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All of this is coming from the point of view of being a student. I took a commutative algebra course last winter and am currently taking a homological algebra course, so maybe I can touch on your differentiation between those two perspectives.

It sounds like you want the motivating ideas, and I think those make the most sense homologically. I completely understand your not wanting to introduce Tor and/or Ext, but you don't have to. $Hom_R (P,-)$ is exact iff $P$ is projective, $Hom_R (-,I)$ is exact iff $I$ is injective, and $F \otimes_R -$ is exact iff $F$ is flat. These can even be taken as definitions (these are equivalent to $Tor_1$ and $Ext^1$ vanishing which is each equivalent to all the higher derived functors vanishing).

Edit: What if you don't know why you should care about exact sequences as Pete asked in a comment? Well, you should still care about when functors preserve injections (tensor-ing with flat modules) and surjections (Hom-ing out of projective modules), and this is precisely what flat and projective do for you. This is the only place where exactness might fail, so just skip it and talk about injections and surjections and when they are preserved.

This is how I feel most comfortable thinking about these concepts. It is pretty concrete and not far away from Commutative Algebra (I think, meaning that this is how we talked about it in my commutative algebra class).

I should also mention I appreciate you posting all of those course notes, it is a great resource for us students out there.(I apologize in advance if my tex is off, I will fix it later at a machine that understands it)

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All of this is coming from the point of view of being a student. I took a commutative algebra course last winter and am currently taking a homological algebra course, so maybe I can touch on your differentiation between those two perspectives.

It sounds like you want the motivating ideas, and I think those make the most sense homologically. I completely understand your not wanting to introduce Tor and/or Ext, but you don't have to. $Hom_R (P,-)$ is exact iff $P$ is projective, $Hom_R (-,I)$ is exact iff $I$ is injective, and $F \otimes_R -$ is exact iff $F$ is flat. These can even be taken as definitions (these are equivalent to $Tor_1$ and $Ext^1$ vanishing which is each eqquivalent equivalent to all th4e the higher derived functors vanishing).

Edit: What if you don't know why you should care about exact sequences as Pete asked in a comment? Well, you should still care about when functors preserve injections (tensor-ing with flat modules) and surjections (Hom-ing out of projective modules), and this is precisely what flat and projective do for you. This is the only place where exactness might fail, so just skip it and talk about injections and surjections and when they are preserved.

This is how I feel most comfortable thinking about these concepts. It is pretty concrete and not far away from Commutative Algebra (I think, meaning that this is how we talked about it in my commutative algebra class).

I should also mention I appreciate you posting all of those course notes, it is a great resource for us students out there. (I apologize in advance if my tex is off, I will fix it later at a machine that understands it)

1

All of this is coming from the point of view of being a student. I took a commutative algebra course last winter and am currently taking a homological algebra course, so maybe I can touch on your differentiation between those two perspectives.

It sounds like you want the motivating ideas, and I think those make the most sense homologically. I completely understand your not wanting to introduce Tor and/or Ext, but you don't have to. $Hom_R (P,-)$ is exact iff $P$ is projective, $Hom_R (-,I)$ is exact iff $I$ is injective, and $F \otimes_R -$ is exact iff $F$ is flat. These can even be taken as definitions (these are equivalent to $Tor_1$ and $Ext^1$ vanishing which is each eqquivalent to all th4e higher derived functors vanishing).

This is how I feel most comfortable thinking about these concepts. It is pretty concrete and not far away from Commutative Algebra (I think, meaning that this is how we talked about it in my commutative algebra class).

I should also mention I appreciate you posting all of those course notes, it is a great resource for us students out there. (I apologize in advance if my tex is off, I will fix it later at a machine that understands it)