Pairwise Compatibility for 2-Simple Minded Collections

In $\tau$-tilting theory, it is often difficult to determine when a set of bricks forms a 2-simple minded collection. The aim of this paper is to determine when a set of bricks is contained in a 2-simple minded collection for a $\tau$-tilting finite algebra. We begin by extending the definition of mutation from 2-simple minded collections to more general sets of bricks (which we call semibrick pairs). This gives us an algorithm to check if a semibrick pair is contained in a 2-simple minded collection. We then use this algorithm to show that the 2-simple minded collections of a $\tau$-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) are given by pairwise compatibility conditions if and only if every vertex in the corresponding quiver has degree at most 2. As an application, we show that the classifying space of the $\tau$-cluster morphism category of a $\tau$-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) is an Eilenberg-MacLane space if every vertex in the corresponding quiver has degree at most 2.Comment: v4: changed title, updated and added references, changes to exposition for clarity and readability, corrected typos. v3: changes to introduction. v2: Made a correction to Corollary 5.12 (Theorem D in the introduction). 27 page

{\tau}-Cluster Morphism Categories and Picture Groups

$\tau$-cluster morphism categories were introduced by Buan and Marsh as a generalization of cluster morphism categories (as defined by Igusa and Todorov) to $\tau$-tilting finite algebras. In this paper, we show that the classifying space of such a category is a cube complex, generalizing results of Igusa and Todorov and Igusa. We further show that the fundamental group of this space is isomorphic to a generalized version of the picture group of the algebra, as defined by Igusa, Todorov, and Weyman. We end this paper by showing that if the algebra is Nakayama, then this space is locally $\mathsf{CAT}(0)$, and hence a $K(\pi,1)$. We do this by constructing a combinatorial interpretation of the 2-simple minded collections for Nakayama algebras. A key step in the proof is to show that, for Nakayama algebras, 2-simple minded collections are characterized by pairwise compatibility conditions, a fact not true in general.Comment: 29 pages, 14 figures. v3: Numerous improvements have been made following suggestions of an anonymous referee. v2: Lemma 4.13 has been combined with Lemma 4.12 (now Lemma 4.11) and its proof has been changed. The introduction has been rewritten and minor typos have been fixe

A uniqueness property of {\tau} exceptional sequences

Recently, Buan and Marsh showed that if two complete $\tau$-exceptional sequences agree in all but at most one term, then they must agree everywhere, provided the algebra is $\tau$-tilting finite. They conjectured that the result holds without that assumption. We prove their conjecture. Along the way, we also show that the dimension vectors of the modules in a $\tau$-exceptional sequence are linearly independent.Comment: 7 page

Exact structures for persistence modules

We discuss applications of exact structures and relative homological algebra to the study of invariants of multiparameter persistence modules. This paper is mostly expository, but does contain a pair of novel results. Over finite posets, classical arguments about the relative projective modules of an exact structure make use of Auslander-Reiten theory. One of our results establishes a new adjunction which allows us to "lift" these arguments to certain infinite posets over which Auslander-Reiten theory is not available. We give several examples of this lifting, in particular highlighting the non-existence and existence of resolutions by upsets when working with finitely presentable representations of the plane and of the closure of the positive quadrant, respectively. We then restrict our attention to finite posets. In this setting, we discuss the relationship between the global dimension of an exact structure and the representation dimension of the incidence algebra of the poset. We conclude with our second novel contribution. This is an explicit description of the irreducible morphisms between relative projective modules for several exact structures which have appeared previously in the literature.Comment: v2: corrected typos and minor erros, 25 page

B780: A Cost Analysis of Pruning Procedures in Lowbush Blueberry Production

Burning fields with fuel oil is currently the most practical method of pruning blueberries but is costly and destructive to the organic material on the surface of the soil. Fuel oil is a nonrenewable resource that is rapidly increasing in cost and, in the future, may become less readily available for this use. The need to develop alternative means of pruning lowbush bleuberries is evident. This bulletin compares the economics of six pruning procedures on operations of three sizes. The budgets are based on certain assumptions and costs which will change over time. The results will allow blueberry growers to compare procedures to determine which one is most economically feasible for their particular operation and its resources.https://digitalcommons.library.umaine.edu/aes_bulletin/1066/thumbnail.jp