Let T be a set of elements in a ring A. The set T is right permutable if for any a ∈ A and t ∈ T, there exist b ∈ A, u ∈ T such that au = tb. A multiplicative set in a ring A is any subset T of A such that 1 ∈ T,0 ∉ T and T is closed under multiplication. A completely prime ideal in a ring A is any proper ideal B such that A\B is a multiplicative set (i.e. A/Bis a domain). A minimal prime ideal (resp. minimal completely prime ideal) in a ring A is any prime (resp. completely prime) ideal P such that P contains no properly any other prime ideal (resp. completely prime ideal) of A. Let I be any proper ideal of a ring A. The set of all elements a ∈ A such that a + I is a regular element of A/I is denoted by c(I). In particular, c(0) is the set of all regular elements of A.

Rings of quotients

Basic concepts of enriched category theory

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The Theory Of Categories

Discrete derived categories were studied initially by Vossieck (J Algebra 243:168–176, 2001) and later by Bobinski et al. (Cent Eur J Math 2:19–49, 2004). In this article, we describe the homomorphism hammocks and autoequivalences on these categories. We classify silting objects and bounded t-structures.

/pdf/discrete-derived-categories-i-homomorphisms-autoequivalences-132rqfue58.pdf

Discrete derived categories I: homomorphisms, autoequivalences and t-structures

Cyclic posets are generalizations of cyclically ordered sets. In this article, we show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The stable category of a Frobenius category is always triangulated and has a cluster structure in many cases. The continuous cluster categories of [14], the infinity-gon of [12], and the m-cluster category of type A ∞ (m ≥ 3) [13] are examples of this construction as well as some new examples such as the cluster category of ℤ2. An extension of this construction and further examples are given in [16].

https://people.brandeis.edu/~igusa/Papers/PosetsC140709.pdf

Cluster Categories Coming from Cyclic Posets

We introduce thread quivers as an (infinite) generalization of quivers, and show that every k-linear (k algebraically closed) hereditary category with Serre duality and enough projectives is equivalent to the category of finitely presented representations of a thread quiver. In this way, we obtain an explicit construction of a new class of hereditary categories with Serre duality.

/pdf/representations-of-thread-quivers-bt8x0z3lnp.pdf

Representations of Thread Quivers

https://pure.mpg.de/pubman/item/item_3122404_1/component/file_3122405/berg_hereditary_oa_2011.pdf

Hereditary categories with Serre duality which are generated by preprojectives

An abelian Krull-Schmidt category is said to be uniserial if the isomorphism classes of subobjects of a given indecomposable object form a linearly ordered poset. In this paper, we classify the hereditary uniserial categories with Serre duality. They fall into two types: the first type is given by the representations of the quiver A_n with linear orientation (and infinite variants thereof), the second type by tubes (and an infinite variant). These last categories give a new class of hereditary categories with Serre duality, called big tubes.

Hereditary uniserial categories with Serre duality

An abelian Krull-Schmidt category is said to be uniserial if the isomorphism classes of subobjects of a given indecomposable object form a linearly ordered poset. In this paper, we classify the hereditary uniserial categories with Serre duality. They fall into two types: the first type is given by the representations of the quiver A n with linear orientation (and infinite variants thereof), the second type by tubes (and an infinite variant). These last categories give a new class of hereditary categories with Serre duality, called big tubes.

/pdf/hereditary-uniserial-categories-with-serre-duality-2ygps4wdfs.pdf

Hereditary Uniserial Categories with Serre Duality

Using the framework of noncommutative Kahler structures, we generalise to the noncommutative setting the celebrated vanishing theorem of Kodaira for positive line bundles. The result is established under the assumption that the associated Dirac-Dolbeault operator of the line bundle is diagonalisable, an assumption that is shown to always hold in the quantum homogeneous space case. The general theory is then applied to the covariant Kahler structure of the Heckenberger-Kolb calculus of the quantum Grassmannians allowing us to prove a direct q-deformation of the classical Grassmannian Bott-Borel-Weil theorem for positive line bundles.

Adam-Christiaan van Roosmalen

Papers

Representations of Thread Quivers

Hereditary categories with Serre duality which are generated by preprojectives

Hereditary uniserial categories with Serre duality

Hereditary Uniserial Categories with Serre Duality

A Kodaira Vanishing Theorem for Noncommutative Kahler Structures