Skip to main content
MathOverflow

Return to Question

added definition of stable module category
Source Link
Dylan Wilson
Dylan Wilson
  • 13.5k
  • 9
  • 64
  • 108

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).

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!)

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).

Source Link
Dylan Wilson
Dylan Wilson
  • 13.5k
  • 9
  • 64
  • 108

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!)