Pure weight perfect Modules on divisorial schemes

Abstract

In this paper, we introduce a notion of weight r pseudo-coherent Modules associated to a regular closed immersion i : Y ֒→ X of codimension r, and prove that there is a canonical derived Morita equivalence between the DG-category of perfect complexes on a divisorial scheme X whose cohomological support are in Y and the DG-category of bounded complexes of weight r pseudo-coherent O X -Modules supported on Y . The theorem implies that there is the canonical isomorphism between the Bass-Thomason-Trobaugh nonconnected K-theory [TT90], [Sch06] (resp. the Keller-Weibel cyclic homology [Kel98], [Wei96]) for the immersion and the Schlichting nonconnected K-theory [Sch04] associated to (resp. that of) the exact category of weight r pseudo-coherent Modules. For the connected K-theory case, this result is just Exercise 5.7 in [TT90]. As its application, we will decide on a generator of the topological filtration on the non-connected K-theory (resp. cyclic homology theory) for affine Cohen-Macaulay schemes. * Supported by the JSPS Fellowships for Young Scientists. † This research is supported by JSPS core-to-core program 18005 pseudo-coherent O X -Modules K S (Wt(X on Y )) (resp. HC(Wt(X on Y ))). That is, we have isomorphismsfor each q ∈ Z. For the connected K-theory this result is nothing other than Exercise 5.7 in [TT90]. For Grothendieck groups (q = 0), there is a detailed proof if X is the spectrum of a Cohen-Macaulay local ring and Y is the closed point of X ([RS03], Prop. 2). For K-theory, as mentioned in Exercise 5.7, this problem is related with the works [Ger74], [Gra76] and [Lev88]. Namely the problem about describing the homotopy fiber of K B (X) → K B (X Y ) (or rather than K Q (X) → K Q (X Y )) by using the K-theory of a certain exact category. As described in [Ger74], there is an example due to Deligne which suggests difficulty of the problem for a general closed immersion. Conversely, the example indicate that for an appropriate scheme X, there is a good class of pseudo-coherent O X -Modules. That is, Modules of pure weight. This concept is intimately related to Weibel's K-dimensional conjecture [Wei80] (see Conj. 6.4), Gersten's conjecture [Ger73] and its consequences. These subjects will be treated in [HM08], [Moc08]. Notice that there are different notions of pure weight by Grayson [Gra95] and Walker [Wal00] and these two notions are compatible in a particular situation [Wal96]. In a future work, the authors hope to compare the Walker weight with the Thomason-Trobaugh one by utilizing the (equidimensional ) bivariant algebraic K-theory [GW00]. Now we explain the structure of the paper. In §2, we describe to our motivational picture. After reviewing the fundamental facts in §3, we will define the notion of weight and state the main theorem in §4. The proof of the main theorem will be given in §5. Finally we will give applications of the main theorem in §6.Convention. Throughout this paper, we use the letter X to denote a scheme. A complex means a chain complex whose boundary morphism is increase level of te...

Generalized Koszul Resolution

“…Remark 2.2. The definition above is compatible with that in [HM10]. To prove this we shall notice that the notion of torsion and projective dimension are equivalent for a finitely generated modules on a noetherian ring (see [Wei94, Proposition 4.1.5]) and that in the notation above, Spec A/I ֒→ Spec A is a regular closed immersion.…”

“…In this section, we start from reviewing a notion of pure weight perfect modules over noetherian rings. For more information of pure weight perfect modules over any schemes, see [HM10]. Mainly we intend to study fundamental properties of pure weight modules over a Cohen-Macaulay local ring.…”

“…(1) M is pure weight two modules supported on V (f, g) in the sense of [HM10]. Namely, M is supported on V (f, g) and of projective dimension less than two.…”

“…The main point of this paper is to deal with new theory of complexes and resolutions. More precisely, the theory of generalized Koszul resolutions and related this notion with pure weight modules defined in [HM10] and Koszul cubes defined in [Moc11]. To state the main theorem now we define the notion of generalized Koszul resolution and Koszul cubes.…”

Higher $K$-theory of Koszul cubes

“…Remark 4.7. In [HM10], a pseudo-coherent O X -Module on a scheme X is said to be of Thomason-Trobaugh weight q if it is supported on a regular closed immersion Y ֒→ X of codimension q and if it is of Tor-dimension ≤ q. The following are equivalent for any finitely generated A-module M and any ideal I which is generated by an A-regular sequence…”