The Auslander bijections: How morphisms are determined by modules

Let A be an artin algebra. In his seminal Philadelphia Notes published in 1978, M. Auslander introduced the concept of morphisms being determined by modules. Auslander was very passionate about these ivestigations (they also form part of the final chapter of the Auslander-Reiten-Smaloe book and could and should be seen as its culmination), but the feedback until now seems to be somewhat meager. The theory presented by Auslander has to be considered as an exciting frame for working with the category of A-modules, incorporating all what is known about irreducible maps (the usual Auslander-Reiten theory), but the frame is much wider and allows for example to take into account families of modules - an important feature of module categories. What Auslander has achieved is a clear description of the poset structure of the category of A-modules as well as a blueprint for interrelating individual modules and families of modules. Auslander has subsumed his considerations under the heading of "morphisms being determined by modules". Unfortunately, the wording in itself seems to be somewhat misleading, and the basic definition may look quite technical and unattractive, at least at first sight. This could be the reason that for over 30 years, Auslander's powerful results did not gain the attention they deserve. The aim of this survey is to outline the general setting for Auslander's ideas and to show the wealth of these ideas by exhibiting many examples

Efficient and Modular Coalgebraic Partition Refinement

We present a generic partition refinement algorithm that quotients coalgebraic systems by behavioural equivalence, an important task in system analysis and verification. Coalgebraic generality allows us to cover not only classical relational systems but also, e.g. various forms of weighted systems and furthermore to flexibly combine existing system types. Under assumptions on the type functor that allow representing its finite coalgebras in terms of nodes and edges, our algorithm runs in time $\mathcal{O}(m\cdot \log n)$ where $n$ and $m$ are the numbers of nodes and edges, respectively. The generic complexity result and the possibility of combining system types yields a toolbox for efficient partition refinement algorithms. Instances of our generic algorithm match the run-time of the best known algorithms for unlabelled transition systems, Markov chains, deterministic automata (with fixed alphabets), Segala systems, and for color refinement.Comment: Extended journal version of the conference paper arXiv:1705.08362. Beside reorganization of the material, the introductory section 3 is entirely new and the other new section 7 contains new mathematical result

Taylor's modularity conjecture and related problems for idempotent varieties

We provide a partial result on Taylor's modularity conjecture, and several related problems. Namely, we show that the interpretability join of two idempotent varieties that are not congruence modular is not congruence modular either, and we prove an analogue for idempotent varieties with a cube term. Also, similar results are proved for linear varieties and the properties of congruence modularity, having a cube term, congruence $n$-permutability for a fixed $n$, and satisfying a non-trivial congruence identity.Comment: 27 page

On modular homology in projective space

AbstractFor a vector space V over GF(q) let Lk be the collection of subspaces of dimension k. When R is a field let Mk be the vector space over it with basis Lk. The inclusion map ∂:Mk→Mk−1 then is the linear map defined on this basis via ∂(X)≔∑Y where the sum runs over all subspaces of co-dimension 1 in X. This gives rise to a sequenceM:0←M0←M1←⋯←Mk−1←Mk←⋯which has interesting homological properties if R has characteristic p>0 not dividing q. Following on from earlier papers we introduce the notion of π-homological, π-exact and almost π-exact sequences where π=π(p,q) is some elementary function of the two characteristics. We show that M and many other sequences derived from it are almost π-exact. From this one also obtains an explicit formula for the Brauer character on the homology modules derived from M. For infinite-dimensional spaces we give a general construction which yields π-exact sequences for finitary ideals in the group ring RPΓL(V)