REGULAR RING SPECTRA AND WEIGHT STRUCTURES

Transcription

1 EGULA ING SPECTA AND WEIGHT STUCTUES ADEEL A. KHAN AND VLADIMI SOSNILO Abstract. We prove a gluing result for weight structures and adjacent t-structures, in the setting of a semi-orthogonal decomposition of stable -categories. We apply this to the study of regular ring spectra, by interpreting the property of regularity in terms of the existence of a t-structure adjacent to the weight structure on the category of perfect complexes. Namely, we use this to construct many examples of regular ring spectra from existing ones, as upper triangular ring spectra. arxiv: v1 [math.kt] 8 Jan 2019 Contents 1. Introduction 1 2. egular ring spectra Preliminaries Bounded regular ring spectra A noncommutative counterexample 4 3. Adjacent structures Weight structures Adjacent weight and t-structures Gluing weight structures Upper triangular matrix rings Proof of Theorem eferences Introduction Let be a left noetherian E 1 -ring spectrum. We will say that is regular if every coherent left -module is perfect. By Serre s criterion, this agrees with the usual definition when is discrete (an Eilenberg MacLane spectrum). The notion was first studied in [BL14], where it was used to construct certain localization fiber sequences for algebraic K-theory. As shown there, examples of regular E 1 -ring spectra include ko, ku, BP n, and tmf. In all the known examples, regular ring spectra are either discrete or have infinitely many non-trivial homotopy groups. According a result of J. Lurie [SAG, Lemma ], this is in fact a general phenomenon, at least if we restrict our attention to E -ring spectra. As we note here (Theorem 2.2.1), Lurie s result holds (with the same proof) more generally for E 1 -ring spectra satisfying a much weaker commutativity condition: namely, that π 0 () is central in π (). However, in the noncommutative case, i.e., when π 0 () is not central in π (), this is no longer the case: we construct a simple example of a non-discrete ring spectrum which is regular but has only finitely many homotopy groups (Sect. 2.3). 1

2 EGULA ING SPECTA AND WEIGHT STUCTUES 2 Our example is an instance of a general construction. Given ring spectra and S, and an -S-bimodule M, we consider the upper triangular ring spectrum T=( M 0 S ). If and S are regular, we prove that T is regular as long as M is connective and perfect as an S-module (Theorem 3.4.5). To prove Theorem it is useful to shift perspective. ecall that, for any connective ring spectrum, the category of perfect complexes over is endowed with a canonical weight structure (Example 3.1.5). The condition of regularity of can be reformulated as the existence of a t-structure on this category which is adjacent to the weight structure; that is, its non-negative part coincides with the non-negative part of the weight structure. Adjacent t-structures have been studied in general contexts by Bondarko (see [Bon18]). Our main result (Theorem 3.3.3) deals with semi-orthogonal decompositions or split short exact sequences of stable -categories i C C E D. L C When C and D are endowed with weight structures, we will explain how to construct a weight structure on E extending those on C and D, if the essential images of D w 0 and ΣC w 0 in E are orthogonal. Moreover, if there are adjacent t-structures on C and D, then under a mild (and necessary) condition, we also construct an adjacent t-structure on E. This easily specializes to Theorem Acknowledgments. We would like to thank Benjamin Antieau, Mikhail Bondarko, and Denis- Charles Cisinski for interesting discussions. L D i D 2. egular ring spectra 2.1. Preliminaries. Let be a connective E 1 -ring, and write Mod for the category of left -modules. We recall a few finiteness conditions from [HA, 7.2.4] The perfect left -modules are those built out of under finite (co)limits and direct summands. These span a thick subcategory Mod perf Mod Suppose that is left noetherian, i.e. that π 0 () is left noetherian as an ordinary ring and the homotopy groups π i () are finitely generated as left π 0 ()-modules. A left -module M is almost perfect if it is bounded below and each π i (M) is finitely generated as a left π 0 ()- module [HA, Prop ], and coherent if it is almost perfect and bounded above. We write Mod coh for the full subcategory of Mod spanned by coherent left -modules; this is also a thick subcategory by [HA, Prop ]. emark Note that the noetherian hypothesis guarantees that is almost perfect as a left -module. Thus if is bounded above, then it is moreover coherent as a left -module. It follows then that every perfect left -module is coherent, i.e., that there is an inclusion Mod perf Modcoh when is bounded above. Definition Let be a left noetherian E 1 -ring. We will say that is regular if every coherent left -module is perfect.

