Modules¶
Job: linear algebra over rings — free/projective/injective modules, structure over PIDs, and exactness of \(\operatorname{Hom}\) and \(\otimes\).
Core¶
Definition. A (left) \(R\)-module is an abelian group \(M\) with \(R\times M\to M\) bilinear, associative, unital. Morphisms: \(R\)-linear maps. Category \(R\text{-Mod}\) is abelian.
| Type | Characterization |
|---|---|
| Free | \(M\cong R^{(I)}\); has a basis |
| Projective | direct summand of a free; \(\operatorname{Hom}(P,-)\) exact |
| Injective | \(\operatorname{Hom}(-,I)\) exact; Baer’s criterion for ideals |
| Flat | \(-\otimes_R M\) exact; directed colimits of projectives (Lazard) |
| Finitely presented | coker of \(R^m\to R^n\) |
Over a field \(k\): all modules are free (vector spaces); projective = injective = flat = free.
Exactness of Hom and Tensor¶
For fixed \(R\)-modules \(A,B\) and a short exact sequence \(0\to M'\to M\to M''\to 0\):
| Functor | Exactness | Failure mode |
|---|---|---|
| \(\operatorname{Hom}_R(A,-)\) | left exact | coker not preserved; \(\operatorname{Ext}^1(A,-)\) measures |
| \(\operatorname{Hom}_R(-,B)\) | left exact (contravariant) | same; \(\operatorname{Ext}^1(-,B)\) |
| \(-\otimes_R B\) | right exact | ker not preserved; \(\operatorname{Tor}_1(-,B)\) measures |
| \(A\otimes_R -\) | right exact | same |
Projective resolution \(P_\bullet\to M\to 0\) free/projective: \(\operatorname{Tor}_i(M,N)=H_i(P_\bullet\otimes N)\), \(\operatorname{Ext}^i(M,N)=H^i(\operatorname{Hom}(P_\bullet,N))\).
Structure over a PID¶
Theorem. Let \(R\) be a PID, \(M\) f.g. \(R\)-module. Then $$ M\cong R^r\oplus R/(d_1)\oplus\cdots\oplus R/(d_k),\qquad d_i\mid d_{i+1}, $$ uniquely (invariant factors), or primary decomposition as for abelian groups (\(R=\mathbb{Z}\)). In particular f.g. torsion-free \(\Leftrightarrow\) free over a PID.
Corollary (rational canonical / Jordan). Apply to \(F[t]\)-modules \(V\) with \(t\) acting as \(T\in\operatorname{End}_F(V)\): invariant factors \(\leftrightarrow\) rational canonical form; elementary divisors over alg. closed \(F\) \(\leftrightarrow\) Jordan form.
Nakayama’s lemma¶
Theorem (Nakayama). \(R\) local with max ideal \(\mathfrak{m}\), \(M\) f.g. If \(\mathfrak{m}M=M\), then \(M=0\). Equivalently: lifts of generators of \(M/\mathfrak{m}M\) generate \(M\).
Used constantly for: free modules of constant rank on connected Spec, smoothness criteria, completions.
Snake lemma¶
A commutative diagram with exact rows yields a six-term exact sequence relating \(\ker\) and \(\operatorname{coker}\) of the three vertical maps — the workhorse for long exact sequences in homology.
See it¶
Ext and group extensions. For abelian groups, \(\operatorname{Ext}^1_{\mathbb{Z}}(A,B)\) classifies extensions \(0\to B\to E\to A\to 0\) up to equivalence. \(\operatorname{Ext}^1(\mathbb{Z}/n,B)\cong B/nB\). Non-split extension: \(0\to\mathbb{Z}/2\to\mathbb{Z}/4\to\mathbb{Z}/2\to 0\).
Tor and torsion. \(\operatorname{Tor}_1^{\mathbb{Z}}(\mathbb{Z}/m,\mathbb{Z}/n)\cong\mathbb{Z}/\gcd(m,n)\). Tensoring the resolution \(0\to\mathbb{Z}\xrightarrow{\times m}\mathbb{Z}\to\mathbb{Z}/m\to 0\) with \(\mathbb{Z}/n\) produces the kernel \(n\)-torsion in \(\mathbb{Z}/m\).
Check¶
Prove \(\mathbb{Q}\) is flat as \(\mathbb{Z}\)-module but not projective.
Flat: localization \(S^{-1}\mathbb{Z}\) with \(S=\mathbb{Z}\setminus\{0\}\) is flat over \(\mathbb{Z}\). Not projective: projective \(\mathbb{Z}\)-modules are free; \(\mathbb{Q}\) is not free (any two elements are \(\mathbb{Z}\)-linearly dependent over \(\mathbb{Q}\), or: \(\operatorname{Hom}(\mathbb{Q},\mathbb{Z})=0\) so cannot be a summand of \(\mathbb{Z}^{(I)}\) nontrivially in the required way — more cleanly: if free, basis would be uncountable or contradict finite generation of relations). Standard: nonzero projectives over \(\mathbb{Z}\) are free of positive rank, and free modules of rank \(\ge 1\) admit nonzero maps to \(\mathbb{Z}\).
Give \(R\)-modules where \(M\otimes N=0\) but \(M,N\neq 0\).
\(\mathbb{Z}/2\otimes_{\mathbb{Z}}\mathbb{Z}/3=0\). Or \(R=k[x,y]/(xy)\), \(M=R/(x)\), \(N=R/(y)\).
Pitfalls¶
- Free \(\Rightarrow\) projective \(\Rightarrow\) flat; converses fail: \(\mathbb{Q}\) flat not projective over \(\mathbb{Z}\); over Dedekind domains, f.g. projective = f.g. torsion-free, not always free (ideal class group).
- Infinite direct products of free modules need not be free (e.g. \(\mathbb{Z}^{\mathbb{N}}\) is not free abelian — Specker).
- \(\operatorname{Hom}_R(M,N)\) is an abelian group; \(R\)-module structure requires \(R\) commutative (or care with bimodules).
- Right vs left modules matter when \(R\) is noncommutative; \(\otimes\) needs balanced products \(M_R\) and \({}_RN\).
Refs¶
- Lang, Algebra
- Rotman, An Introduction to Homological Algebra
- Atiyah–Macdonald, Introduction to Commutative Algebra (chs. 2–3)
Next¶
Galois theory · Rings and fields · Group theory · Category theory · Homological ideas in alg. top.