How to calculate the pullback of sheafs I am trying to look at a concrete example of the pullback of sheaves. The easiest example to consider might be the closed embedding. $i: \operatorname{Spec} k[x] / (x) \to \operatorname{Spec} k[x]$. The definition of pullback sheaf is not as straightforward as the pushforward as it involves sheafification, so I am struggling to see what should the pullback sheaf be in this case.
I guess a better question would be in general how do you work with pullback sheaves. I find the definition so abstract that makes it very difficult to comprehend concretely.
 A: It is actually very straightforward to pull back a quasi-coherent sheaf along a map of affine schemes:
Proposition. If $\operatorname{Spec} B\to \operatorname{Spec} A$ is a map of affine schemes corresponding to the ring map $A\to B$, then the pullback of the quasi-coherent sheaf $\widetilde{M}$ on $\operatorname{Spec} A$ is the sheaf $\widetilde{M\otimes_A B}$ on $\operatorname{Spec} B$.
Proof. See Hartshorne proposition II.5.2(e), Vakil section 16.3, Qing Liu proposition 5.1.14(2), Stacks 01I9, etc. $\blacksquare$
In your specific case of the closed immersion $\operatorname{Spec} k[x]/(x) \to \operatorname{Spec} k[x]$, this means that for any quasi-coherent sheaf $\widetilde{M}$ the pullback is the sheaf associated to $M/xM$. In general, the answer to "how do we deal with pullbacks" depends on the presentation of the sheaf that we're given and what we hope to do with it, but the above lemma is a great basic tool in our toolbox: assuming we're working with a quasi-coherent sheaf, we can always break our map $X\to Y$ up in to maps of affines glued together suitably, use the above procedure to calculate the result on each piece, then patch everything together. (This would generally be somewhat tedious and surpassed by other technology when available - for instance, pulling back line bundles can sometimes be a little easier by doing calculations with divisors using the line bundle/divisor correspondence.)