3 EGULA ING SPECTA AND WEIGHT STUCTUES 3 It suffices to require that every discrete coherent left -module is perfect, in view of the exact triangles Σ i π i (M) τ i (M) τ i 1 (M) for M Mod and i Z. Moreover, one has the following useful criterion (see [BL14, Prop. 1.3]): Proposition Let be a left noetherian E 1 -ring such that π 0 () is a regular commutative ring. Then is regular if and only if π 0 () is perfect as an -module. emark Let be a left noetherian E 1 -ring. The -category Mod admits a canonical t-structure, where the non-negative part(mod ) t 0 is spanned by the connective modules. This always restricts to the full subcategories of almost perfect and coherent modules [HA, Prop ]. One might ask whether it also restricts to the full subcategory Mod perf of perfect -modules. This is the case if and only if π i (M) belongs to Mod perf for any M Mod perf, i.e., precisely when is regular. emark For every M Mod perf and N Mod coh, there exists an integer n such that the mapping space Mod (Σ n M,N) is contractible. Indeed, this is true for M=Σ j since N is bounded above, and therefore for an arbitary M, since it is built out of Σ j using finite colimits and retracts. For regular, this says that the space Mod (M,N) has finitely many non-zero homotopy groups, for any M,N Mod perf. Examples (i) If is discrete, then it is regular if and only if it is regular in the sense of ordinary commutative algebra. This follows from Serre s criterion. (ii) Let k be a regular commutative ring and =k[t] with T in degree 2. Then is regular. (iii) The E -ring spectra ku, ko, and tmf are regular [BL14] We say that is quasi-commutative if, for every x π 0 () and y π n (), the equality xy= yx holds in π n (). Under this assumption, we can show that regularity is stable under localizations: Lemma Let be a quasi-commutative connective E 1 -ring spectrum. If is regular, then for any set S π 0 (), the localization S 1 is also regular. Proof. It will suffice to show that, for every perfect S 1 -module N, we have π 0 (N) Mod perf Denote by L the extension of scalars functor Mod perf Mod perf S 1. S 1. Since any perfect S 1 -, we may restrict our, it follows from [HA, Prop ] that we have module N is a retract of an object of the form L(M) for some M Mod perf attention to N=L(M). For M Mod perf π n (L(M)) S 1 π n (M) π n (M) π0()s 1 π 0 (). This implies in particular that π 0 (L(M)) L(π 0 (M)) belongs to Mod perf S 1, as claimed Bounded regular ring spectra. This subsection is dedicated to the following result, due to J. Lurie; see [SAG, Lemma ], where it is provenfor E -ring spectra. Here we only note that the same proof works more generally for quasi-commutative E 1 -ring spectra. Note that this is optimal in the sense that the statement is false without the quasi-commutativity assumption (see Subsect. 2.3). Theorem Let be a quasi-commutative connective E 1 -ring spectrum. If is regular and has only finitely many nontrivial homotopy groups, then is discrete.

