Comptes Rendus Mathématique

. We give a condition which characterises those weight structures on a derived category which come from a Thomason filtration on the underlying scheme. Weight structures satisfying our condition will be called ⊗ c -weight structures. More precisely, for a Noetherian separated scheme X , we give a bijection between the set of compactly generated ⊗ c -weight structures on D (Qcoh X ) and the set of Thomason filtrationsof X .Weachievethisclassificationintwosteps.First,weshowthatthebijection[12,Theorem4.10] restricts to give a bijection between the set of compactly generated ⊗ c -weight structures and the set of compactlygeneratedtensort-structures.Wethenuseourearlierclassificationofcompactlygeneratedtensor t-structurestoobtainthedesiredresult.Wealsostudysomeimmediateconsequencesoftheseclassifications in the particular case of the projective line. We show that in contrast to the case of tensor t-structures, there are no non-trivial tensor weight structures on D b (Coh P 1 k ). Résumé


Introduction
Weight structures on triangulated categories were introduced by Bondarko [4] as an important natural counterpart of t-structures with applications to Voevodsky's category of motives. Pauksztello independently came up with the same notion while trying to obtain a dual version of a result due to Hoshino, Kato and Miyachi; he termed it co-t-structures, see [11]. It has been observed by Bondarko that the two notions, t-structures and weight structures, are connected by interesting relations. In this vein, Št'ovíček and Pospíšil have proved for a certain class of triangulated categories, the collection of compactly generated t-structures and compactly generated weight structures are in bijection [12,Theorem 4.10] with each other, where the bijection goes via a duality at the compact level. In particular, this bijection holds in the derived category of a Noetherian ring R and since in this case, we have the classification of compactly generated t-structures in terms of Thomason filtrations of Spec R [1, Theorem 3.11], they obtain a classification of compactly generated weight structures of D(R).
Our aim in this short article is twofold: first to generalize the theorem of Št'ovíček and Pospíšil [12,Theorem 4.15] to the case of separated Noetherian schemes, and second to understand the two types of notions, in the simplest non-affine situation: the derived category of the projective line over a field k. Our interest in this special case arose partly from the work of Krause and Stevenson [10], where the authors study the localizing subcategories of D(Qcoh P 1 k ), and partly from our desire to better understand the general results.
In our earlier work [7], we have shown that a t-structure on D(Qcoh X ) supported on a Thomason filtration of a Noetherian scheme X satisfies a tensor condition. We call them tensor t-structures. In this article, we introduce the analogous notion of tensor weight structures, also a slightly weaker notion which we call ⊗ c -weight structures. We then show that the bijection [12,Theorem 4.10] restricts to a bijection between tensor t-structures and ⊗ c -weight structures; this can be seen as a consequence of our Lemma 10 and Theorem 15. Next, we specialize to the case of the derived categories of separated Noetherian schemes and classify compactly generated ⊗ cweight structures in this case, see Theorem 21.
In the last section, we apply all the general theory and the classification results to the derived category of the projective line over a field k. By our Theorem 15 classifying compactly generated tensor t-structures of D(Qcoh P 1 k ) is equivalent to classifying thick ⊗-preaisles of D b (Coh P 1 k ), so we restrict our attention to ⊗-preaisles of D b (Coh P 1 k ). We give a complete description of the ⊗preaisles in Proposition 24, and in Proposition 28 we determine which of these are aisles or in other words give rise to t-structures on D b (Coh P 1 k ). The result of Proposition 28 is not new, it can possibly be deduced from [3]; also in [8], the authors describe the bounded t-structures on D b (Coh P 1 k ) by using the classification of t-stabilities on D b (Coh P 1 k ). Finally, we consider the same question for tensor weight structures, and to our surprise, we discovered that on D b (Coh P 1 k ) there are no non-trivial tensor weight structures.

Preliminaries
Let T be a triangulated category and T c denote the full subcategory of compact objects. We recall the definition of t-structures which was introduced in [2].

Definition 1.
A t-structure on T is a pair of full subcategories (U , V ) satisfying the following properties: We call such a triangle truncation decomposition of T .
Next, we quote the definition of weight structures from [5].

Definition 2.
A weight structure on T is a pair of full subcategories (X , Y ) satisfying the following properties: The above triangle is called a weight decomposition of T .
For any subcategory U of T , we denote U ⊥ to be the full subcategory consisting of objects B ∈ T such that Hom(A, B ) = 0 for all A ∈ U . Analogously we define ⊥ U to be the full subcategory of objects B ∈ T such that Hom(B, A) = 0 for all A ∈ U .

