Question

When geometrically flavoured words like "mapping cone" or "chain homotopy" crop up in homological algebra, there's usually a good reason. (In this case, looking at the chain complexes associated with each geometric construction gives the algebraic construction).

But, for the life of me, I cannot find a reason why the "stable module category" deserves to be called "stable." It doesn't seem, superficially at least, to be related to the stable homotopy category. So:

1. Are these categories related in any nontrivial manner that gives merit to the terminology?

2. If not, is there some other reason why the stable module category deserves to be called stable? (Mind you, this sounds like it might have a higher-categorical answer, and my knowledge of higher category theory is very sketchy... so do be gentle if that's the direction your answer is taking!)

EDIT: The stable module category for a ring $R$ has $R$-modules for objects and maps "modulo projectives" for morphisms. That is, we put an equivalence relation on $Hom_{R}(M,N)$ by declaring $f \sim 0$ if $f$ factors through a projective module. For certain rings (e.g. group rings), this category is tensor triangulated (which is why I'm interested).

Answer 1

The answer to the first question is yes, the two categories are nontrivially related: Both the traditional stable homotopy category and the stable module category are examples of stable homotopy categories in the sense of Hovey-Palmieri-Strickland, Axiomatic Stable Homotopy Theory. See the first few pages of Schwede's Stable model categories are categories of modules for a convenient overview. Of special interest are Definition 2.1.1 and the subsequent paragraph on page 107, and Example 2.4.(v) on page 111. You may also find it useful to consult Hovey's Model Categories, especially Chapters 2 and 7, as well as Chapter I.2 in Quillen's Homotopical Algebra.

As for the second question, observe the following extract from Example 2.4.(v):

Fortunately, the two different meanings of ‘stable’ fit together nicely; the stable module category is the homotopy category associated to an underlying stable model category structure [21, Section 2].

so it sounds like Schwede would agree with Mariano and Tom that, at least originally, the selection of the word 'stable' in stable module category likely had nothing to do with stable homotopy and all that.

Corrections are welcome.

Answer 2

In stable homotopy theory we have inverted the suspension map, this is what we are stabilizing with respect to (at least that is part of it, we are inverting some other things as well, the quasi-isomorphisms). We can also do this in most graded category, if we have a graded module simply define $(\Sigma M)_n = M_{n-1}$. What is mentioned above in the comments is the stable homotopy category or even just the homotopy category. The homotopy category is essentially the same as the derived homotopy category, the stable homotopy category is the homotopy category where we have inverted the suspension map mentioned above. We can do this easily by thinking of $Hom(M,N)$ as a graded object where the degree n part is chain maps that lower degree by n, that is $f_n: M_n \to N_{n-1}$ would be a map of degree 1 (I may have this backwards, this should coincide with the grading of the Ext groups). This does in fact come from a geometric construction, recall that $H^{n-1}(\Sigma X) \cong H^n(X)$.

so if you are interested in $ku$, your objects will be graded modules over $E[Q_0,Q_1]$ (the exterior algebra generated by Milnor's $Q_0$ and $Q_1$ with coefficients in $\mathbb(F)_2$, and just in case $Q_0=sq^1$ and $Q_i=[sq^{2^i},Q_{i-1}]$ (commutator)). Note that this is not the categopry of $ku$-modules, this is the category we need to understand for use in the appropriate Adams spectral sequence after the change of rings isomorphism. Since the steenrod operations are stable, they commute with the suspension isomorphism above, the suspension described above is exactly what we want. So we can study this "nice" little category and learn quite a bit about $ku$-modules. We will eventually take the stable category, where we do not distinguish between quasi isomorphic modules or suspensions. These will work in the expected fashion with respect to the ext calculations that you feed into the Adams spectral sequence that will then turn into "homotopy theory".

any corrections are very welcome

Answer 3

This is how I've always interpreted it, but I've never seen direct evidence. I'll talk about Tate cohomology, which is a fragment of the stable module category.

The modern point of view is that cohomology is the derived functor of invariants. But Cartan and Eilenberg talked more in terms of satellite functors. The right satellite of $H^i$ is $H^{i+1}$ meaning that $H^{i+1}(X)=H^i(Y)$, where $Y$ is the quotient of an injective by $X$. We can ask about satellites for more functors than derivable ones. In the setting of modules over a finite group, $H^1$ is the left satellite of $H^2$. Tate cohomology is what we get by going back and forth, taking both left and right satellites, until it the process stabilizes and we get functors indexed by $\mathbb Z$ such that each is its neighbors' satellites, in both directions.