4 EGULA ING SPECTA AND WEIGHT STUCTUES 4 Proof. We can clearly assume that is nonzero. To show that is discrete, it will suffice to show that its localization p, at any prime ideal p π 0 (), is discrete. Since p is regular by Lemma , we may replace by p and thereby assume that π 0 () is local. Denote by m π 0 () the maximal ideal, and κ the residue field. Let n 0 be the largest integer such that π n () 0. For the sake of contradiction, we will assume that n>0. Since is noetherian, the π 0 ()-module π n () is finitely generated. Let p π 0 () be a minimal prime ideal such that p Supp π0()(π n ()). By Lemma , the localization p is regular. Therefore, replacing by p, we may assume that π n () is supported at m, so that in particular m π 0 () is an associated prime ideal of π n (). Since is regular, the coherent left -module κ is perfect. Therefore, the following claim will yield the desired contradiction: ( ) Let N be a nonzero connective perfect left -module. Let k 0 be the smallest natural number such that N is of tor-amplitude k. Then we have π n+k (N) 0. To prove( ) we argue by induction. In the case k=0, N is flat with π 0 (N) 0, so that π n (N) π n () π0() π 0 (N) 0. Assume therefore that k>0. Choose elements of π 0 (N) that induce a basis of the κ-vector space π 0 (N) π0() κ and consider the surjective -module morphism u m N they determine. Its fiber, which we denote N, is a connective perfect left -module fitting in an exact triangle N v m u N. Considering the induced exact sequence of abelian groups 0=π n+k ( m ) π n+k (N) π n+k 1 (N ) ϕ π n+k 1 ( m ), it will suffice to show that ϕ is not injective. Since N is of tor-amplitude k, with k minimal and> 0, it follows that N is nonzero and of tor-amplitude k 1. Therefore by the inductive hypothesis, π n+k 1 (N ) 0. If k>1, then π n+k 1 ( m )=0, so ϕ is not injective. It remains to consider the case k=1. In this case, the perfect left -module N is flat, and hence free of finite rank r 0, since π 0 () is local. Therefore the map v N m is determined up to homotopy by a matrix{v i,j } 1 i r,1 j m of elements v i,j π 0 (). Since m π 0 () is an associated prime of π n (), we may choose a nonzero element x π n () such that multiplication by x kills m. We claim that the element (x,x,...,x) π n () r π n (N ) is sent by ϕ to zero in π n ( m ). For this purpose it will suffice to show that the elements v i,j π 0 () all belong to m. Consider the exact sequence π 0 (N )/m v π 0 ( m )/m u π 0 (N)/m 0. Since the map u π 0 ( m )/m π 0 (N)/m is injective, it follows that v π 0 (N )/m π 0 ( m )/m is the zero map. In other words, the image of v π 0 (N ) π 0 ( m ) lands in m m π 0 () m, as desired A noncommutative counterexample. We now give an example of a noncommutative regular ring spectrum that is bounded but not discrete. This shows that the quasi-commutativity hypothesis in Theorem is necessary. Construction Let k be a field. Consider the graded ring, concentrated in degrees 0 and 1, where 0 =k k and 1 =k with multiplication defined by formulas (a,b) (c,d)=(ac,bd) x (a,b)=bx

5 EGULA ING SPECTA AND WEIGHT STUCTUES 5 where(a,b),(c,d) 0, x 1. (a,b) x=ax, The graded ring is associative but not commutative, and gives rise to a connective E 1 -ring spectrum, which we denote again by, that is not quasi-commutative. However, it has regular π 0, and is also itself regular: Proposition Let be the E 1 -ring spectrum defined by Construction Then is regular. Proof. By Proposition it suffices to show that π 0 () is perfect. Denote by X the direct summand of the free -module corresponding to the projector (0,1) π 0 ()=k k. We first note that X k is discrete. Indeed we have by definition π 0 (X)=k and π i (X)=0 for i 0,1. The only x k satisfying(0,1) x=x is zero, so π 1 (X) is also zero. Since x (0,1)=x π 1 (), the element 1 k π 1 () gives rise to a map X[1] 1. This map induces an isomorphism π 0 (X) π 1 (), and therefore exhibits X[1] as the 1-connective cover τ 1. It follows that its cofiber is an object of Mod perf equivalent to τ 0 =π 0 (). emark In fact it is easy to see using the Künneth spectral sequence that any E 1 -ring spectrum is regular as soon as the ring π ()= n π n () is regular. In the example above, the ring π () is isomorphic to the ring of triangular matrices, which is regular. 3. Adjacent structures In this section we recall the notion of adjacent structures and show its relation to regularity. An adjacent structure consists of a pair of a t-structure and a weight structure compatible in a strict sense. We begin by recalling the latter notion Weight structures. Definition A weight structure w on a stable -category C is the data of two Karoubi closed full subcategories C w 0,C w 0 satisfying the following axioms. (i) Semi-invariance with respect to translations. We have the inclusions ΣC w 0 C w 0, ΩC w 0 C w 0. (ii) Orthogonality. For any X C w 0, Y C w 0, we have π 0 Maps C (X,ΣY)=0. (iii) Weight decompositions. For any X C, there exists an exact triangle where w 0 X C w 0 and w 1 X ΣC 0. w 0 X X w 1 X, The heart Hw of a weight structure is the full subcategory whose objects belong to both C w 0 and C w 0. A weight structure is bounded if any object X is an object of Σ n C w 0 and of Ω n C w 0 for some n. A functor between stable -categorieswith weight structures is weight-exact if it preserves both classes. The following construction gives rise to a variety of examples of weight structures:

