Hom, Tensor and Limits

Pontryagin duals: see Poonen's notes on Tate's thesis, for instance.

Flatness to-do

• Equivalent definitions of flatness, see e.g. Qing Liu... when is locally free a good way to think about it?
• Use both algebra (e.g. number theory) and geometry intuition
• MO: Why are flat morphisms "flat?"

My version/interpretation of Critch's answer: In topology, if a map $X\to Y$ is continuous, limiting behavior in $X$ corresponds to limiting behavior in $Y$. This can be useful if, say, $Y$ is simpler than $X$, and/or if we want to say something about $X$ based on properties of $Y$, say by pulling back a closed or open set of interest from $Y$ to $X$; or say (on the algebraic/analytic side) by pulling back functions of interest from $Y$ to $X$, like idempotents or curves or something.

Now, say $Y = \operatorname{Spec}(A)$ is flat over $X = \operatorname{Spec}(B)$; then e.g. the injection $A\to \operatorname{Frac}(A)$ (generic analysis/functions on base $Y$) induces an injection $B\to B\otimes \operatorname{Frac}(A)$ (allowing generic fiber analysis on $X$; see Qing Liu Chapter 4, Lemma 3.7).

Or say $A,B$ are local and we want to do differential calculus (tangent/cotangent spaces, Taylor series, ramification, etale, smoothness, etc.). Then the short exact sequence $$0\to \mathfrak{m}_y^r \to A \to A/\mathfrak{m}_y^r\to 0$$ induces a short exact sequence $$0\to \mathfrak{m}_y^r\otimes_A B \to B \to B/ \mathfrak{m}_y^rB\to 0.$$ In particular, as Critch mentions, the usually-just-a-surjective-map $$\mathfrak{m}_y^r\otimes_A B \to \mathfrak{m}_y^rB$$ is an isomorphism, so $\mathfrak{m}_y^rB$ should have "no surprise relations" (my currently attempted/preferred wording: it is an ideal of $B$ "cut out and/or described in terms of $A$, at least as much as one can hope for").

Remark: For all of the above, I'm not sure how much intuition (geometric or otherwise) is missing if one pretends $B$ is free over $A$.