Finiteness conditions and cotorsion pairs

Daniel Bravo; Marco A. Pérez

doi:10.1016/j.jpaa.2016.09.008

Representations of the fundamental group and the torsor of deformations. Global aspects

10.23943/princeton/9780691170282.003.0003 ◽

2017 ◽

Author(s):

Ahmed Abbes ◽

Michel Gros

Keyword(s):

Fundamental Group ◽

Koszul Complex ◽

Finiteness Conditions ◽

Inverse Systems ◽

Logarithmic Geometry

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

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Finiteness conditions and relative derived categories

Journal of Algebra ◽

10.1016/j.jalgebra.2020.12.034 ◽

2021 ◽

Vol 573 ◽

pp. 270-296

Author(s):

Lingling Tan ◽

Dingguo Wang ◽

Tiwei Zhao

Keyword(s):

Derived Categories ◽

Finiteness Conditions

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MODEL STRUCTURES ON TRIANGULATED CATEGORIES

Glasgow Mathematical Journal ◽

10.1017/s0017089514000299 ◽

2014 ◽

Vol 57 (2) ◽

pp. 263-284 ◽

Cited By ~ 5

Author(s):

XIAOYAN YANG

Keyword(s):

Proper Class ◽

Triangulated Category ◽

Triangulated Categories ◽

General Study ◽

Proper Class Of Triangles ◽

Cotorsion Pairs ◽

One To One ◽

The Relationship ◽

Model Structures

AbstractWe define model structures on a triangulated category with respect to some proper classes of triangles and give a general study of triangulated model structures. We look at the relationship between these model structures and cotorsion pairs with respect to a proper class of triangles on the triangulated category. In particular, we get Hovey's one-to-one correspondence between triangulated model structures and complete cotorsion pairs with respect to a proper class of triangles. Some applications are given.

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Homological finiteness conditions for a class of metabelian groups

Bulletin of the London Mathematical Society ◽

10.1112/blms.12093 ◽

2017 ◽

Vol 50 (1) ◽

pp. 17-25

Author(s):

Peter H. Kropholler ◽

Joseph P. Mullaney

Keyword(s):

Finiteness Conditions ◽

Metabelian Groups

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INJECTIVE MODULES OVER DOWN-UP ALGEBRAS

Glasgow Mathematical Journal ◽

10.1017/s0017089510000261 ◽

2010 ◽

Vol 52 (A) ◽

pp. 53-59 ◽

Cited By ~ 6

Author(s):

PAULA A. A. B. CARVALHO ◽

CHRISTIAN LOMP ◽

DILEK PUSAT-YILMAZ

Keyword(s):

Roots Of Unity ◽

Finiteness Conditions ◽

Simple Modules ◽

Injective Modules ◽

Injective Hulls

AbstractThe purpose of this paper is to study finiteness conditions on injective hulls of simple modules over Noetherian down-up algebras. We will show that the Noetherian down-up algebras A(α, β, γ) which are fully bounded are precisely those which are module-finite over a central subalgebra. We show that injective hulls of simple A(α, β, γ)-modules are locally Artinian provided the roots of X2 − αX − β are distinct roots of unity or both equal to 1.

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New model structures and projective (injective) cotorsion pairs

Journal of Algebra and Its Applications ◽

10.1142/s0219498823500366 ◽

2021 ◽

Author(s):

Aimin Xu

Keyword(s):

Positive Integer ◽

Triangulated Categories ◽

Model Structure ◽

Cotorsion Pair ◽

Flat Dimension ◽

Gorenstein Projective Module ◽

Chain Complexes ◽

Cotorsion Pairs ◽

Gorenstein Injective ◽

Model Structures

Let [Formula: see text] be either the category of [Formula: see text]-modules or the category of chain complexes of [Formula: see text]-modules and [Formula: see text] a cofibrantly generated hereditary abelian model structure on [Formula: see text]. First, we get a new cofibrantly generated model structure on [Formula: see text] related to [Formula: see text] for any positive integer [Formula: see text], and hence, one can get new algebraic triangulated categories. Second, it is shown that any [Formula: see text]-strongly Gorenstein projective module gives rise to a projective cotorsion pair cogenerated by a set. Finally, let [Formula: see text] be an [Formula: see text]-module with finite flat dimension and [Formula: see text] a positive integer, if [Formula: see text] is an exact sequence of [Formula: see text]-modules with every [Formula: see text] Gorenstein injective, then [Formula: see text] is injective.

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Subgroups of Finite Index in Free Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1949-017-2 ◽

1949 ◽

Vol 1 (2) ◽

pp. 187-190 ◽

Cited By ~ 72

Author(s):

Marshall Hall

Keyword(s):

Free Group ◽

Finite Index ◽

Free Groups ◽

Recursion Formula ◽

Independent Interest ◽

Finiteness Conditions ◽

Total Length ◽

Free Generators

This paper has as its chief aim the establishment of two formulae associated with subgroups of finite index in free groups. The first of these (Theorem 3.1) gives an expression for the total length of the free generators of a subgroup U of the free group Fr with r generators. The second (Theorem 5.2) gives a recursion formula for calculating the number of distinct subgroups of index n in Fr.Of some independent interest are two theorems used which do not involve any finiteness conditions. These are concerned with ways of determining a subgroup U of F.

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Rings whose additive endomorphisms are ring endomorphisms

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700028252 ◽

1990 ◽

Vol 42 (1) ◽

pp. 145-152 ◽

Cited By ~ 6

Author(s):

Gary Birkenmeier ◽

Henry Heatherly

Keyword(s):

Subdirectly Irreducible ◽

Finiteness Conditions ◽

Chain Conditions ◽

Open Questions ◽

Open Case ◽

Ring Endomorphism ◽

Substantial Progress ◽

Additive Endomorphism ◽

Made In

A ring R is said to be an AE-ring if every additive endomorphism is a ring endomorphism. In this paper further steps are made toward solving Sullivan's Problem of characterising these rings. The classification of AE-rings with. R3 ≠ 0 is completed. Complete characterisations are given for AE-rings which are either: (i) subdirectly irreducible, (ii) algebras over fields, or (iii) additively indecomposable. Substantial progress is made in classifying AE-rings which are mixed – the last open case – by imposing various finiteness conditions (chain conditions on special ideals, height restricting conditions). Several open questions are posed.

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