The articles below have links to where they can be read for free. Most are on the arXiv, and at least one that isn't is in an open access journal. The articles go from oldest to newest, since it makes them easier to talk about.
The second article from my thesis, wherein I prove a version of the Green Correspondence for virtually pro-$p$ groups. The statement of the main theorem is complicated by the following question, which to this day I don't know the answer to: need a finitely generated indecomposable module have a finitely generated source? Answers on a postcard!
The first article from my thesis and I'm still super proud of it. The article develops what I've since discovered is called the "local-global" theory of modular representations of profinite groups. Relative projectivity and vertices are introduced for arbitrary profinite groups, before restricting to the class of virtually pro-$p$ groups to discuss the much more delicate issues of sources and Green's indecomposability theorem. The numbering on the arXiv version is inconsistent with the published version -- if you're curious why I don't just update it on the arXiv: me too!
We study what Serre calls the "$cde$ triangle" for finite groups, in order to extend it as best we can to profinite groups. The trick was to define several Grothendieck groups for finite groups as functors. It then follows formally that they make sense for profinite groups. Cooler yet: the maps "$d$" and "$e$" are natural transformations between these functors, so they ALSO make sense for profinite groups, for free! For me the most interesting thing to come out of this approach is that the famous Cartan matrix (the "$c$") is NOT a natural object -- it's more-or-less a coincidence that its definition makes sense! For me this is a convincing explanation why the Cartan matrix is so damn hard.
There was a conference in Ubatuba to celebrate César Polcino's birthday, which I attended and loved. I wrote this little introduction to the above article with Peter for a proceedings for the conference. I guess I didn't put it on the arXiv, so if you want it let me know and I'll try and root it out of storage!
The article coming out of my post-doc in Brasilia and the start of an abusive long-term relationship with Weiss' Theorem (who's abusing who I leave to the jury). We prove a weaker version of Weiss' Theorem for certain infinitely generated modules -- a result strong enough to classify the profinite groups in the title.
My first article with Kostia, and based on one big idea: the path algebra and Gabriel quiver constructions should be functors and should form an adjoint pair. All very well on paper, and basically true, but the details are quite tricky. Since then we've refined the categories and functors, so the constructions in this article seem maybe a bit clumsy in retrospect, but an exciting (to me) line of enquiry, with a lot still to do, began here.
We give lots of basic homological properties of pseudocompact algebras and modules over a complete discrete valuation ring. The main goal of the paper was to give a clear and complete generalization of Weiss' Theorem to infinitely generated pseudocompact modules. Although most of the work was going from finitely to infinitely generated modules, most of the citations come from the fact that Weiss' Theorem hadn't been proved before over a complete discrete valuation ring (fine by me! ♥)
Here we refine the categories considered in the article with Kostia above, and having done so observe that this done, everything holds for arbitrary pointed coalgebras and pseudocompact algebras. The article suggests that useful properties of an algebra can be deduced formally by looking at algebras via their radical layers (cf. the final theorem).
Weiss' Theorem gives a sufficient condition for a $\Z_p$-lattice for a finite $p$-group to be a permutation module in terms of modules coming from a normal subgroup $N$, but the condition isn't necessary. Pavel and I give a necessary and sufficient condition here, in the special but important case where $N$ has order $p$. There is plenty more to be done in this direction but it's hard and not always clear what "should be" true. This is my only published article that doesn't have inverse limits (unless you count $\Z_p$ which you could but it would be unsporting).
It's really amazing how tractable block theory for profinite groups is compared to general representation theory. My feeling is that representation theory gets very hard when an object has "lots of conjugates", and then the topology gets involved and makes a mess. Since blocks are central, no conjugates! We define the defect group of a block, prove some characterizations, and prove an absolutely clean Brauer Correspondence for arbitrary profinite groups. As well as the usual profinite machinery, the "trick" was to define a sneaky variant of the trace map. This map probably has more to give.
Blocks for finite groups are usually not very well understood, which is understandable given that group algebras are usually hard. But blocks whose defect group is cyclic (that is, the blocks of finite type) have a very clean description as "Brauer tree algebras". We classify the blocks of a profinite group whose defect group is cyclic, in the profinite sense. Spoiler: blocks with finite cyclic defect group are just blocks of finite groups with cyclic defect group, and blocks with defect group $\Z_p$ are even easier!
Whether the finitistic dimension (that is, the supremum of the projective dimensions of those modules whose projective dimension is finite) is always finite for a finite dimensional algebra, is a very hard question that has been open for a long time. We show that if a finite dimensional algebra $A$ has a subalgebra $B$ such that $A/B$ has finite projective dimension as a $B$-bimodule, then the finitude of the finitistic dimension passes from $A$ to $B$. The subalgebra $B$ can be ''harder'' than $A$, so new examples of algebras with finite finitistic dimension can be constructed this way.
While studying pseudocompact algebras (♥) Kostia and I found the literature very tricky to navigate and widely spread out, so it was hard to even know if some basic fundamental property is known to hold. This made the study of pseudocompact algebras much more difficult than it needed to be. So half as public service and half as part of our pseudocompact propaganda campaign, we collect and prove here many fundamental results about semisimple and separable pseudocompact algebras.
In a sequence of articles, Claude Cibils, Maria Redondo and Andrea Solotar studied coverings of $k$-categories. I was curious about what happens with normal categories ("no $k$") so Claude and I investigated. In the end the referee complained quite fairly that plenty was already known, and we decided it would be a lot of trouble to figure out what's really new and what follows from results in the literature, so I guess this article won't be published. But I learned loads from this work and I still kind of love it.
In a sequence of articles, Cibils, Lanzillota, Marcos and Solotar develop the theory of a class of extensions of algebras that preserve the finiteness of the global dimension and of the Hochschild homology of the algebras. We generalize this class of extensions so that it holds for algebras of interest to us, and prove that the homological properties are still preserved. We also show that the finitistic dimension is preserved. We have a suspicion that our extensions give interesting examples even for finite dimensional algebras, and give a "proof-of-concept" example. To do this, we give a "relative version" of the fact that an acyclic algebra has finite global dimension.
In the article A characterization of permutation modules extending a theorem of Weiss with Pavel, the main theorem has the form ``The lattice $U$ is a permutation module if, and only if, Conditions A and B are satisfied''. Small examples show that the result is false if we remove Condition A, but it was not clear whether Condition B is superfluous. In this short article, we give a first application of a new method we're developing to analyze permutation modules, using a correspondence due to Butler: we prove the necessity of Condition B by providing (what seem to be highly non-trivial) examples of lattices satisfying Condition A which are not permutation modules.