Commutative Algebra: Foundations¶
Job: Spec, localization, Noetherian finiteness, primary decomposition, dimension and depth, Serre’s criteria, completion — the standard local toolkit.
Core¶
Rings are commutative with \(1\); modules unital. Write \(\operatorname{Spec} R\) for the set of prime ideals with Zariski closed sets \(V(I)=\{\mathfrak{p}:I\subset\mathfrak{p}\}\).
Spec and localization¶
Basic dictionary.
| Algebra | Geometry |
|---|---|
| prime \(\mathfrak{p}\) | point of \(\operatorname{Spec} R\) |
| radical ideal | closed set (Nullstellensatz over alg. closed fields for max ideals) |
| \(D(f)=\operatorname{Spec} R\setminus V(f)\) | basic open; \(\Gamma(D(f),\mathcal{O})=\operatorname{R}_f\) |
| stalk at \(\mathfrak{p}\) | \(R_\mathfrak{p}\) |
| residue field | \(\kappa(\mathfrak{p})=\operatorname{Frac}(R/\mathfrak{p})\) |
Localization. For multiplicative \(S\subset R\), \(S^{-1}R\) is universal for inverting \(S\). Special cases: \(R_f=S^{-1}R\) with \(S=\{1,f,f^2,\ldots\}\); \(R_\mathfrak{p}=S^{-1}R\) with \(S=R\setminus\mathfrak{p}\) (local ring, max ideal \(\mathfrak{p}R_\mathfrak{p}\)).
Exactness. Localization is exact (flat): \(S^{-1}\) preserves injections. For f.g. modules over Noetherian rings, many properties are local: \(M=0\) iff \(M_\mathfrak{p}=0\) all \(\mathfrak{p}\) (or all max \(\mathfrak{m}\)).
Nakayama. \(R\) local, \(\mathfrak{m}\) max, \(M\) f.g.: \(\mathfrak{m}M=M\Rightarrow M=0\). Generators of \(M/\mathfrak{m}M\) lift to generators of \(M\).
Noetherian rings and modules¶
\(R\) Noetherian: ACC on ideals \(\Leftrightarrow\) every ideal f.g. \(M\) Noetherian: ACC on submodules \(\Leftrightarrow\) f.g. and \(R\) Noetherian (for modules over Noeth. rings: f.g. \(\Leftrightarrow\) Noetherian).
| Theorem | Statement |
|---|---|
| Hilbert basis | \(R\) Noeth. \(\Rightarrow R[x_1,\ldots,x_n]\) Noeth. |
| Hilbert Nullstellensatz | \(k\) alg. closed: max ideals of \(k[x_1,\ldots,x_n]\) are \((x_1-a_1,\ldots,x_n-a_n)\); \(I(V(J))=\sqrt{J}\) |
| Artin–Rees | \(R\) Noeth., \(I\) ideal, \(M\) f.g., \(N\subset M\): \(I^n M\cap N=I^{n-k}(I^k M\cap N)\) for \(n\gg 0\) |
| Krull intersection | \(R\) Noeth. local, \(M\) f.g.: \(\bigcap_n \mathfrak{m}^n M=0\) |
Going up / going down / incomparability for integral extensions: \(\dim A=\dim B\) when \(B\) integral over \(A\) (under standard hyp.); lying-over for integral extensions.
Primary decomposition¶
Ideal \(\mathfrak{q}\) is primary if \(xy\in\mathfrak{q}\Rightarrow x\in\mathfrak{q}\) or \(y^n\in\mathfrak{q}\) some \(n\); then \(\sqrt{\mathfrak{q}}\) is prime. Primary decomposition: \(I=\mathfrak{q}_1\cap\cdots\cap\mathfrak{q}_r\) with \(\mathfrak{q}_i\) primary.
Theorem (Lasker–Noether). In a Noetherian ring every ideal has a primary decomposition. Minimal primes over \(I\) are uniquely determined (associated primes \(\operatorname{Ass}(R/I)\)); embedded components need not be unique as ideals but associated primes are.
For modules: \(\operatorname{Ass}(M)=\{\mathfrak{p}:\mathfrak{p}=\operatorname{Ann}(m)\text{ some }m\in M\}\); support \(\operatorname{Supp}(M)=V(\operatorname{Ann} M)=\{\mathfrak{p}:M_\mathfrak{p}\neq 0\}\).
Dimension theory¶
Krull dimension \(\dim R=\sup\{\text{lengths of chains of primes}\}\). For \(M\), \(\dim M=\dim(R/\operatorname{Ann} M)\).
| Theorem | Statement |
|---|---|
| Krull principal ideal | \(\mathfrak{p}\) minimal over \((a_1,\ldots,a_r)\) \(\Rightarrow\operatorname{ht}\mathfrak{p}\le r\) |
| Generalized PIT | converse bounds under catenary / CM hyp. |
| Noether normalization | affine domain over field: finite over a polynomial subring of rank \(=\dim\) |
| Dimension formula | for catenary domains / finitely gen. over fields: \(\operatorname{ht}\mathfrak{p}+\dim R/\mathfrak{p}=\dim R\) |
Regular local ring: \((R,\mathfrak{m},k)\) Noetherian local with \(\dim_k\mathfrak{m}/\mathfrak{m}^2=\dim R\) (minimal number of generators of \(\mathfrak{m}\) equals dim). Regular \(\Rightarrow\) domain (in fact UFD — Auslander–Buchsbaum).