Definition 4. A subcategory U of T is a preaisle if it is closed under positive shifts and extensions. Dually, we say
A preaisle is called thick if it is closed under direct summands. We say a preaisle is cocomplete if it is closed under coproducts in T , and complete if it is closed under products. Similarly, we define thick, cocomplete and complete copreaisles.
For a t-structure (U , V ) the subcategory U is a cocomplete preaisle of T , and for a weight structure (X , Y ) the subcategory X is a cocomplete copreaisle of T .
We need the notion of stable derivators to formulate the next theorem but our requirement of the theory of derivators is the bare minimum. We will not go into the precise lengthy definition here, instead, we refer the reader to [12, Section 2.1] and references therein.
(ii) There is a bijection between the set of compactly generated weight structures of T and the set of thick copreaisles of T c given by

Tensor weight and t-structures
We recall the definition of tensor triangulated category from [9, Definition A.

Definition 7.
Let T be a tensor triangulated category given with a preaisle T ≤0 satisfying Let S ⊂ T be a class of objects. We denote the smallest cocomplete preaisle containing S by 〈S 〉 ≤0 and call it the cocomplete preaisle generated by S . If T does not have coproducts we denote 〈S 〉 ≤0 to be the smallest preaisle containing S . Lemma 8. Let T ≤0 be generated by a set of objects K , that is, T ≤0 = 〈K 〉 ≤0 . Then Since U is a cocomplete preaisle we can observe that B is also a cocomplete preaisle. Now, by our assumption preaisle. Completeness of X implies B is cocomplete. Now, by following a similar argument as in (i) we get X is ⊗-copreaisle.
□ An immediate consequence of the above lemma is if T ≤0 = 〈1〉 ≤0 then every cocomplete preaisle of T is a ⊗-preaisle and every complete copreaisle of T is a ⊗-copreaisle. In particular, for a commutative ring R, all cocomplete preaisle and complete copreaisle of D(R) satisfy the tensor condition.

Definition 9.
We say an object X ∈ T is rigid or strongly dualizable if for each Y ∈ T the natural is rigidly compactly generated if the following conditions hold: Let T be a rigidly compactly generated tensor triangulated category. For such a triangulated category the tensor product on T restricts to T c therefore (T c , ⊗, 1) is also a tensor triangulated category. Suppose T is given with a preaisle T ≤0 satisfying the condition of Definition 7 then so does the preaisle T c ∩ T ≤0 of T c . So we can define ⊗-preaisles and ⊗-copreaisles of T c with respect to T c ∩ T ≤0 .

Lemma 10. Let T be a rigidly compactly generated tensor triangulated category given with a preaisle T ≤0 satisfying the condition of Definition 7.
Then, there is a one-to-one correspondence between the set of ⊗-preaisles and the set of ⊗copreaisles of T c .
Proof. Let U be a full subcategory of T c . We denote U * the full subcategory given by The assignment U → U * induces an equivalence between the preaisles and copreaisles of T c ; see [12,Lemma 4.9]. We only need to show that the above assignment preserves the tensor condition.
Let U be a preaisle of T c , X ∈ U * and T ∈ T c ∩ T ≤0 . We have for a set of compact objects S .

Definition 12.
We say a triangulated category T has the property ( * ) if: For a triangulated category T having the property ( * ), we define a weaker notion than ⊗-(co)preaisle.
Remark 14. This weaker notion gives something new only for preaisles(resp. copreaisles) which are not cocomplete(resp. complete) since by Lemma 8 it can easily be observed that for T having the property ( * ): (i) every cocomplete ⊗ c -preaisles of T is a ⊗-preaisle of T , and (ii) every complete ⊗ c -copreaisle of T is a ⊗-copreaisle of T .
With this weaker notion, we now prove the tensor analogue of Theorem 5.
Theorem 15. Let T have the property ( * ) (see Definition 12) and T = D(e), where D is a stable derivator such that for each small category I , D(I ) has all small coproducts. Then, (i) There is a bijective correspondence between the set of compactly generated tensor tstructures of T and the set of thick ⊗-preaisles of T c given by (ii) There is a bijective correspondence between the set of compactly generated ⊗ c -weight structures of T and the set of thick ⊗-copreaisles of T c given by Before proving the theorem, we will make some comments about the lack of symmetry in the above statement: Remark 16. It is easy to observe that given a subcategory P the subcategory ⊥ (P ⊥ ) is always cocomplete, that is, closed under coproducts. Therefore, by Remark 14 saying ⊥ (P ⊥ ) is a ⊗ cpreaisle of T is equivalent to saying it is a ⊗-preaisle. However, compactly generated copreaisles are not closed under products in general, so Remark 14 is not applicable here.
Example 17. 1 Consider the derived category D(Z), it is equivalent to the homotopy category of Kprojectives or dg-projectives K(d g -proj Z). The copreaisle K ≥0 (d g -proj Z) is compactly generated but it is not closed under products. Indeed, if K ≥0 (d g -proj Z) is closed under product then the countable product of Z, that is, i Z ∈ K ≥0 (d g -proj Z). Let P · be a projective (hence K-projective) resolution of i Z. Now, take the brutal truncations of P · at degree 0, σ ≥0 (P · ) and σ <0 (P · ). As σ <0 (P · ) ∈ K <0 (d g -proj Z), we have Hom( i Z, σ <0 (P · )) = 0 which implies σ ≥0 (P · ) is isomorphic to i Z. Since σ ≥0 (P · ) is projective we get i Z is projective, which is a contradiction; see for instance [6, Corollary to Theorem 3.1, p. 466].
Proof of Theorem 15. Part (i). It has already been shown in [12, Theorem 4.5 (i)] that the above assignments are bijections between the set of compactly generated t-structures of T and the set of thick preaisles of T c . We only need to show that the assignments preserve the tensor conditions.
From the definition of ⊗-preaisle, it is easy to observe that if U is a ⊗-preaisle of T then We note that B is a cocomplete preaisle containing P . Since ⊥ (P ⊥ ) is the smallest cocomplete preaisle containing P by [7, Lemma 1.9] we get B = ⊥ (P ⊥ ).

