/sci/ - Science & Math

[Anonymous No.6540215]

Hi /sci/,

Does anyone know any homological algebra here?

I was just wondering if anyone could explain intuitively what projective, injective, and flat modules are. Is there any type of geometric reasoning as to why they get those names? I find this subject to be really abstract and unmotivated, so I was hoping for some insight. Thanks.

[Anonymous No.6540243]

>>6540215
vector spaces over a field are projective, injective & flat (of course projective implies flat). recast the definitions in this context and hopefully that should help.

[Anonymous No.6540262]

>>6540243
Could you elaborate a little?

[Anonymous No.6540289]

>>6540262
what more is there to say? you have the definitions, you have an example. now go nuts, I'm watching footy.

[Anonymous No.6540298]

>>6540289
For example, what is the intuition/motivation for caring about when surjective homomorphisms can be lifted to projective modules (as per their definition)?

[Anonymous No.6540916]

bump

[Anonymous No.6540948]

Projectivity comes out of the idea of defining an algebraic object in terms of generators and relations.

For example, define the dihedral group D_8 in the usual way via generators and relations < x, y | x^4 = y^2 = (xy)^2 = 1>. This implicitly puts D_8 into a short exact sequence...

0 --> <x^4, y^2, (xy)^2> --> <x, y> --> D_8 --> 0

...and in some sense this is the One Correct Way to express D_8 as a quotient of a free group--which is something you very frequently want to be able to do if you want to understand homomorphisms to D_8.

Projectivity captures, in category-theoretic terms, the One Correct Way of expressing an algebraic object as a quotient. Specifically, a projective module is what you need for a projective *cover*, which is a covering map that's so good it *also covers all other possible covering maps*.

Injectivity is the dual notion, and flatness is...more complicated.

[Anonymous No.6540988]

"projection" always means reducing the number of dimensions via linear combinations.
"Injection" always means one-to-one.
"flat" means it does not permute sequences.

You are dealing with transformations that reduces the dimension of your modules, has unique mappings, and preserves order.

[Anonymous No.6541047]

Sorry, SES above should have 1s not 0s since it's groups, and I should have presented a SES of modules anyway. D_8 is just the first thing that came to mind for an example of generators and relations.

Another example would be something like...

0 --> <2> --> Z --> Z/2 --> 0

Projectivity means that if you have any other map Z/2n --> Z/2 --> 0, you can set up a diagram

Z --> Z/2 --> 0
v
Z/2n --> Z/2 --> 0

where the top and bottom edges are the given homomorphisms and the right edge is the identity. Then projectivity gives you a nice map that completes it to a commutative square.

Over Z this is way too much formalism, because Z is free. But if (for example) your base ring is complicated enough to decompose into direct summands, there are a lot of contexts where you only care about the weaker condition of projectivity.

[Anonymous No.6541049]

sigh, 4chan ate the spaces. in the diagram above that 'v' should be going from Z/2 to Z/2.

[Anonymous No.6541354]

Here's one connection to topology: a projective cover is (very roughly) analogous to the idea of a universal covering space. In both cases, you have a surjective map that in some sense can be seen as "standing above" all other surjective maps. Not all spaces can be universal covering spaces, and not all modules can be projective covers.

Like I said earlier, injectivity is just the formal dual of projectivity, but it's frequently harder to work with. Just as an example from abelian groups (i.e. Z-modules): the projective cover of Z/p is just Z, but the injective hull of Z/p is a Prüfer group--a much more complicated object.