[Paper Review] Grothendieck groups and tilting objects

Idun Reiten, Michel Van den Bergh|ArXiv.org|May 10, 2000

Algebraic structures and combinatorial models14 references4 citations

TL;DR

This paper establishes a precise equivalence between the finite generation of the Grothendieck group and the existence of a tilting object in connected, noetherian, hereditary, Ext-finite abelian categories over an algebraically closed field that possess a Serre functor and an object of infinite length. Using classification results from prior work, it proves that under these conditions, the Grothendieck group is finitely generated if and only if a tilting object exists, thereby linking a homological invariant (Grothendieck group) to a structural property (tilting object) in a broad class of categories including coherent sheaves on weighted projective lines and quotient categories of graded singularities.

ABSTRACT

Let C be a connected noetherian hereditary abelian Ext-finite category with Serre functor over an algebraically closed field k, with finite dimensional homomorphism and extension spaces. Using the classification of such categories from math.RT/9911242, we prove that if C has some object of infinite length, then the Grothendieck group of C is finitely generated if and only if C has a tilting object.

Motivation & Objective

• To determine when the Grothendieck group of a connected, noetherian, hereditary, Ext-finite abelian category over an algebraically closed field is finitely generated.
• To clarify the relationship between the existence of a tilting object and the finite generation of the Grothendieck group in such categories.
• To extend understanding of the role of coherent sheaves on weighted projective lines and quotient categories of graded singularities within the broader class of derived categories.
• To provide a criterion for tilting objects in terms of homological vanishing and generation, applicable in the context of categories with Serre duality.
• To establish that under the given conditions, finite generation of the Grothendieck group implies the existence of a tilting object, and vice versa.

Proposed method

• Leverages the classification of connected, noetherian, hereditary, Ext-finite abelian categories with Serre duality over an algebraically closed field, as established in [31].
• Applies the theory of Grothendieck groups in the context of Ext-finite, hereditary categories to analyze finite generation under the presence of objects of infinite length.
• Introduces a new criterion for an object to be a tilting object based on vanishing of Hom and Ext¹ functors and the generation condition.
• Constructs exceptional collections of modules over hereditary orders over discrete valuation rings to facilitate the construction of tilting objects in categories of coherent sheaves on P¹ with sheaves of orders.
• Uses derived equivalence invariance of key properties (tilting object, Grothendieck group finite generation, Serre duality) to extend results beyond noetherian categories to derived-equivalent ones.
• Employs localization and injective resolution techniques in the context of graded rings and their quotient categories to analyze global dimension and hereditariness of quotient categories.

Experimental results

Research questions

• RQ1 Under what conditions is the Grothendieck group of a connected, noetherian, hereditary, Ext-finite abelian category over an algebraically closed field finitely generated?
• RQ2 Does the finite generation of the Grothendieck group imply the existence of a tilting object in such categories, particularly when some object has infinite length?
• RQ3 How does the presence of a Serre functor influence the interplay between tilting objects and Grothendieck group structure?
• RQ4 What is the precise relationship between the Grothendieck group and the existence of tilting objects in categories of coherent sheaves on weighted projective lines or quotient categories of graded singularities?
• RQ5 Can a new criterion for tilting objects be formulated that is independent of projective dimension and instead based on homological vanishing and generation?

Key findings

• The Grothendieck group of a connected, noetherian, hereditary, Ext-finite abelian category with a Serre functor and an object of infinite length is finitely generated if and only if the category admits a tilting object.
• The existence of a tilting object implies that the Grothendieck group is free abelian of finite rank, confirming a known result in a broader context.
• A new criterion for a tilting object is established: an object T is tilting if Ext¹(T,T)=0 and the conditions Hom(T,X)=0=Ext¹(T,X) imply X=0.
• The construction of exceptional collections over hereditary orders over discrete valuation rings enables the explicit construction of tilting objects in the category of coherent sheaves on P¹ with a sheaf of orders.
• For quotient categories of graded modules over commutative noetherian isolated singularities of Krull dimension two, the Grothendieck group is finitely generated precisely when a tilting object exists.
• The results extend to categories derived equivalent to noetherian hereditary categories with Serre duality, showing that the equivalence between finite generation of the Grothendieck group and existence of a tilting object is preserved under derived equivalence.