The object is replaced; the invariants survive
A resolution replaces a module by an exact complex of projectives or injectives, which behave perfectly under the functor of interest. The construction is always possible, and the result is unique up to chain homotopy equivalence — enough to make derived functors well defined. The shortest possible length is the projective or injective dimension, and the supremum over all modules is the global dimension of the ring, a single number bounding all higher Ext.
Learning objectives
- Construct projective and injective resolutions.
- State the horseshoe lemma and its role.
- Define projective, injective and global dimension.
- Relate global dimension to vanishing of Ext.
- Quote the global dimension of the standard rings.
Section 01Construction and uniqueness
Existence is immediate given enough projectives: surject a projective onto the module, then onto the kernel, and repeat. Uniqueness holds only up to chain homotopy equivalence, by the comparison theorem — and that is exactly the right amount, since additive functors preserve homotopies.
| Projective | Injective | |
|---|---|---|
| Shape | … → P1 → P0 → M → 0 | 0 → M → I0 → I1 → … |
| Used for | Left derived functors; Ext in the first variable | Right derived functors; Ext in the second variable |
| Exists when | Enough projectives | Enough injectives |
| Module categories | Both available | Both available |
| Sheaf categories | Generally not | Available — hence sheaf cohomology |
| Minimal version | Projective cover — may not exist | Injective hull — always exists |
Section 02The horseshoe lemma
- Given 0 → A → B → C → 0 and projective resolutions P• of A and R• of C.
- Set Qn = Pn ⊕ Rn. The direct sum is the candidate resolution of B.
- Construct the differential and the augmentation using projectivity of Rn to lift maps into B.
- The result is a projective resolution of B fitting into a degreewise split short exact sequence of complexes 0 → P → Q → R → 0.
- Applying an additive functor preserves the degreewise splitting, so the sequence of complexes stays exact and the long exact sequence follows.
The long exact sequences of Ext and Tor are not separate constructions. They are the long exact homology sequence applied to the short exact sequence of complexes produced by the horseshoe lemma.
Section 03Dimension
The projective dimension of M is the least length of a projective resolution, or ∞. Equivalently:
The global dimension of a ring is the supremum of pd(M) over all modules, and it equals the corresponding supremum of injective dimensions.
| Ring | Global dimension | Consequence |
|---|---|---|
| Field, or any semisimple ring | 0 | Every module projective; all higher Ext vanishes |
| PID, Dedekind domain | 1 | Extn = 0 for n ≥ 2 |
| k[x1, …, xn] | n | Hilbert's syzygy theorem |
| Regular local ring | Its Krull dimension | Serre's characterisation of regularity |
| ℤ/p²ℤ | ∞ | Non-reduced; resolutions never terminate |
| k[G], char k dividing |G| | ∞ | Modular representation theory has arbitrarily high cohomology |
It makes every derived functor vanish beyond a fixed degree, which converts infinite computations into finite ones. Serre's theorem — that a Noetherian local ring is regular exactly when its global dimension is finite — is the model result showing homological finiteness detecting geometric regularity.
ReferenceFrequently asked questions
Are minimal resolutions unique?
Minimal injective resolutions are, since injective hulls are unique. Minimal projective resolutions require projective covers, which exist only over perfect rings — over a local ring they do, and the Betti numbers they produce are genuine invariants.
What is a syzygy?
The kernel at each stage of a resolution — the relations among the chosen generators. Higher syzygies are obtained by iterating, and Hilbert's syzygy theorem bounds how far the process runs over a polynomial ring.
Does finite projective dimension imply finite injective dimension?
Not for individual modules, but the two suprema over all modules coincide, which is why global dimension is unambiguous. Rings where the finiteness conditions interact well — Gorenstein rings — are studied precisely because of this.
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