Depth and regular sequences¶
\(x_1,\ldots,x_r\in R\) is an \(M\)-regular sequence if \(x_i\) is a nonzerodivisor on \(M/(x_1,\ldots,x_{i-1})M\) and \(M/(x)M\neq 0\) (conventions vary on the last).
Depth \(\operatorname{depth}_I M=\) length of maximal \(M\)-regular sequence in ideal \(I\) (well-defined for Noetherian \(R\), \(M\) f.g., via Ext or Koszul homology). Local depth: \(\operatorname{depth} M=\operatorname{depth}_\mathfrak{m} M\).
Always \(\operatorname{depth} M\le\dim M\). Cohen–Macaulay: equality for \(M\) (or for \(R\) as module over itself). Gorenstein local: CM of type \(1\) (equiv. finite inj dim, etc.).
Serre’s criteria¶
For Noetherian \(R\):
| Condition | Meaning |
|---|---|
| \((R_k)\) | \(R_\mathfrak{p}\) regular for all \(\mathfrak{p}\) with \(\operatorname{ht}\mathfrak{p}\le k\) |
| \((S_k)\) | \(\operatorname{depth} R_\mathfrak{p}\ge\min(k,\dim R_\mathfrak{p})\) all \(\mathfrak{p}\) |
Theorem (Serre). \(R\) normal (integrally closed domain, universally Japanese / excellent hyp. as needed for geometric cases) \(\Leftrightarrow\) \((R_1)+(S_2)\). Reduced \(\Leftrightarrow\) \((R_0)+(S_1)\).
Smoothness / regularity of schemes is checked stalkwise via \((R_k)\) and geometric fibers; see schemes.
Completion¶
\(\mathfrak{m}\)-adic topology: \(I\)-adic completion \(\hat M=\varprojlim M/I^n M\). For Noetherian local \((R,\mathfrak{m})\), \(\hat R\) is local Noetherian, flat over \(R\); Artin–Rees controls kernels.
Henselian (one-liner). \((R,\mathfrak{m})\) is henselian if simple roots of polynomials over \(k=R/\mathfrak{m}\) lift to \(R\) (equiv. unique lifting of étale neighborhoods of the closed point). Complete local rings are henselian; henselization \(R^h\) is the “algebraic” approximation to completion used in étale topology.
Cohen structure (pointer). Complete regular local rings containing a field are power series rings over a coefficient field (with hyp.).
See it¶
Normalization of a cusp. \(R=k[t^2,t^3]\cong k[x,y]/(y^2-x^3)\) is a \(1\)-dimensional domain, not regular at \(\mathfrak{m}=(x,y)\): \(\dim_k\mathfrak{m}/\mathfrak{m}^2=2>1=\dim R\). Integral closure is \(k[t]\), finite over \(R\). Depth at \(\mathfrak{m}\) is \(1\) (CM curve singularity) but \((R_1)\) fails at \(\mathfrak{m}\) — Serre’s criterion detects non-normality of the un-normalized cusp ring before normalization.
DVRs. Discrete valuation rings: regular local of dim \(1\) \(\Leftrightarrow\) PID local \(\Leftrightarrow\) integrally closed Noetherian local dim \(1\). Valuation rings dominate local domains; completions give \(k[[t]]\) or mixed-char Witt-type pictures.
Check¶
Prove: if \(R\) is Noetherian local regular, then \(R\) is a domain (sketch).
Induct on \(\dim R\). If \(\dim=0\), regular \(\Rightarrow\) field. If \(\dim\ge 1\), choose \(x\in\mathfrak{m}\setminus\mathfrak{m}^2\); then \(R/(x)\) is regular of dim \(\dim R-1\), hence a domain by induction, so \((x)\) is prime. Regular local rings are UFDs (deeper: Auslander–Buchsbaum); already \(x\) nonzerodivisor and \(\bigcap\mathfrak{m}^n=0\) prevent nilpotents.
Why is \(k[x,y]/(xy)\) not Cohen–Macaulay at the origin? Or is it?
\(R=k[x,y]/(xy)\) has \(\dim R_\mathfrak{m}=1\) at \(\mathfrak{m}=(x,y)/(xy)\). Associated primes \((x),(y)\) are minimal — reduced nodal cross. Depth is \(1\) (e.g. \(x+y\) is regular). So this cross is CM; non-CM examples need embedded components or higher-depth failure (e.g. \(k[x,y,z]/(x)\cap(y,z)\)).
Pitfalls¶
- Height vs codimension in geometric fibers can differ without catenary / universally catenary hypotheses (pathological Noetherian rings).
- Primary vs prime: primary decomposition is not unique; associated primes are.
- Flatness \(\neq\) freeness for modules; localization is flat but \(R_\mathfrak{p}\) is rarely free over \(R\).
- Henselian \(\neq\) complete: henselization is algebraic; completion adds analytic/limit data (Artin approximation bridges them).
Refs¶
- Matsumura, Commutative Ring Theory
- Eisenbud, Commutative Algebra
- Atiyah–Macdonald (concise basics)
- Stacks Project, Tag 00ET and following
- Serre, Local Algebra
Next¶
Homological CA · CA index · Modules · AG schemes · Homological algebra · ANT foundations · Prismatic cohomology (δ-rings)