6 EGULA ING SPECTA AND WEIGHT STUCTUES 6 Proposition Let A be an additive -category. Denote by  the stable -category Fun add (A op,spt) of additive presheaves of spectra on A. Then the full subcategory Âc of compact objects admits a bounded weight structure w with the following properties: (i) The full subcategory(âc ) w 0 is spanned by presheaves with values in connective spectra. (ii) The Yoneda embedding A  factors through a canonical functor A Hw which exhibits the heart Hw as the idempotent completion of A. In fact, this is essentially the only example, as the following proposition shows. Proposition Let C be a idempotent complete stable -category endowed with a bounded weight structure. Then there is an equivalence of -categories C Ĥwc, where Ĥw denotes the construction of Proposition Both Propositions and follow from [Sos17, Cor. 3.4]. The following example will be of special interest for us: Example Let be a connective E 1 -ring spectrum and let A be the additive -category Mod free of finitely generated free -modules. Then the construction Âc of Proposition is canonically equivalent to the stable -category Mod perf. It follows that Modperf admits a canonical bounded weight structure w such that the non-negative part(mod perf ) w 0 is spanned by the connective perfect -modules Adjacent weight and t-structures. Weight structures look quite similar to t-structures. However, the orthogonality conditions and the directions of the maps in the weight and t- decompositions are opposite to each other. A less direct relation between the two notions is given by the following result of Bondarko [Bon18, Thm. 0.1]. Theorem Let C be a stable -category having all coproducts and satisfying the Brown representability theorem (e.g., compactly generated). Suppose(C w 0,C w 0 ) is a weight structure such that C w 0 is closed under arbitrary coproducts. Then there exists a t-structure on C such that C t 0 =C w 0. When C does not have all coproducts, such a t-structure may not exist. When it does, it is called an adjacent t-structure: Definition Let C be a stable -category. Given a weight structure(c w 0,C w 0 ) and a t-structure(c t 0,C t 0 ), we say that they are adjacent if the equality C t 0 =C w 0 holds. Any non-regular ring spectrum provides an example of a weight structure with no adjacent t-structure: Example Let be a left noetherian E 1 -ring spectrum and consider the canonical weight structure w on Mod perf (Example 3.1.5). Note that there a t-structure t on Mod perf is adjacent to w if and only if it is the restriction of the homotopy t-structure on Mod. In particular, such an adjacent t-structure exists if and only if is regular (see emark 2.1.6). If this t-structure is bounded, then has finitely many homotopy groups and hence, by Theorem 2.2.1, is discrete Gluing weight structures. ecall the following definition from [BGT13]: Definition A split short exact sequence (or semi-orthogonal decomposition) of stable -categories is a diagram i C C E D L C L D i D

7 EGULA ING SPECTA AND WEIGHT STUCTUES 7 satisfying the following conditions: (i) The functors i C and i D are fully faithful. (ii) The functor L C is right adjoint to i C, and i D is right adjoint to L D. (iii) The essential image of i D is the right orthogonal to the essential image of i C. Lemma Given a split short exact sequence of stable -categories as above, the unit and counit maps of the adjunctions form an exact triangle i C L C (X) X i D L D (X). Proof. Condition (iii) of the definition implies that the fiber of the map X i D L D has the form i C Y for some Y. Now Y L C i C Y L C X is an equivalence. Our main result is as follows: Theorem Suppose given a split short exact sequence of stable -categories: i C C E D. L C Assume that C and D admit weight structures such that π 0 E(i D X,i C Y)=0 for any X D w 0 and Y ΣC w 0. Then there exists a unique weight structure on E such that all the functors in the diagram are weight-exact. Moreover, if i D has a right adjoint D and the weight structures on C and D admit adjacent t-structures, then so does the weight structure on E. We postpone the proof to Subsect L D i D 3.4. Upper triangular matrix rings. In this subsection we explain how Theorem can be applied to produce new examples of regular ring spectra Any split short exact sequence as in Definition is determined by C, D, and the C-D-bimodule E(i C ( ),i D ( )), in the following sense. Given stable -categories C, D, and an exact C-D-bimodule M, one can define the upper triangular -category ( C M 0 D ) as the comma category j D M(,C), where j D is the Yoneda embedding D Fun ex (D op,spt) and M(,C) is the functor C Fun ex (D op,spt) sending X C to M(,X). This -category is always stable and is equipped with fully faithful functors from C and D, admitting right and left adjoints, respectively, which fit into a split short exact sequence. Conversely, given a split short exact sequence i C C E D, the -category E is canonically equivalent to the upper triangular category L C L D i D ( C E(i C( ),i D ( )) ). 0 D

