A Geometric Description of Injective Modules

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

The analogous of a module $M$ over a ring $A$ is a (quasi coherent) sheaf of modules over a scheme $X$. Indeed, in the case $X=\operatorname{Spec} A$ is an affine variety, the two notions coincide. When

First point: how one can think of a sheaf $F$ over $X$? Well, you can take the associated étalé space and think of it as a space $p: E \to X$, such that

$$F(U) \simeq \{e \in E: p(e) \in U\}$$

Note that sections are $R$-modules in this case. In other words, you can always think a sheaf as a sheaf of sections. A cool way to think about sections are a sort of "generalized functions" that can take more than one value. For example, take the function $x \mapsto x^2$ over $\mathbb{C}$: the local sections in a neighborhood of 1 are exactly the two functions $\sqrt{x}, -\sqrt{x}$.

Another set of geometric examples of sheaf - despite less general and less common in algebraic geometry - is the sheaf of solutions to a given constraint (as a differential equation) : locally it can have a lot of solutions, but it is not necessarily true that they glue together to a global solution. This point of view is useful because is more common to have an action of a ring $R$ on the space of solutions. For example, if they have some symmetry under the action of a group and stable for multiplication by a real constant, the ring $\mathbb{R}[G]$ is acting on the space of solutions; however in this situation it is much more natural to have a ""continous"" group of symmetries (like rotations) , and one considers the space of solutions as a module over the associated lie algebra.

Note that the example of "generalized functions" above is a special case of the "solutions to a constraint" case: sections of the function $x^2 \mapsto x$ are exactly the complex functions that satisfy $f(x)^2 = x$.

Second point: what does it mean to be an injective sheaf? The most "neat" consequence is that its cohomology groups are all zero (except the zero-th one which coincide with the module).

Intuitively, I think as "having zero cohomology" like being without holes of any dimension. A concrete reincarnation of this intuition is the fact that an injective sheaf is "flasque", or " flabby": this means that any local section extends to a global section (read: any local solution extends tk a global solution!).

Now we see a completely different approach, which is equivalent to be injective but not pictorial as the previous.

Recall the criterion for which is enough to check that every morphism $I \to M$ extends to a morphism $R \to M$, where $I$ is an ideal of $\mathbb{R}$. Suppose the ring is noetherian, so that the ideal $I$ is generated by a finite number of elements $f_1, \ldots, f_n$. This means we have an exact sequence

$$ 0 \to K \to R^n \to I \to 0 $$

And that a map from $I$ to $M$ is the same as choosing n elements $m_1, \ldots, m_n\in M $ satisfying the relations prescribed by $K$. Finding an extension to $R$ means to find an element $m \in M$ such that $f_j m = m_j$. Let us look at an easy example: if you take an ideal generated by one element $(f) $, for any $m_1 \in M$ you can find an element $m$ such that $fm = m_1$. In other words, you have "" $ m =m_1/f$, that is the module is divisible. For PID like $\mathbb{Z}$ this is equivalent to be injective. In general, you can always find a "common divisor" given that the elements satisfy the obvious relations they should satisfy. If for example $R= \mathbb{C}[x, y]$ and you want an element $m$ such that $xm = m_1, ym= m_2$, it's necessary that $m_1, m_2$ satisfy $ym_1 = xym = xm_2$.

In terms of sheaves, this is not much more significant. Given an ideal $I=(f_1, \ldots, f_r) $, it means that if some global sections $m_1, \ldots, m_r$ satisfies the relations analogous to the relations satisfied by $f_1, \ldots, f_r$, you can always find a global section $m$ such that $f_j m = m_j$.