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I was reading the definitions of projective modules and flat modules and found myself a bit unenlightened (by all of their equivalent definitions). At least the Wikipedia articles for these classes of modules did not provide enough (if any) motivation for the definitions. Is there an intuition behind these definitions that could help me understand what's going on here?

Thigh High Crocs
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    If you have a (compact Hausdorff) space $X$, then vector bundles over $X$ are in 1-1 correspondence with projective $C(X)$-modules. This gives a geometric flavor. Algebraically, they are important in homological algebra. – Just a user Jun 14 '22 at 15:59
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    For homology and cohomology of groups, for example, this is very useful. Finding a projective resolution of a $G$-module is very helpful. This arises in number theory, homological algebra, algebraic topology, commutative algebra etc. – Dietrich Burde Jun 14 '22 at 18:07
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    Actually, you have asked yourself about it here, because a free resolution is also a projective resolution. – Dietrich Burde Jun 14 '22 at 18:15
  • @Justauser I like this as an example from outside of homological algebra -- it brings the definition down to Earth. – Thigh High Crocs Jun 14 '22 at 21:00

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Homologically speaking, projectivity/injectivity/flatness are related to exactness of certain functors people are interested in.

Specifically:

  1. $\mathrm{Hom}(-,E)$ is exact precisely when $E$ is injective
  2. $\mathrm{Hom}(P,-)$ is exact precisely when $P$ is projective
  3. $-\otimes L$ is exact precisely when $L$ is flat.

When those functors are exact, you can draw more conclusions when applying them, as opposed to more mundane functors. It is apparently very valuable to people who understand the calculus of exact sequences.

rschwieb
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