Talk:Projective module

Section and split exact sequences

There was a recent concern about an equivalence of sections and direct sum decompositions for modules. First I'll describe the equivalence, and then describe what probably went wrong with the proposed example (an edit summary is not long enough to specify such an example, so one must interpret).

If f:P->M is surjective, and h:M->P is such that f(h(m))=m for all m in M, and both f,h are R-module homomorphisms, then P is the direct sum of ker(f) and im(h), and the image of h is isomorphic to M. An easy proof is that (1) P=ker(f)+im(h) since x = (x - h(f(x))) + h(f(x)), and f( x - h(f(x)) ) = f(x) - f(h(f(x))) = f(x) - f(x) = 0, so the first summand is in the kernel of f, and the second summand is in the image of h. If x is in the intersection of ker(f) and im(h), then x=h(m), but then 0 = f(x) = f(h(m)) = m, and x=h(0)=0. Finally, since f and h are R-module homomorphisms, im(h) and ker(f) are R-submodules of P.

The group ring example might have a few different interpretations, but either the implicit h is not a G-homomorphism, or the composition is not the identity. Since there is a copy of the trivial module Z contained in the center of ZG and a G-homomorphism from Z to that copy, I'll assume we are in the latter case: h:Z->Z(G):n->n*Sum(g,g in G). But then f(h(1)) = |G|, rather than 1. The composition is not the identity, but rather |G| times the identity. Now if |G| is invertible in the ring (changing Z to just some ring R), then one can fix this little snag, but then the trivial module R *is* a direct summand of RG. JackSchmidt (talk) 14:21, 26 March 2008 (UTC)[reply]

Locally free redirects here

Clearly there should be an extra page with at least the definition.T3kcit (talk) 20:40, 16 August 2008 (UTC)[reply]

Examples

Could someone please provide a few examples of non-free projective modules? —Preceding unsigned comment added by 128.40.136.156 (talk) 13:08, 3 March 2009 (UTC)[reply]

Done. Non-principal ideals of Dedekind domains are popular examples, but simpler examples are direct factor of direct product rings, or the natural module of a full matrix ring. JackSchmidt (talk) 14:32, 3 March 2009 (UTC)[reply]

Here is a counter example, a vector bundle which is not a projective module, https://math.stackexchange.com/questions/697681/graded-projective-modules-and-vector-bundles-on-projective-varieties — Preceding unsigned comment added by 65.153.185.86 (talk) 19:37, 9 December 2019 (UTC)[reply]

Basic Examples

The article needs to have examples of projective modules (including explicit ones) and explicit examples of non-projective models. MathKnight-at-TAU (talk) 14:41, 4 December 2012 (UTC)[reply]

There are some examples and non-examples in the section "Properties". Arcfrk (talk) 00:14, 5 December 2012 (UTC)[reply]

Mistake in the subsection titled Exactness

If A is a non-commutative ring, and if M,N are two left A-modules, then the set of A-linear morphisms from M to N is an abelian group. It is not an A-module. At best, it is a module over the center of A! Any text on noncommutative algebra will say it. Therefore, I am reverting once again the edit of Lazard. 85.65.231.247 (talk) 15:39, 1 April 2014 (UTC)[reply]