Amnon Neeman

The chromatic tower for D(R)

Following the conventions of algebraists, we will call triangulated subcategories of D*(R) epaisse if they are full and closed under direct summands. Hopkins’ theorem is a beautiful result. Among other things, it establishes that out of something seemingly nonsensical, like the derived category of R, one can recover a very sensible object, like Spec(R). However, there is a gap in the proof. Without some added hypotheses (e.g. R Noetherian) the theorem is false. A counterexample may be found in Section 4. I should immediately add that Hopkins obtained his result by studying analogous properties in the topological setting, where [6] obtained some really remarkable and powerful results. The theorem quoted above occurs in a paper in a conference proceedings, where he explained the topological result and remarked in passing that the algebraic analogue is also correct. It should be stressed that Hopkins’ result is very intriguing, and possibly very important. He discovered a parallel between stable homotopy theory and algebraic geometry, and this parallel should be explored further. This is perhaps the appropriate point to briefly outline the topological parallel of what we do here. The starting point for us (this is historically quite wrong) is that CmSo can be given the structure of an Em ring spectrum. Therefore, in some sense it may be viewed as a commutative ring, and one may wish to study the algebraic geometry of this curious

The derived category of an exact category

There is some confusion in the literature regarding the derived category of an exact category 8. Thomason gives a satisfactory treatment of the bounded derived category in CT, 1.11.6 (see also Appendix A)]. The result is that provided all weakly split epimorphisms in 8 are admissible, the bounded derived category may be defined as usual. Although Thomason does not say it, the categories D+(6), D-(d) may also be defined in the same way. The definition of the unbounded derived category is more difficult, and is very poorly treated in the literature. For instance, in [BBD] this derived category is only defined provided every morphism in 6 admits a kernel. (See [BBD, 1.1.41; this is very unsatisfactory since very few exact categories satisfy the condition.) In this article we will show that D(B) may be defined whenever 6 is saturated (“Karoubian” in Thomason’s terminology). An exact category is saturated if every idempotent splits; i.e., 6 contains all direct summands of its objects. For the purpose of comparing with Thomason’s result, the bounded derived category is defined for more 8’s. All that is required to define Db(&) is that whenever an idempotent e: A + A factors as A f, BA A with fog=lB, then e is split. Thomason calls such idempotents weakly split, and we will honor his notation. We will show here that these constructions are in some sense best possible (Remark 1.8 for D(6), Remarks 1.9 and 1.10 for D+(d), D-(b), Db(6)). There are two reasons why I wrote this note. One is to correct the misconceptions in the literature. But, more importantly, the proof of the key result, Lemma 1.2, depends on an important characterization of tpaisse subcategories due to Rickard, and this article is intended to highlight Rickard’s criterion: a full triangulated subcategory Y of a triangulated

Some new axioms for triangulated categories

and this mapping cone should, by rights, be a triangle. It is, for instance, true that any homological functor applied to (*) gives a long exact sequence. Unfortunately, the world of triangulated categories is a bad one, and (*) need not be a triangle. It is however true that, givenf and g, there exists an h for which (*) is a triangle (see Theorem 1.8); not all morphisms of triangles are equal. Some are better than others. It turns out that Theorem 1.8 is equivalent to the octahedral axiom. In the first two sections of this article, we study two possible notions of “good” morphisms between triangles, and we quickly decide that neither is satisfactory. The morphisms do not compose well; the composite of good morphisms need not be good. Worse still, the non-category of good morphisms is non-additive; the sum of two good morphisms need not be good. The problem goes right back to the definition of a triangulated category.

On a theorem of Brown and Adams

Let ℸ be the homotopy category of all spectra, Tc ⊂ T the full sub-category of finite spectra. Brown and Adams proved that any homological functor H:Tcop → Ab is the restriction of a representable functor on T. Furthermore, any natural transformation is the restriction of a map in ℸ. One may naturally wonder whether this generalizes to arbitrary triangulated categories, for instance D(R) where R is a ring. We show that the answer is in general no, although for R countable the generalization holds.

