While reading Weibel's "An introduction to homological algebra'', I've noticed that many properties of a module are characterized by the vanishing of some Tor or Ext. Fix a (commutative) ring $R$ and let $\textrm{Mod}_R$ denote the category of $R$-modules, then 2 easy examples of this phenomenon are given below.
$\bullet$ A module $B \in \textrm{Mod}_R$ is flat iff $\textrm{Tor}_1^R(A,B) = 0$ for all $A \in \textrm{Mod}_R$.
$\bullet$ The projective dimension of $A \in \textrm{Mod}_R$ is $\leq d$ iff $\textrm{Ext}_{R}^{d+1}(A,B) = 0$ for all $B \in \textrm{Mod}_R$ (there are analogous statements for flat/injective dimensions).
What other properties of modules can be characterized in this way? Is there a broader 'theme' dictating when these characterizations arise?
Edit: if the given condition can be interpreted as saying something about the geometry of $\textrm{Spec }(R)$, I'd be very interested in hearing it!
Best Answer
Some more examples:
You can sometimes also restrict the class of test-modules: For example, by Baer's criterion, a left $R$-module $M$ is injective if and only if $\text{Ext}^1_R(R/I,M)=0$ for all left ideals $I\lhd R$.
It interesting to note, however, that this is not possible for projective modules: It was shown by Trlifaj that it is consistent with $\textsf{ZFC+GCH}$ that no non-perfect ring has a 'test-module for projectivity'. This is due to the high asymmetry in the properties of module categories, or more generally Grothendieck categories: filtered colimits are required to be exact, but there's no similar requirement for limits.
As a famous example, the Whitehead problem asks whether ${\mathbb Z}$ is a test-module for projectivity over ${\mathbb Z}$.
Over a Noetherian local ring $(R,{\mathfrak m})$ with residue field $k := R/{\mathfrak m}$, you can often restrict to considering $\text{Ext}$ or $\text{Tor}$ groups of an $R$-module with $k$. For example: the projective/flat dimension of a finitely generated $R$-module $M$ is the largest $n\geq 0$ such that $\text{Ext}^n_R(M,k)\neq 0$, and also the largest $n\geq 0$ such that $\text{Tor}_n^R(M,k)\neq 0$. Similarly, the injective dimension is the largest $n\geq 0$ such that $\text{Ext}^n_R(k,M)\neq 0$, and the depth is the smallest $n\geq 0$ such that $\text{Ext}^n_R(k,M)\neq 0$.
Many classes arise as the left- or right-orthogonal with respect to $\text{Ext}^1_R$ of another class. They are studied under the name of cotorsion pairs: A cotorsion pair is a pair $({\mathcal C},{\mathcal D})$ of classes of $R$-modules, say, such that ${\mathcal C} = {^{\perp}}{\mathcal D}$ and ${\mathcal D}={\mathcal C}^{\perp}$, where $(-)^{\perp}$ and ${^{\perp}}(-)$ denote right- and left-orthogonal with respect to $\text{Ext}^1_R$, respectively.