8 EGULA ING SPECTA AND WEIGHT STUCTUES Let and S be E 1 -rings. Then any -S-bimodule M gives rise to a split short exact sequence i C C E D, where C=Mod perf, D=Modperf S, and E is the upper triangular -category L C L D i D ( C M S ). 0 D The latter is compactly generated by the object G=i C () i D (S) E. In particular, there is an equivalence Mod perf End E E ([SS03]; see also Proposition 3.1.4). (G) Definition For, S, M as above, we call the endomorphism spectrum End E (G) the upper triangular matrix ring and denote it by ( M 0 S ). emark When and S are discrete, the above definition is compatible with the classical one. Moreover, we have an isomorphism π ( E(i D(S),i C ()) ) ( π π E(i D (S),i C ()) ). 0 S 0 π S In particular, if and S are left noetherian and M is connective and almost perfect, then the upper triangular matrix ring is left noetherian. In this setting, Theorem yields: Theorem Let and S be regular connective E 1 -rings, and M a connective -S-bimodule which is perfect as an S-module. Then the E 1 -ring spectrum is regular. T=( M 0 S ) Proof. Let C=Mod perf, D=Modperf S, and E=( C M S ). 0 D ecall that C and D are equipped with canonical weight structures(example 3.1.5). To show that T is regular, it will suffice to show that Mod perf T E admits a weight structure with an adjacent t-structure (Example 3.2.3). In order to apply Theorem 3.3.3, note that the first condition translates as connectivity of the spectrum M = E(i D (S),i C ()), while the second condition translates as perfectness of M as a right S-module Proof of Theorem

9 EGULA ING SPECTA AND WEIGHT STUCTUES Weight structure. WesetE w 0 ande w 0 tobetheextension-closuresoftheunionsi C C w 0 i D D w 0 and i C C w 0 i D D w 0, respectively. These classes are semi-invariant with respect to translations, and the orthogonality axiom follows from fully faithfulness of i C and i D and from the extra condition in the statement. It suffices to construct a weight decomposition for an object X E. By Lemma we have an exact triangle i C L C X X i D L D. Take any weight decompositions of L C X and L D X. Since there are no non-zero maps Ωi C w 0 L D X i C w 1 L C X we obtain a homotopy commutative square i C w 0 L D ΩX i C w 0 L C X i C L D ΩX i C Y Using Lemma of [BBD82] we get a commutative diagram i C w 0 L C X X i D w 0 L D X i C Y X i D L D X i C w 1 L C X X i D w 1 L D X in the homotopy category of E, whose rows and columns come from exact triangles. In particular, X E w 0 and X ΣE w 0, so the exact triangle X X X is a weight decomposition Adjacent t-structure. We now assume that the weight structures on C and D admit adjacent t-structures, and that i D admits a right adjoint D. In this case we will show that the weight structure on E constructed above also admits an adjacent t-structure. It sufficesto constructmapsτ τ 0 X X forallx whereτ 0 X E w 0 and π 0 E(Y,Cofib(τ))=0 for any Y E w 0. First consider the case X= i D Y. Then the map i D τ i D τ 0 Y i D Y satisfies the conditions. Indeed, for any Z D w 0 and π 0 E(i D Z,Cofib(i D τ))=π 0 E(i D Z,i D Cofib(τ))=π 0 D(Z,Cofib(τ))=0 π 0 E(i C Z,Cofib(i D τ))=π 0 E(i C Z,i D Cofib(τ))=π 0 C(Z,L C i D Cofib(τ))=0 for any Z C. So, π 0 E(Z,Cofib(i D τ))=0 for any Z from the extension-closure of i C C w 0 i D D w 0. Next consider the case X=i C Y. The cofiber Y of the map i C τ i C τ 0 Y i C Y satisfies π 0 E(i C Z,Y )=π 0 C(Z,Cofib(τ))=0. for any Z C w 0. This orthogonality property is also satisfied by the cofiber Y of the map i D τ 0 D Y i D D Y Y since for any Z C. The map π 0 E(i C Z,i D τ 0 D Y )=π 0 E(Z,L C i D τ 0 D Y )=0 π 0 E(i D Z,i D τ 0 D Y ) π 0 D(Z,i D τ 0 D Y ) π 0 D(Z, D Y ) π 0 E(i D Z,Y ) is an isomorphism for Z D w 0 and an injection for Z ΩD w 0 by orthogonality properties of the t-structure on D. Therefore from the long exact sequence associated to an exact triangle we

