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I've found that when studying commutative algebra, thinking of things in terms of their algebro-geometric interpretation helps them stick as well as motivates otherwise odd and abstract concepts.

(For example, thinking of localization as referring to open sets, completions to infinitesimal neighborhoods, primary decomposition to irreducible components, etc.)

I'd like to do the same for concepts from homological algebra. I'm dimly aware that projective modules correspond geometrically to vector bundles (i.e. to the condition of being locally free) at least in nice cases, and flatness seems to have a deeply geometric interpretation as meaning something difficult to describe about "smoothly varying families". Is there a similarly geometric way to think about injective modules? If not, is there some intuition to why dualizing the concept of a projective module breaks the geometric correspondence?

If you're aware of other nice correspondences between geometric ideas and homological algebra constructions, I'd also love to hear them

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    $\begingroup$ The following is far from precise, but intuitively Injectivity has to do with "flabbiness" geometrically. An injective sheaf $F$ is one for which any morphism $A\to F$ lifts to one $B\to F$ whenever $A$ is a subsheaf of $B$. Unwinding that to a geometric interpretation, for a space X with open subspace $Y$, any function on a space taking values in $F$ on $Y$ comes from one on $X$. As a geometric example, the rational functions $k(x)$ form an injective module over $k[x]$: Any rational function on a (Zariski)-open subset of the affine line is the restriction of a one on the whole line. $\endgroup$ – John Brevik Aug 10 '15 at 10:10

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