operatorname{QCoh}(\operatorname{Spec} A)\simeq A\text{-Mod}$.

Zariski topology — working facts¶

Not Hausdorff in interesting cases; quasi-compact opens $D(f_1)\cup\cdots\cup D(f_n)$ relate to $1\in(f_1,\ldots,f_n)$.
Irreducible $\Leftrightarrow$ generic point unique (for schemes: $\operatorname{Spec} A$ irreducible $\Leftrightarrow$ nilradical prime).
Noetherian topological space if descending chain of closed sets stabilizes — follows when $A$ Noetherian.
Closed immersions $\leftrightarrow$ $A\to A/I$; open immersions to principal $D(f)$ on affines.

Morphisms (first pass)¶

A morphism of schemes $f:X\to Y$ is a morphism of locally ringed spaces: continuous map + sheaf map $f^\sharp:\mathcal{O}_Y\to f_*\mathcal{O}_X$ such that on stalks $f^\sharp_y:\mathcal{O}_{Y,f(y)}\to\mathcal{O}_{X,y}$ is a local homomorphism.

Affine case: $\operatorname{Hom}_{\mathbf{Sch}}(\operatorname{Spec} B,\operatorname{Spec} A)\cong\operatorname{Hom}_{\mathbf{Ring}}(A,B)$.

Construction	Affine description
Open immersion	localization / covering by $D(f)$
Closed immersion	$A\to A/I$
Base change	$B\otimes_A C$ for $\operatorname{Spec}$ maps
Fibers $X_y$	$\operatorname{Spec}\kappa(y)\times_Y X$

Relative viewpoint: study $X\to S$ (family), not absolute $X$. Fibers and flatness control variation — see schemes.

Sheaves of modules (pointer)¶

Presheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules: $\mathcal{F}(U)$ is an $\mathcal{O}_X(U)$-module, restriction compatible. Sheaf axiom: unique gluing on covers. Global sections $\Gamma(X,-)$ left exact; derived functors $H^i(X,-)$ — cohomology, homological algebra.

See it¶

Nodal cubic. $C:y^2=x^2(x+1)$ in $\mathbb{A}^2_k$ ($k$ alg. closed, $\operatorname{char}\neq 2$). As a variety, $C$ is irreducible with a node at origin. Scheme-theoretically, $\mathcal{O}_{C,0}$ is not regular (embedding dimension 2, dimension 1). Normalization $\widetilde{C}\to C$ is finite birational; $\widetilde{C}\cong\mathbb{A}^1$ via parametrization. The conductor ideal and $\delta$-invariant live in the structure sheaves — invisible if one only tracks point-sets.

Arithmetic line. $\operatorname{Spec}\mathbb{Z}$: closed points $(p)$, generic point $(0)$. A morphism $X\to\operatorname{Spec}\mathbb{Z}$ is an arithmetic scheme; fiber over $(p)$ is reduction mod $p$. No classical variety over a field captures this.

Check¶

Prove that $D(f)\cap D(g)=D(fg)$ and that $\{D(f)\}$ is a basis for the Zariski topology.

$\mathfrak{p}\in D(f)\cap D(g)\Leftrightarrow f,g\notin\mathfrak{p}\Leftrightarrow fg\notin\mathfrak{p}$ (prime). Any open is complement of $V(I)=\bigcap_{f\in I}V(f)$, so union of $D(f)$ for $f\in I$.

Why does a nonzero nilpotent in $A$ not change the underlying topological space of $\operatorname{Spec} A$ vs $\operatorname{Spec} A_{\mathrm{red}}$?

Primes of $A$ biject with primes of $A/\sqrt{0}$: every prime contains $\sqrt{0}$. Topology via $V(I)$ likewise factors through the radical. The structure sheaf changes: $\Gamma(\operatorname{Spec} A,\mathcal{O})=A$ may have nilpotents.

Pitfalls¶

Maximal spectrum $\operatorname{mSpec}$ loses generic points; fiber dimension and specialization need full $\operatorname{Spec}$.
“Polynomial functions = global sections” fails on non-affine varieties ($H^0(\mathbb{P}^n,\mathcal{O})=\text{const}$).
Local homomorphism condition on morphisms is not automatic from continuous + ring maps on opens; it enforces “points map to points with residue field extensions,” not weird stalk maps.
Irreducible $\neq$ connected; integral = reduced + irreducible.

Refs¶

Atiyah–Macdonald, Introduction to Commutative Algebra
Hartshorne, Algebraic Geometry (I–II)
Mumford, The Red Book of Varieties and Schemes
Eisenbud–Harris, The Geometry of Schemes

Next¶

Schemes · Cohomology · AG index · Rings and fields · Modules · Category theory