Properties characterized by a vanishing Ext or Tor module

Question

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!

Answer 1

Some more examples:

1. 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 ${\mathbb Z}$.

2. 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 $\text{Ext}^n_R(k,M)\neq 0$.

3. 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.

Answer 2

Here are some more specific examples from representation theory.

I will take examples from representations of reductive algebraic groups, but most of this generalizes to for example complex semisimple Lie algebras.

So let $G$ be a reductive algebraic group over the algebraically closed field $k$ and let $B$ be a fixed Borel subgroup.

If $\lambda$ is a $1$-dimensional representation of $B$, we define $\nabla(\lambda)$ to be the $G$-module induced from $\lambda$ (which is finite dimensional). We also define $\Delta(\lambda)$ to be the dual of $\nabla(\lambda')$ where $\lambda' = -w_0(\lambda)$ and $w_0$ is the longest element of the Weyl group for $G$ (for full details of this, one should see for example Jantzen's book).

A filtration of a $G$-module is called good if the successive quotients are isomorphic to modules of the form $\nabla(\lambda)$ for suitable $\lambda$. It is then a result of Donkin that a finite dimensional $G$-module $M$ has a good filtration if and only if $\operatorname{Ext}_G^1(\Delta(\lambda),M) = 0$ for all $\lambda$.

As I mentioned, the above also works for complex Lie algebras, where the $\Delta(\lambda)$ correspond to Verma modules, and the $\nabla(\lambda)$ are the dual Verma modules.

Getting back the the case of algebraic groups, note that if $k$ has characteristic $0$ then this whole story is largely uninteresting, since all the modules involved are semisimple. But in positive characteristic, it becomes a lot more interesting.
A further example of the above can be obtained if we introduce some further modules. We now assume that $k$ has characteristic $p> 0$ (this is becoming rather long and the next example is rather specific, I just happen to like it a lot because I have done a lot of research on this topic).

Among the modules of the form $\nabla(\lambda)$ there are some which are particularly nice, namely $\nabla((p^r-1)\rho)$ where $\rho$ is half the sum of the positive roots (I am omitting all details of what this actually means and how this is a $B$-module). We call these the Steinberg modules and denote them by $\operatorname{St}_r$. What we need to now about them for now is that they are in fact simple and also isomorphic to $\Delta((p^r-1)\rho)$.

The example I have in mind is that $\operatorname{St}_r\otimes M$ has a good filtration if and only if $\operatorname{Ext}_G^1(\Delta^{(p,r)}(\lambda),M) = 0$ for all $\lambda$. I will not go into the definition of the $\Delta^{(p,r)}(\lambda)$ here, as that would require way too much additional details (they can all be found in my paper with Dan Nakano http://arxiv.org/abs/1403.7011). One of the main interests of this criterion is that conjecturally it is equivalent to the existence of a certain type of filtration on $M$, though so far this is only a conjecture. One direction is known when the prime is large enough (a result of Andersen) or in some small rank cases. In the other direction, the best so far is that a consequence of it (that the $\nabla(\lambda)$ have this type of filtration) is known to be true when the prime is large enough that the Lusztig conjecture holds (a result of Parshall and Scott).

Answer 3

My favorite is the Castelnuovo-Mumford Regularity, which can be characterized by the vanishing of Ext or Tor equivalently. I will not describe it here as there are many references, e.g. Eisenbud or Bruns and Herzog. Many other sources online too. To give you an idea though, the CM regularity has to do with the "complexity" of graded modules.