# What makes the stable module category stable?

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).

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The root meaning of the English word stable is something like "unchanged/unchanging/unmoved/unmoving" (or "resistant to change/motion". The word (or a related word "stably" or "stabilize") has entered mathematics many times, mostly independently. I would not expect any two the mathematical meanings to have much to do with other, except in the sense that they are related (at least distantly) to the general meaning. By the way, what is a (or the) stable module category? – Tom Goodwillie Jul 29 '10 at 12:48
In this case, seeing as the stable module category and stable homotopy category are both tensor triangulated, I was just wondering if "stable" meant something similar in both cases. If not, then part 2 of my question still applies. Also, if anyone knows the history of the term, that could be useful. – Dylan Wilson Jul 29 '10 at 18:34
I would say that the name comes from the fact that you are, in part, looking at modules modulo projective summands, much as one speaks of, say, vector bundles being stably parallelizable and so on. – Mariano Suárez-Alvarez Jul 29 '10 at 18:38
Sorry, but I'm not "one"- can you explain what stably parallelizable is? Also, I thought every (smooth) vector bundle was a projective module already, so what are you factoring out? – Dylan Wilson Jul 29 '10 at 18:47
Well, in the case of vector bundles one uses "stably" to mean "up to free summands", because those are the uninteresting ones; when you have a Frobenious algebra, experience has shown that the uninteresting summands are (in some contextes!) the projective modules, and so on. The word "stable" is even used when one constructs the Grothendieck group of an abelian semigroup (where the equivalence relation $(a,b)\sim(a'b')$ that is used is that $a+b'=a'+b$ stably, that is, up to adding something to both sides) – Mariano Suárez-Alvarez Jul 29 '10 at 18:57

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.

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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.

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