Quasi-hereditary Ring Research Articles

Quasi-hereditary rings are closed under taking block extensions

In this paper, we give a sufficient condition for Morita context rings to be quasi-hereditary. As an application, we show that each block extension of a quasi-hereditary ring is also quasi-hereditary.

Gabriel–Roiter Measure, Representation Dimension and Rejective Chains

Abstract The Gabriel–Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel–Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel–Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and any Gabriel–Roiter measure $\mu$ in an abelian length category $\mathcal{A}$, there exists an object $X^{\prime}$ that depends on $X$ and $\mu$, such that $\Gamma =\operatorname{End}_{\mathcal{A}}(X\oplus X^{\prime})$ has finite global dimension. Analogously to Iyama’s original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.

Singular Curves and Quasi-hereditary Algebras

In this article we construct a categorical resolution of singularities of an excellent reduced curve X, introducing a certain sheaf of orders on X. This categorical resolution is shown to be a recollement of the derived category of coherent sheaves on the normalization of X and the derived category of finite length modules over a certain artinian quasi–hereditary ring Q depending purely on the local singularity types of X. Using this technique, we prove several statements on the Rouquier dimension of the derived category of coherent sheaves on X. Moreover, in the case X is rational and projective we construct a finite dimensional quasi–hereditary algebra Λ such that the triangulated category Perf(X) embeds into D(Λ −mod) as a full subcategory.

Finitistic dimension and Igusa–Todorov algebras

The notion of Igusa–Todorov algebras is introduced in connection with the (little) finitistic dimension conjecture, and the conjecture is proved for those algebras. Such algebras contain many known classes of algebras over which the finitistic dimension conjecture holds, e.g., algebras with the representation dimension at most 3, algebras with radical cube zero, monomial algebras and left serial algebras, etc. It is an open question whether all artin algebras are Igusa–Todorov. We provide some methods to construct many new classes of (2-)Igusa–Todorov algebras and thus obtain many algebras such that the finitistic dimension conjecture holds. In particular, we show that the class of 2-Igusa–Todorov algebras is closed under taking endomorphism algebras of projective modules. Hence, if all quasi-hereditary algebras are 2-Igusa–Todorov, then all artin algebras are 2-Igusa–Todorov by [V. Dlab, C.M. Ringel, Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring, Proc. Amer. Math. Soc. 107 (1) (1989) 1–5] and have finite finitistic dimension.

The Cartan determinant and generalizations of quasihereditary rings

The Cartan determinant conjecture for left artinian rings was verified for quasihereditary rings showing detC(R) = detC(R/I), where I is a protective ideal generated by a primitive idempotent. This article identifies classes of rings generalizing the quasihereditary ones, first by relaxing the “projective” condition on heredity ideals. These rings, called left k-hereditary are all of finite global dimension. Next a class of rings is defined which includes left serial rings of finite global dimension, quasihereditary and left 1-hereditary rings, but also rings of infinite global dimension. For such rings, the Cartan determinant conjecture is true, as is its converse. This is shown by matrix reduction. Examples compare and contrast these rings with other known families and a recipe is given for constructing them.

A generalization of quasi-hereditary rings

We give a generalization of quasi-hereditary rings called QH-1 rings. We show that a left QH-1 ring R is quasi hereditary if and only if R has finite global dimension if and only if R has Cartan determinant +1. We then give a bound on the global dimension of the quasi-hereditary serial rings, and charac terize the QH-1 serial rings of infinite global dimension in termi of the admissible sequence.

Quasi-hereditary orders

Quasi-hereditary rings are defined as generalizations of quasi-hereditary algebras; it is shown that some of the main properties of quasi-hereditary algebras carry over to quasi-hereditary rings. In particular, a generalization of the concept of highest weight categories gives a class of classical orders which satisfy our definition of quasi-heredity. For this class of orders the category of good lattices is investigated and it is shown that this category has almost split sequences.

Every Semiprimary Ring is the Endomorphism Ring of a Projective Module Over a Quasi-Hereditary Ring

Every Semiprimary Ring is the Endomorphism Ring of a Projective Module Over a Quasi-Hereditary Ring

Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring

The paper provides a proof of the following statement: Given a semiprimary ring R R , there is a quasi-hereditary ring A A and an idempotent e ∈ A e \in A such that R ≃ e A e R \simeq eAe .