Related papers: How big is a tiling's return module?

The moduli space of rank-n commutative algebras equipped with an ordered basis is an affine scheme B_n of finite type over Z, with geometrically connected fibers. It is smooth if and only if n <= 3. It is reducible if n >= 8 (and the…

We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective…

Let $\Lambda $ be an artin algebra and $T$ a $\tau$-tilting $\Lambda$-module. We prove that $T$ is a tilting module if and only if ${\rm Ext}_{\Lambda}^{i}(T,\Fac T)=0$ for all $i\geq 1$, where $\Fac T$ is the full subcategory consisting of…

Tilings of a surface of negative Euler characteristic by n-gons with n\ge 7 is a finite problem. We develop the algorithm for finding all the tilings for fixed number of tiles and present the calculation for tilings of surfaces of small…

For $n\geq 3$ and $r\geq n$, we show that there are rank-$r$ vector bundles on $\mathbb{P}^n$ with arbitrary homological dimension. We apply the Bernstein-Gel'fand-Gel'fand correspondence to translate the vector bundle question into a…

In a compactly generated triangulated category, we introduce a class of tilting objects satisfying certain purity condition. We call these the decent tilting objects and show that the tilting heart induced by any such object is equivalent…

We study infinite dimensional tilting modules over a concealed canonical algebra of domestic or tubular type. In the domestic case, such tilting modules are constructed by using the technique of universal localization, and they can be…

A module $M$ over the tropical semifield $T$ is analogous to a module over a field. We assume that $M$ is straight reflexive, and define the dimension of $M$ to the number of elements of a basis. We study the dimension of a straight…

For a $P$-indexed persistence module ${\sf M}$, the (generalized) rank of ${\sf M}$ is defined as the rank of the limit-to-colimit map for the diagram of vector spaces of ${\sf M}$ over the poset $P$. For $2$-parameter persistence modules,…

Consider a complex affine variety $\tilde V$ and a real analytic Zariski-dense submanifold V of $\tilde V$. We compare modules over the ring $O (\tilde V)$ of regular functions on $\tilde V$ with modules over the ring $C^\infty (V)$ of…

This work builds on earlier work of the first three authors where a notion of congruence modules in higher codimension is introduced. The main new results are a criterion for detecting regularity of local rings in terms of congruence…

This paper deals with the notion of grade of ideals with respect to torsion theories defined via some homological tools such as Ext-modules, Koszul cohomology modules, \v{C}ech and local cohomology modules over commutative rings which are…

Here we survey several results and conjectures on the cohomology of the total space of the Hitchin system: the moduli space of semi-stable rank n and degree d Higgs bundles on a complex algebraic curve C. The picture emerging is a dynamic…

We generalize the notion of length to an ordinal-valued invariant defined on the class of finitely generated modules over a Noetherian ring. A key property of this invariant is its semi-additivity on short exact sequences. We show how to…

The study of homological invariants such as Tor, Ext and local cohomology modules constitutes an important direction in commutative algebra. Explicit descriptions of these invariants are notoriously difficult to find and often involve…

We analyze and compare different dynamical systems and groupoids which can be obtained from projection point patterns. We define the cohomology of a point pattern as the cocycle cohomology of the pattern groupoid. We describe this…

One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and…

Over a commutative local Cohen--Macaulay ring, we view and study the category of maximal Cohen--Macaulay modules as a ring with several objects. We compute the global dimension of this category and thereby extend a result of Leuschke to the…

Over $d$-dimensional Cohen-Macaulay rings with a canonical module, $d$-cotilting classes containing the maximal and balanced big Cohen-Macaulay modules are classified. Particular emphasis is paid to the direct limit closure of the balanced…

In this paper, we study the structure of the set of tilings produced by any given tile-set. For better understanding this structure, we address the set of finite patterns that each tiling contains. This set of patterns can be analyzed in…