10 EGULA ING SPECTA AND WEIGHT STUCTUES 10 see that π 0 E(i D Z,Y )=0 for any Z D w 0. The fiber F of the map X Y is an extension of i D τ 0 D Y = Cofib(Y Y ) by i C τ 0 Y = Cofib(X Y ), so it belongs to E w 0 and the map F X satisfies the desired conditions. Finally let X be arbitrary. By Lemma there is an exact triangle Ωi D L D X p i C L C X X. The already constructed maps τ 0 Ωi D L D Ωi D L D X and τ 0 i C L C X i C L C X fit into a homotopy commutative diagram τ 0 Ωi D L D X τ 0p τ 0 i C L C X τ D τ C Ωi D L D X i C L C X. p Applying Lemma of [BBD82] we get a homotopy commutative diagram τ 0p τ 0 Ωi D L D X τ 0 i C L C X P τ D τ C p Ωi D L D X i C L C X X τ 1 Ωi D L D X τ 1 i C L C X N whose rows and columns are exact triangles. We note that P belongs to E w 0 and π 0 E(Y,N)=0 for any Y E w 1. Moreover, τ 1 Ωi D L D X=i D τ 1 ΩL D X by construction of τ D, so π 0 E(Y,N)= 0 for any Y i C C w 0. However, the group π 0 E(i D Y,N) might be non-zero for Y Hw D. The issue is fixed using the following. Claim. There exists an object I Ht D and a map τ 0 i D I N such that the map π 0 E(i D Y,τ 0 ) is an isomorphism for any Y Hw D. Indeed, assume the claim is proven. Then it implies the vanishing of π 0 E(i D Y,Cofib(τ 0 )) for any Y D w 0. Moreover, π 0 E(i C Y,Cofib(τ 0 )) is just isomorphic to π 0 E(i C Y,N) for Y C, so π 0 (Y,Cofib(τ 0 ))=0 for any Y E w 0. The fiber τ 0 X of the map X Cofib(τ 0 ) belongs to E w 0 as it is an extension of i D I by P. So τ 0 X X gives us the desired map. It suffices to prove the claim. From the bottom and the middle exact trianglesin the diagramabovewesee that fory Hw D π 0 E(i D Y,N) im(π 0 E(i D Y,X) u π 0 E(i D Y,Στ 1 Ωi D L D X)) and the isomorphism is induced by the map N Στ 1 Ωi D L D X. Sincei D isfullyfaithful, thecompositionoftheunit andthecouniti D D X X i D L D X defines a map γ D X L D X. We define I to be the image in the abelian categoryht D of the map π 0 γ π 0 D X π 0 L D X. There is an obvious map f from i D I to Στ 1 Ωi D L D X= i D τ 0 L D X. The following commutative diagram shows that f maps π 0 E(i D Y,i D I) isomorphically to the subset of π 0 E(i D Y,i D τ 0 L D X) isomorphic to π 0 E(i D Y,N).

