A Noether–deuring Theorem for Derived Categories

Abstract

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.

Silting, cosilting, and extensions of commutative ring

“…The proof of the following lemma follows the lines of the proofs presented in [30, Lemma 2.2], [42,Lemma 3] for noetherian rings, where C may be a complex of finitely generated modules. Lemma 2.5.…”