arXiv subjects

Matthew Grime

Publications and source records attributed to Matthew Grime.

6 recordsLinked to original sources

arXiv2008

Some relative stable categories are compactly generated

Let G be a finite group. The stable module category of G has been applied extensively in group representation theory. In particular, it has been used to great effect that it is a triangulated category which is compactly generated. Let H be a subgroup of G. It is possible to define a stable module category of G relative to H. It too is a triangulated category, but no non-trivial examples have been known where this relative stable category was compactly generated. We show here that the relative stable category is compactly generated if the group algebra of H has finite representation type. In characteristic p, this is equivalent to the Sylow p-subgroups of H being cyclic.

math.GR↗

arXiv2007

The correct relatively stable category for idempotent modules

We answer a question posed by Carlson, Peng, and Wheeler, and demonstrate that in general Rickard modules in relatively stable categories are not idempotent modules even if one localizes with respect to a tensor ideal subcategory. We also show that there is a modification one can make so as to recover the idempotent behaviour.

math.RT↗

arXiv2007

Precovers, localizations and stable homotopy

We prove a new localization theorem for stable model categories if the localizing subcategory is generated by a precovering class in the model category. We use this to show how one may explicitly realize certain Bousfield localization functors that arise naturally in the study of relative homological algebra for group algebras.

math.CT↗

arXiv2007

Finite dimensional modules and perpendicular subcategories

We explain how, under some hypotheses, one can construct a sequence of finite dimensional $kG$-modules that lie in certain prescribed additive subcategories, but whose direct limits do not. We use these to show that many of the triangulated quotients of $\Mod$ are not generated, as triangulated categories, by the corresponding quotient of $\mod$ considered as a full subcategory.

math.RT↗

arXiv2007

Adjoint functors and triangulated categories

We give a construction of triangulated categories as quotients of exact categories where the subclass of objects sent to zero is defined by a triple of functors. This includes the cases of homotopy and stable module categories. These categories naturally fit into a framework of relative derived categories, and once we prove that there are decent resolutions of complexes, we are able to prove many familiar results in homological algebra.

math.CT↗

arXiv2006

Bousfield localization in quotients of module categories

We examine various triangulated quotients of the module category of a finite group. We demonstrate that these are not compactly generated by the simple modules and present a modification of Rickard's Idempotent Module construction that accounts for this. When the localizing subcategories are sufficiently nice we give an explicit description of the objects in the Bousfield triangles for modules that are direct limits of sequences of finite dimensional modules in terms of homotopy colimits.

math.RT↗