11 EGULA ING SPECTA AND WEIGHT STUCTUES 11 π 0 E(i D Y,i D π 0 D X) π 0 E(i D Y,i D I) π 0 E(i D Y,i D π 0 L D X) (1) π 0 E(i D Y,i D τ 0 D X) π 0 E(i D Y,i D τ 0 L D X) (3) (4) π 0 E(i D Y,i D D X) π 0 E(i D Y,X) π 0 E(i D Y,i D τ 0 L D X) Indeed, considering the long exact sequences the maps (1) and (2) fit into, we see that they are isomorphisms by the orthogonality axiom for weight structures. The map (3) is a surjection for the same reason. The map (4) becomes the map π 0 D(Y, D i D D X) π 0 D(Y, D X) under the adjunction isomorphism for i D and D. The latter map is an isomorphism because id D D i D is an equivalence. Therefore we see that the images of π 0 E(i D Y,i D I) and of π 0 E(i D Y,X) in π 0 E(i D Y,i D τ 0 L D X) coincide and the former maps injectively onto the image. Now it suffices to construct a map i D I N that makes the triangle u f = (2) N i D I f i D τ 0 L D X commute. Since N is a fiber of the map i D τ 0 L D X Στ 1 i C L C X, it suffices to prove that the composite map i D I Στ 1 i C L C X is null-homotopic. We will show that the element f in the upper left corner of the following commutative diagram becomes trivial after applying the horizontal map. π 0 E(i D I,i D τ 0 L D X) π 0 E(i D I,Στ 1 i C L C X) (1) (2) π 0 E(i D π 0 D X,i D τ 0 L D X) π 0 E(i D π 0 D X,Στ 1 i C L C X) (3) π 0 E(i D τ 0 D X,i D τ 0 L D X) π 0 E(i D τ 0 D X,Στ 1 i C L C X) (4) (5) π 0 E(i D D X,i D τ 0 L D X) π 0 E(i D D X,Στ 1 i C L C X) The long exact sequence associated to the exact triangle i D K i D π 0 D X i D I where K denotes the kernel of the surjection π 0 D X I in the abelian category Ht D, together with the vanishing property in the definition of τ C and τ D, yields injectivity of the maps (1) and (2). So it suffices to show that the image of f in π 0 E(i D π 0 D X,i D τ 0 L D X) is mapped to zero via the horizontal map. By definition of f the image lifts via (3) to a map f= i D τ 0 (γ). The long exact sequence associated with the exact triangle i D τ 1 D X i D D X i D τ 0 D X together with the vanishing property in the definition of τ C and τ D implies that the maps (4) and (5) are isomorphisms. So it suffices to show that the image of f via (4) is mapped to zero via the horizontal map. The image of f is the composite i D D X i D τ 0 D X i D τ 0 L D X.

12 EGULA ING SPECTA AND WEIGHT STUCTUES 12 This factorizes as i D D X i D L D X i D τ 0 L D X. Therefore, the result follows from the fact that the composite of the two maps in a exact triangle is null-homotopic. eferences [BBD82] A. Beilinson, J. Bernstein, and P. Deligne. Faisceaux pervers. Asterisque, 100:5 171, [BGT13] Andrew J Blumberg, David Gepner, and Gonçalo Tabuada. A universal characterization of higher algebraic K-theory. Geometry & Topology, 17(2): , [BL14] Clark Barwick and Tyler Lawson. egularity of structured ring spectra and localization in K-theory. arxiv preprint arxiv: , [Bon18] Mikhail Bondarko. On torsion pairs, (well generated) weight structures, adjacent t-structures, and related (co)homological functors. arxiv preprint arxiv: v6, [HA] Jacob Lurie. Higher algebra. Preprint, available at www. math. harvard. edu/ ~ lurie/ papers/ HigherAlgebra. pdf, Version of [SAG] Jacob Lurie. Spectral algebraic geometry. Preprint, available at www. math. harvard. edu/ ~ lurie/ papers/ SAG-rootfile. pdf, Version of [Sos17] Vladimir Sosnilo. Theorem of the heart in negative K-theory for weight structures. arxiv preprint arxiv: , [SS03] Stefan Schwede and Brooke Shipley. Stable model categories are categories of modules. Topology, 42: , Fakultät für Mathematik, Universität egensburg, Universitätsstr. 31, egensburg, Germany address: adeel.khan@mathematik.uni-regensburg.de UL: Laboratory of Modern Algebra and Applications, St. Petersburg State University, 14th line, 29B, Saint Petersburg, ussia St. Petersburg Department of Steklov Mathematical Institute of ussian Academy of Sciences, Fontanka, 27, Saint Petersburg, ussia address: vsosnilo@gmail.com