Failure of Brown representability in derived categories

Abstract Let T be a triangulated category with coproducts, T c ⊂ T the full subcategory of compact objects in T . If T is the homotopy category of spectra, Adams (Topology 10 (1971) 185–198), proved the following: All homological functors { T c } op → A b are the restrictions of representable functors on T , and all natural transformations are the restrictions of morphisms in T . It has been something of a mystery, to what extent this generalises to other triangulated categories. In Neeman (Topology 36 (1997) 619–645), it was proved that Adams’ theorem remains true as long as T c is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in T . A puzzling open problem remained: Is every homological functor the restriction of a representable functor on T ? In a recent paper, Beligiannis (Relative homological and purity in triangulated categories, 1999, preprint) made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories T =D(R) of rings, and homological functors { T c } op → A b which are not restrictions of representables.

Noncommutative localisation in algebraic K-theory I

This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic K-theory. The main result goes as follows. Let A be an associative ring and let A → B be the localisation with respect to a set σ of maps between finitely generated projective A-modules. Suppose that Tor A n (B,B) vanishes for all n > 0. View each map in a as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes D p e r f (A). Denote by the thick subcategory generated by these complexes. Then the canonical functor D p e r f (A) → D p e r f (B) induces (up to direct factors) an equivalence D p e r f (A)/ → D p e r f (B). As a consequence, one obtains a homotopy fibre sequence K(A,σ) → K(A) → K(B) (up to surjectivity of K 0 (A) → K 0 (B)) of Waldhausen K-theory spectra. In subsequent articles [26, 27] we will present the K- and L-theoretic consequences of the main theorem in a form more suitable for the applications to surgery. For example if, in addition to the vanishing of Tor A n (B,B), we also assume that every map in σ is a monomorphism, then there is a description of the homotopy fiber of the map K(A) → K(B) as the Quillen K-theory of a suitable exact category of torsion modules.

Stable homotopy as a triangulated functor

Let J-I be the homotopy category of all finite spectra, ~the homotopy category of all spectra and Y--[ 89 be the homotopy category of all prime-to-2 spectra. Let H: ~--y ~ (Abelian groups) be a homological functor (i.e. a generalized homology theory). A natural question is whether H can be lifted to a triangulated functor; is there a functor /-t: ~--y~D (Abelian groups) so that /~ composed with the usual homology functor on D(Abelian groups) is H. Here D (Abelian groups) = D(Z) is the derived category of Z-modules. If H is ordinary homology, the answer is clearly yes. After all, the ordinary homology of a space is given by the singular chain complex; almost by definition one has a lifting/~. By the Brown Representability Theorem, the general statement follows from the special case H = H . , the stable homotopy functor. Let us be a little more precise:

Representations of algebras as universal localizations

Every finitely presented algebra S is shown to be Morita equivalent to the universal localization \sigma^{-1}R of a finite dimensional algebra R. The construction provides many examples of universal localizations which are not stably flat, i.e. Tor^R_i(\sigma^{-1}R,\sigma^{-1}R) is non-zero for some i>0. It is also shown that there is no algorithm to determine if two Malcolmson normal forms represent the same element of \sigma^{-1}R.

Oddball Bousfield classes

Abstract We produce a commutative ring R, whose spectrum, Spec(R), is the one point space. Nevertheless, there is a huge set of distinct Bousfield classes in its derived category.

ALGEBRAIC GEOMETRY OF THE THREE-STATE CHIRAL POTTS MODEL

For more than a decade now, the chiral Potts model in statistical mechanics has attracted much attention. A number of mathematical physicists have written quite extensively about it. The solutions give rise to a curve over ℂ, and much effort has gone into studying the curve and its Jacobian.In this article, we give yet another approach to this celebrated problem. We restrict attention to the three-state case, which is simplest. For the first time in its history, we study the model with the tools of modern algebraic geometry. Aside from simplifying and explaining the previous results on the periods and Theta function of this curve, we obtain a far more complete description of the Jacobian.