In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. Every free module is a projective module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains. However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a (multivariate) polynomial ring over a field (this is the Quillen–Suslin theorem). Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan and Samuel Eilenberg.
Definitions
Lifting property The usual category theoretical definition is in terms of the property of lifting that carries over from free to projective modules: a module P is projective if and only if for every surjective module homomorphism f : N ↠ M and every module homomorphism g : P → M, there exists a module homomorphism h : P → N such that fh = g. (We don't require the lifting homomorphism h to be unique; this is not a universal property.)
The advantage of this definition of "projective" is that it can be carried out in categories more general than module categories: we don't need a notion of "free object". It can also be dualized, leading to injective modules. The lifting property may also be rephrased as every morphism from P {\displaystyle P} to M {\displaystyle M} factors through every epimorphism to M {\displaystyle M} . Thus, by definition, projective modules are precisely the projective objects in the category of R-modules.
Split-exact sequences A module P is projective if and only if every short exact sequence of modules of the form
0 → A → B → P → 0 {\displaystyle 0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0}
is a split exact sequence. That is, for every surjective module homomorphism f : B ↠ P there exists a section map, that is, a module homomorphism h : P → B such that fh = idP. In that case, h(P) is a direct summand of B, h is an isomorphism from P to h(P), and hf is a projection on the summand h(P). Equivalently,
B = Im ( h ) ⊕ Ker ( f ) where Ker ( f ) ≅ A and Im ( h ) ≅ P . {\displaystyle B=\operatorname {Im} (h)\oplus \operatorname {Ker} (f)\ \ {\text{ where }}\operatorname {Ker} (f)\cong A\ {\text{ and }}\operatorname {Im} (h)\cong P.}
Direct summands of free modules A module P is projective if and only if there is another module Q such that the direct sum of P and Q is a free module.
Exactness An R-module P is projective if and only if the covariant functor Hom(P, -): R-Mod → Ab is an exact functor, where R-Mod is the category of left R-modules and Ab is the category of abelian groups. When the ring R is commutative, Ab is advantageously replaced by R-Mod in the preceding characterization. This functor is always left exact, but, when P is projective, it is also right exact. This means that P is projective if and only if this functor preserves epimorphisms (surjective homomorphisms), or if it preserves finite colimits.
Dual basis A module P is projective if and only if there exists a set { a i ∈ P ∣ i ∈ I } {\displaystyle \{a_{i}\in P\mid i\in I\}} and a set { f i ∈ H o m ( P , R ) ∣ i ∈ I } {\displaystyle \{f_{i}\in \mathrm {Hom} (P,R)\mid i\in I\}} such that for every x in P, fi(x) is only nonzero for finitely many i, and x = ∑ f i ( x ) a i {\displaystyle x=\sum f_{i}(x)a_{i}} .
Elementary examples and properties The following properties of projective modules are quickly deduced from any of the above (equivalent) definitions of projective modules:
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