Why the stable module category?

Question

Let $R$ be a ring (usually assumed to be Frobenius). The stable module category is what you get when you take the category $\mathsf{Mod}_R$ of $R$-modules, and kill the projective modules. (Of course, if $R$ is Frobenius, these are the same as the injective modules.)

Question 1: Why do we kill the projective modules? What did they in particular do to provoke our wrath?

Obviously, one motivation for the construction is that it happens to result in a triangulated category (and I think, if done correctly, a stable $\infty$-category). But on the one hand, there are other ways of extracting a stable $\infty$-category from $R$, such as taking the derived category. And on the other hand, surely there are other classes of modules we could have chosen to kill to get the same effect? For instance, why not kill all dualizable modules or something (assuming $R$ is a Hopf algebra)?

Question 2: What is the geometric interpretation of the stable module category?

Question 3: What is the relationship between the stable module category and the singularity category of $R$?

Answer 1

To question 1: One big motivation for me is that two Frobenius algebras can be stable equivalent but not Morita equivalent and a classification up to stable equivalence can be very nice.

For example a classification of representation-finite Frobenius algebras up to Morita equivalence is quite messy (and only known for algebraically closed field, so for simplicitly assume that field are algebraically closed of characteristic 0).

But the classification up to stable equivalence is very nice and can be used to prove things about a large class of algebras by looking only at a much smaller and simpler class of algebras.

For example if you want to prove that the stable endomorphism ring of every indecomposable module of a Brauer tree algebras is isomorphic to $K[x]/(x^n)$ (for some $n$) you can use that all Brauer tree algebras are stable equivalent to symmetric Nakayama algebras where the calculation is almost trivial as even the usual endomorphism rings are isomorphic to $K[x]/(x^n)$ for some $n$. As a bit more non-trivial example you can also determine all stable endomorphism rings or Extension groups (or any other stable invariance) for general representation-finite Frobenius algebras in that way.

Classification results for certain important classes of modules like cluster tilting modules for selfinjective algebras also only depend on the stable module category.

Another example of nice stable equivalence is the rings of simple singularities of type $A_n$, where the stable category of maximal Cohen-Mcaulay modules is stable equivalent to the stable module category of $K[x]/(x^n)$.

Another answer to question 1 might be that certain important functors are only well-defined in the stable module category but not the whole module category, such as the syzygy functor or the Auslander-Reiten translate.

To question 2: To get a "geometric" picture at least for a representation-finite stable category you can take the stable endomorphism ring, which is often a finite dimensional quiver algebra. For example taking a ring of simple singularities of Dynkin type you will get the preprojective algebra of that Dynkin type as the stable endomorphism ring of the direct sum of all maximal Cohen-Macaulay modules.

To question 3: The stable category of maximal Cohen-Macaulay modules of a Gorenstein ring $R$ is equivalent to the singularity category by a famous result of Buchweitz, see for example in the book https://bookstore.ams.org/view?ProductCode=SURV/262 . When $R$ is Frobenius, then this is just the whole stable module category.

Answer 2

One reason is just that $\text{stab}_{kG}(k,M)_*=\widehat{H}^{-*}(g;M)$ (the Tate cohomology of $G$ with coefficients in $M$). I think that Tate invented Tate cohomology for applications in Class Field Theory, and there are applications in various other places as well; if you are interested in any of these then it is natural to consider the stable module category. As another example, one can consider the Adams spectral sequence $$ \text{Ext}^{st}_{\mathcal{A(1)}}(H^*(X;\mathbb{F}_2),\mathbb{F}_2) \Longrightarrow kO_{t-s}(X) $$ It is often convenient to treat the $\text{Ext}^0$ term separately and think of the higher $\text{Ext}$ groups as morphism sets in the stable module category. This is because there are often formulae like $M\otimes N=L\oplus F$ where $F$ is free but large and $L$ is small and given by a tidy formula. I think that similar things happen with $\text{tmf}$ and $\mathcal{A}(2)$.

In equivariant stable homotopy theory, the counterpart of Tate cohomology is the functor $X\mapsto \widetilde{EG}\wedge F(EG_+,X)$ (or $X\mapsto (\widetilde{EG}\wedge F(EG_+,X))^G$) on $G$-spectra. More generally, if $\mathcal{F}$ is a family of subgroups of $G$ closed under subconjugacy, there is an essentially unique $G$-space $E\mathcal{F}$ with $E\mathcal{F}^H$ contractible (for $H\in\mathcal{F}$) or empty (for $H\not\in\mathcal{F}$). We also write $\widetilde{E\mathcal{F}}$ for the unreduced suspension of $E\mathcal{F}$. Given any two such families, one can consider the functor $X\mapsto\widetilde{E\mathcal{E}}\wedge F(E\mathcal{F}_+X)$, which is a generalisation of Tate cohomology. Functors of this type appear frequently in the literature. I would guess that people have also considered similar things in purely algebraic categories. A key property of $\widetilde{E\mathcal{F}}$ is that it is a smashing localisation of the unit object and so is idempotent, and the corresponding properties have certainly been used in other tensor-triangulated contexts.

Answer 3

At least when (in addition to OP assumptions) $R$ is a finite-dimensional algebra over a field, there is a theorem of Rickard which says that if we consider the bounded derived category $D^b(R\text{-mod})$ of finitely generated $R$-modules, its Verdier quotient by the perfect complexes is precisely the stable module category (of finitely generated $R$-modules): $$D^b(R\text{-mod})\, / \,D^{\text{perf}}(R) \cong \operatorname{stmod}(R) \, .$$

If $R$ further is a cocommutative bialgebra, these triangulated categories are tensor-triangulated (tt) categories in Balmer's sense, so have an associated tt-spectrum $\operatorname{Spc}$ which is often considered a geometric object. With this point of view $D^{\text{perf}}(R)$ is often "small", or put another way the derived category and the stable module category are "close". For example when $R=kG$ for a finite group $G$, these spectra can be interpreted (via work of Benson-Carlson-Rickard in the 90s) in terms of the cohomology algebra $\operatorname{H}^*(G;k)$ as $$\operatorname{Spc}(D^b(kG\text{-mod})) \cong \operatorname{Spec}^h(\operatorname{H}^*(G;k))$$ $$\operatorname{Spc}(\operatorname{stmod}(kG)) \cong \operatorname{Proj}(\operatorname{H}^*(G;k))$$ where the former has only one more (closed) point.