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Morita duality for Grothendieck categories.

José L. Gómez Pardo, Francisco de A. Guil Asensio (1992)

Publicacions Matemàtiques

We survey some recent results on the theory of Morita duality for Grothendieck categories, comparing two different versions of this concept, and giving applications to QF-3 and Qf-3' rings.

n -angulated quotient categories induced by mutation pairs

Zengqiang Lin (2015)

Czechoslovak Mathematical Journal

Geiss, Keller and Oppermann (2013) introduced the notion of n -angulated category, which is a “higher dimensional” analogue of triangulated category, and showed that certain ( n - 2 ) -cluster tilting subcategories of triangulated categories give rise to n -angulated categories. We define mutation pairs in n -angulated categories and prove that given such a mutation pair, the corresponding quotient category carries a natural n -angulated structure. This result generalizes a theorem of Iyama-Yoshino (2008) for...

Natural dualities between abelian categories

Flaviu Pop (2011)

Open Mathematics

In this paper we consider a pair of right adjoint contravariant functors between abelian categories and describe a family of dualities induced by them.

Non-singular covers over monoid rings

Ladislav Bican (2008)

Mathematica Bohemica

We shall introduce the class of strongly cancellative multiplicative monoids which contains the class of all totally ordered cancellative monoids and it is contained in the class of all cancellative monoids. If G is a strongly cancellative monoid such that h G ⊆ G h for each h ∈ G and if R is a ring such that a R ⊆ R a for each a ∈ R , then the class of all non-singular left R -modules is a cover class if and only if the class of all non-singular left R G -modules is a cover class. These two conditions are also equivalent whenever...

Non-singular covers over ordered monoid rings

Ladislav Bican (2006)

Mathematica Bohemica

Let G be a multiplicative monoid. If R G is a non-singular ring such that the class of all non-singular R G -modules is a cover class, then the class of all non-singular R -modules is a cover class. These two conditions are equivalent whenever G is a well-ordered cancellative monoid such that for all elements g , h ∈ G with g < h there is l ∈ G such that l g = h . For a totally ordered cancellative monoid the equalities Z ( R G ) = Z ( R ) G and σ ( R G ) = σ ( R ) G hold, σ being Goldie’s torsion theory.

Non-singular precovers over polynomial rings

Ladislav Bican (2006)

Commentationes Mathematicae Universitatis Carolinae

One of the results in my previous paper On torsionfree classes which are not precover classes, preprint, Corollary 3, states that for every hereditary torsion theory τ for the category R -mod with τ ≥ σ , σ being Goldie’s torsion theory, the class of all τ -torsionfree modules forms a (pre)cover class if and only if τ is of finite type. The purpose of this note is to show that all members of the countable set 𝔐 = { R , R / σ ( R ) , R [ x 1 , ⋯ , x n ] , R [ x 1 , ⋯ , x n ] / σ ( R [ x 1 , ⋯ , x n ] ) , n < ω } of rings have the property that the class of all non-singular left modules forms a (pre)cover...

Obstructions for deformations of complexes

Frauke M. Bleher, Ted Chinburg (2013)

Annales de l’institut Fourier

We develop two approaches to obstruction theory for deformations of derived isomorphism classes of complexes of modules for a profinite group G over a complete local Noetherian ring A of positive residue characteristic.

On a certain generalization of spherical twists

Yukinobu Toda (2007)

Bulletin de la Société Mathématique de France

This note gives a generalization of spherical twists, and describe the autoequivalences associated to certain non-spherical objects. Typically these are obtained by deforming the structure sheaves of ( 0 , - 2 ) -curves on threefolds, or deforming ℙ -objects introduced by D.Huybrechts and R.Thomas.

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