Acyclicity versus total acyclicity for complexes over noetherian rings

By S. Iyengar, Henning Krause. Documenta Mathematica, 2005

140 citations recorded by the source corpus.

It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is equivalent, as a triangulated category, to the homotopy category of injective modules. Restricted to compact objects, this statement is a reinterpretation of Grothendieck's duality theorem. Using this equiva- lence it is proved that the (Verdier) quotient of the category of acyclic complexes of projectives by its subcategory of totally acyclic com- plexes and the corresponding category consisting of injective modules are equivalent. A new characterization is provided for complexes in Auslander categories and in Bass categories of such rings.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/df1ae5d78700cbc871f0eaac12153ee39b69940a

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Citations in the corpus

THE GORENSTEIN DEFECT CATEGORY (2012, background): “ection. The dimension of the Gorenstein defect category, in the sense of Roquier, is therefore in some sense a measure of “how far” the ring is from being Gorenstein. We note that the construction in [IyeKra] yields an interesting alternate defect category characterizing Gorenstein rings, namely the”
Relative singularity categories and Gorenstein‐projective modules (2007, methodology): “Theorem 6.7 and [7], Theorem 6.6), and Iyengar-Krause ([ 15 ], Theorem 5.4 (2)) using different”
Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence (2009, background): “m Dco(B–mod) ≃Dctr(B–mod) when B is a CDG-algebra over a ﬁeld ksuch that the underlying graded algebra B# is ﬁnite-dimensional; see 5.2. Further (and deeper) results in this direction can be found in [22]. For a discussion of the assumptions under which the derived and absolute derived categories D(”
Homotopy Equivalences induced by Balanced Pairs (2008, background): “Theorem B is related to a recent result by Iyengar and Krause ([21]).”
Beyond totally reflexive modules and back (2008, background): “In the terminology of [75], part (ii)/(iii) above says that every acyclic complex of projective/injective modules is totally acyclic.”
Some adjoints in homotopy categories (2010, background): “Dedicated to Max Karoubi, who started me on this project by encouraging me to read the remarkable article [8]”
Recollement of Homotopy Categories and Cohen-Macaulay Modules (Representation Theory of Finite Groups and Algebras, and Related Topics) (2009, methodology): “table t-structure in K∞,b(ProjR). (2) (K−,b(InjR),K∞,∅(InjR)) is a stable t-structure in K∞,b(InjR). Proof. Let V be a two-sided injective resolution of R, then V is a dualizing complex. According to [IK] Theorem 4.2 and 4.8, G = −⊗R V : K(ProjR) →K(InjR) is a triangle equivalence. Also [IK] Theorem”
The homotopy category of flat modules, and Grothendieck duality (2008, background): “The relation of dualizing complexes, with compactly generated triangulated categories, comes from two recent theorems, which appeared in a couple of beautiful articles in 2005: Theorem 2.6 (Krause [9]).”
Coherent analogues of matrix factorizations and relative singularity categories (2011, background): “Notice that our proof is more akin to the arguments in [35, 34] than those of [15, 21] in that”
On algebras of finite Cohen-Macaulay type (2011, background): “t the complex HomRpP‚,Pq is acyclic, @P P ProjR. Hence any acyclic complex of projective R-modules is totally acyclic. Since R is complete, it is well-known that R admits a dualizing complex. Then by [55] it follows that R is Gorenstein. Then R is Artinian by Proposition 5.14. (ii) ñ (i) Clearly R i”
An Auslander-Type result for Gorenstein-projective modules (2007, unknown): “R T ] 7 A pr 2”
Approximations and adjoints in homotopy categories (2010, background): “Using fp-injective modules, a result from [19] takes the following form.”
Totally acyclic complexes over noetherian schemes (2009, methodology): “As examples of the technique, we generalise results of Jorgensen [Jor07] and Iyenga r-Kr ause [IK06].”