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The Classification of Triangulated Subcategories 3 Compositio Mathematica 105: 1±27, 1997. 1 c 1997 Kluwer Academic Publishers. Printed in the Netherlands. The classi®cation of triangulated subcategories R. W. THOMASON ? CNRS URA212, U.F.R. de Mathematiques, Universite Paris VII, 75251 Paris CEDEX 05, France email: thomason@@frmap711.mathp7.jussieu.fr Received 2 August 1994; accepted in ®nal form 30 May 1995 Key words: triangulated category, Grothendieck group, Hopkins±Neeman classi®cation Mathematics Subject Classi®cations (1991): 18E30, 18F30. 1. Introduction The ®rst main result of this paper is a bijective correspondence between the strictly full triangulated subcategories dense in a given triangulated category and the sub- groups of its Grothendieck group (Thm. 2.1). Since every strictly full triangulated subcategory is dense in a uniquely determined thick triangulated subcategory, this result re®nes any known classi®cation of thick subcategories to a classi®cation of all strictly full triangulated ones. For example, one can thus re®ne the famous clas- si®cation of the thick subcategories of the ®nite stable homotopy category given by the work of Devinatz±Hopkins±Smith ([Ho], [DHS], [HS] Thm. 7, [Ra] 3.4.3), which is responsible for most of the recent advances in stable homotopy theory. One can likewise re®ne the analogous classi®cation given by Hopkins and Neeman (R ) ([Ho] Sect. 4, [Ne] 1.5) of the thick subcategories of D parf, the chain homotopy category of bounded complexes of ®nitely generated projective R -modules, where R is a commutative noetherian ring. The second main result is a generalization of this classi®cation result of Hop- kins and Neeman to schemes, and in particular to non-noetherian rings. Let X be a quasi-compact and quasi-separated scheme, e.g. any commutative ring or (X ) algebraic variety. Denote by D parf the derived category of perfect complexes, O the homotopy category of those complexes of sheaves of X -modules which are O locally quasi-isomorphic to a bounded complex of free X -modules of ®nite type. D (X ) I say a thick triangulated subcategory A parf is a -subcategory if for D (X ) A A E A each object E in parf and each in , the derived tensor product A X L is also in A. This is a mild condition on .If has an ample line bundle (e.g. for X a classical, hence quasi-projective variety), it suf®ces by 3.11.1 below 1 A A X = (R ) R to check that L .If Spec for a commutative ring, all D (R ) thick subcategories A parf are -subcategories. The second main result ? Deceased October 1995. Downloaded from https://www.cambridge.org/core. IP address: 170.106.34.90, on 23 Sep 2021 at 10:35:43, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1017932514274 *INTERPRINT* PREPROOF CRCs Corr M/c 2 Pips: 87450 MATHKAP comp3750.tex; 8/05/1997; 8:14; v.5; p.1 2 R. W. THOMASON (Thm. 3.15) gives for X a quasi-compact and quasi-separated scheme a bijective D (X ) correspondence between the thick triangulated -subcategories of parf and Y X Y X X Y the subspaces which are unions of closed subspaces with quasi-compact. To such a Y corresponds the thick -subcategory of those perfect Y X complexes acyclic at each point of X .For noetherian, all subspaces are quasi-compact, and the remaining condition that Y be a union of subspaces closed Y in X is usually expressed by saying that ` is closed under specialization'. For a detailed comparison with the previous work of Hopkins and Neeman see 3.17 below. Re®ning the second main result by the ®rst, I obtain (Thm. 4.1) a classi®cation A D (X ) of all strictly full triangulated -subcategories of parf : they correspond Y; H ) Y X H K (X bijectively to data ( where is a subspace as above and 0 on ) K (X ) Y is a 0 -submodule of the Grothendieck group of perfect complexes acyclic A Y off Y . Thus is determined by a condition on the supports and a condition H on the multiplicities. This classi®cation has been my personal motivation for developing the results of this paper. I seek to de®ne a good intersection ring of `algebraic cycles' on schemes X where the classical construction of the Chow ring fails, for example on singular algebraic varieties or on regular schemes ¯at and of ®nite type over Z. Inspired by the superiority of Cartier divisors over Weil divisors and by recent progress in local intersection theory, I believe the good notion of `algebraic n-cycle' is that of those perfect complexes in some