Degenerations for derived categories

“…A degeneration theory for complexes of projective modules has been developed in [9]. In particular, an analogue of the Riedtmann-Zwara theorem for complexes has been proved there.…”

Section: Suppose Now That W Is Anmentioning

confidence: 99%

Tilting, deformations and representations of linear groups over Euclidean algebras

Bekkert

Drozd

Futorny

2013

Journal of Pure and Applied Algebra

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Abstract. We consider the dual space of linear groups over Dynkinian and Euclidean algebras, i.e. finite dimensional algebras derived equivalent to the path algebra of Dynkin or Euclidean quiver. We prove that this space contains an open dense subset isomorphic to the product of dual spaces of full linear groups and, perhaps, one more (explicitly described) space. The proof uses the technique of bimodule categories, deformations and representations of quivers.

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“…REMARK 10. In [3], we developed a theory to roughly speaking parameterise geometrically objects in D b (A) by orbits of a group action on a variety. For this purpose, we need to assume that A is a finite dimensional algebra over an algebraically closed field K, so that it is possible to use arguments and constructions from algebraic geometry.…”

Section: S ⊗ R (R/rad(r)) S/(s ⊗ R Rad(r))mentioning

confidence: 99%

A Noether–deuring Theorem for Derived Categories

Zimmermann¹

2012

Glasgow Math. J.

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Abstract. We prove a Noether-Deuring theorem for the derived category of bounded complexes of modules over a Noetherian algebra.2010 Mathematics Subject Classification. Primary 16E35; Secondary 11S36, 13J10, 18E30, 16G30. Introduction.The classical Noether-Deuring theorem states that given an algebra A over a field K and a finite extension field L of K, two A-modules M andIn 1972, Roggenkamp gave a nice extension of this result to extensions S of local commutative Noetherian rings R and modules over Noetherian R-algebras.For the derived category of A-modules no such generalisation was documented before. The purpose of this note is to give a version of the Noether-Deuring theorem, in the generalised version given by Roggenkamp, for right bounded derived categories of A-modules. If there is a morphism α ∈ Hom D( ) (X, Y ), then it is fairly easy to show that for a faithfully flat ring extension S over R the fact that id S ⊗ α is an isomorphism implies that α is an isomorphism. This is done in proposition (1). More delicate is the question if only an isomorphism in Hom D(S⊗ R ) (S ⊗ R X, S ⊗ R Y ) is given. Then, we need further finiteness conditions on and on R and proceed by completion of R and then a classical going-down argument. This is done in theorem (4) and corollary (8).For the notation concerning derived categories, we refer to Verdier [6]. In particular, D(A) (resp D − (A), resp D b (A)) denotes the derived category of complexes (resp. right bounded complexes, resp. bounded complexes) of finitely generated A-modules,is the homotopy category of right bounded complexes (resp. bounded complexes, resp. right bounded complexes with bounded homology) of finitely generated projective A-modules. For a complex Z, we denote by H i (Z) the homology of Z in degree i, and by H(Z) the graded module given by the homology of Z.

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Degenerations for derived categories

Cited by 24 publications

References 10 publications

Remarks on Hall algebras of triangulated categories

Remarks on Hall algebras of triangulated categories

Tilting, deformations and representations of linear groups over Euclidean algebras

A Noether–deuring Theorem for Derived Categories

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