triangulated n D (X ) subcategory A parf which remains to be de®ned. Technological secrets n about `moving lemmas' demand that A should be a -subcategory, and show it cannot be thick in general. The classi®cation has proved to be very helpful here in clarifying the issues to be resolved. Other results presented here worth mentioning are the Tensor Nilpotence Theo- rem (Thms. 3.6 and 3.8), generalizing to schemes a result of Hopkins and Neeman for noetherian rings ([Ho] Thm. 10, [Ne] 1.1) analogous to the Nilpotence Theorem of Devinatz±Hopkins±Smith ([DHS] Thm. 1) in stable homotopy, and a useful but little-known necessary and suf®cient condition for the equality of two classes in the Grothendieck group of a triangulated category (Lemma 2.4), due to Landsburg inspired by Heller. I have tried to make each theorem, proposition, and lemma resistant to misinterpretation on being `zapped' out of its context by endlessly restating, or at least referencing, all hypotheses and de®nitions explicitly. However, I have followed a customary global convention that `ring' always means `commutative ring'. 2. `Rappels' For the convenience of the reader and to eliminate ambiguities, I will brie¯y recall some basic de®nitions and results of the theory of triangulated categories. Useful general references are the classic treatments [Ha] I and II, and [Ve], as well as Downloaded from https://www.cambridge.org/core. IP address: 170.106.34.90, on 23 Sep 2021 at 10:35:43, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1017932514274 comp3750.tex; 8/05/1997; 8:14; v.5; p.2 THE CLASSIFICATION OF TRIANGULATED SUBCATEGORIES 3 [SGA4] XVII Section 1.2, [SGA5] VIII, [SGA6], and the more recent [BBD] Section 1, [Ri], [Sp], and [BN]. T 1.1. Let T be a triangulated category. A full triangulated subcategory of is a full subcategory A with the structure of a triangulated category such that the inclusion A!T functor i: is a triangulated (a.k.a. `exact') functor, i.e. such that it preserves distinguished (`exact') triangles and commutes with the suspension (`translation') endofunctors. Since the axioms of a triangulated category imply that any morphism is an edge of an exact triangle which is unique up to isomorphism of the opposite vertex ([Ha] I Sect. 1 TR1, TR3, Prop. 1.1.c, or [Ve] I Sect. 1 no. 1), and since the A T functor i is fully faithful so two triangles in are isomorphic in if and only if A theyareisomorphicin A, a triangle in is exact if and only if its image is exact A in T . Thus the triangulated category structure of is uniquely determined by that on T . Thus the de®nition above is equivalent to: a full triangulated subcategory of T T T is a full non-empty subcategory A of such that for every exact triangle of of A which two vertices are in A, then the third vertex is isomorphic to an object of (c.f. [Ve] I Sect. 1 no. 2±3). T 1.2. A strictly full triangulated subcategory A of is a full triangulated subcate- T gory such that A contains every object of which is isomorphic to an object of A. T 1.3. A thick (a.k.a epaisse ) triangulated subcategory A of is a strictly full tri- A angulated subcategory such that every direct summand in T of an object of is itself an object of A. Rickard ([Ri] Prop. 1.3) showed that this de®nition is equiv- alent to the somewhat more complicated classic de®nition given by Verdier ([Ve] T I Sect. 2 1±1). Using his de®nition Verdier showed that a full subcategory A of is a thick triangulated subcategory if and only if there exists a triangulated category 0 0 T !T A T and a triangulated functor such that is the full subcategory of those 0 objects whose image in T is isomorphic to 0. (c.f. [Ve] I Sect. 2 nos. 1, 2, 3). D D 1.4. A full triangulated subcategory A of a triangulated category is dense in A if each object of D is a direct summand of an object isomorphic to an object in . REMARK 1.5. If A is a full triangulated subcategory of a triangulated category T A T , the intersection of all the thick triangulated subcategories of containing D is a thick triangulated subcategory D .This is the smallest thick triangulated A subcategory of T which contains . On the other hand, one checks easily that e T the strictly full subcategory A of all objects of which are summands of objects e T DA isomorphic to objects in A is a thick triangulated subcategory of . Thus . e D But by Rickard's de®nition of thick one has ADsince is thick and contains e D = A A.
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