Part (ii).
In view of [12, Theorem 4.5 (ii)], again we only need to show that the assignments preserve the appropriate tensor conditions. If X is a ⊗ c -copreaisle of T then it is easy to observe that X ∩ T c is a ⊗-copreaisle of T c . Suppose P is a ⊗-copreaisle of T c we need to show that ⊥ (P ⊥ ) is a ⊗ c -copreaisle of T . By [12,Theorem 3.7] an object A of T belongs to ⊥ (P ⊥ ) if and only if A is a summand of a homotopy colimit of a sequence.
First, we observe that for any compact object T the functor H om(T, −) preserves small coproducts therefore H om(T, −) takes homotopy sequences to homotopy sequences. Since P is ⊗-copreaisle of T c for any T ∈ T c ∩ T ≤0 we have H om(T, S i ) ∈ Add P . Thus applying [12, Theorem 3.7] again we get H om(T, A) ∈ ⊥ (P ⊥ ). , O X ) is a tensor triangulated category with the derived tensor product ⊗ L O X and the structure sheaf O X as the unit. The full subcategory of complexes whose cohomologies vanish in positive degree D ≤0 (Qcoh X ) is a preaisle of D(Qcoh X ) satisfying the conditions of Definition 7. We define the ⊗-preaisles and ⊗-copreaisles of D(Qcoh X ) with respect to D ≤0 (Qcoh X ). Similarly, the ⊗-preaisles and ⊗copreaisles of Perf (X ) are defined with respect to Perf ≤0 (X ). Note that D(Qcoh X ) has the property ( * ) (see Definition 12), so we can define ⊗ c -(co)preaisles of D(Qcoh X ).

Definition 18. A subset Z is a specialization closed subset of X if for each x ∈ Z the closure of the singleton set {x} is contained in Z , that is,{ x} ⊂ Z . Note that a specialization closed subset is a union of closed subsets of X .
A subset Y is a Thomason subset of X if Y = α Y α is a union of closed subsets Y α such that X \ Y α is quasi compact. Note that if X is Noetherian then the two notions coincide.

Definition 19. A Thomason filtration of X is a map
In our earlier work we have mentioned without proof (see [7,Remark 4.13]) about the following result, here we explicitly state it for future reference. This is a generalization of Thomason's classification [13,Theorem 3.15] of ⊗-ideals to ⊗-preaisles of Perf (X ), for separated Noetherian scheme X .
Proposition 20. Let X be a separated Noetherian scheme. The assignment sending a Thomason filtration φ to S φ = {E ∈ Perf (X ) | Supp(H i (E )) ⊂ φ(i )} provides a one-to-one correspondence between the following sets:

(i) the set of Thomason filtrations of X ;
(ii) the set of thick ⊗-preaisles of Perf (X ).

Proof. In [7, Theorem 4.11] we have shown that sending φ to
provides a bijection between the set of Thomason filtration of X and the set of compactly generated tensor t-structure of D(Qcoh X ). From part (i) of Theorem 15, we conclude that the above assignment provides a bijection between Thomason filtrations of X and thick ⊗-preaisles of Perf (X ).

Theorem 21. Let X be a separated Noetherian scheme. There is a one-to-one correspondence between the following sets:
(i) the set of Thomason filtrations of X ; (ii) the set of compactly generated ⊗ c -weight structures of D(Qcoh X ).

The assignment is given by
Proof. Let φ be a Thomason filtration of X . By Proposition 20 we know φ → S φ is a bijection. Now sending S φ to S * φ is again a bijection by Lemma 10. Since S * φ is a ⊗-copreaisle of Perf (X ), is a bijection by Theorem 15. We only need to show that B φ = (S * φ ) ⊥ which is the consequence of the tensor-hom adjunction.

In the case of projective line
In this section, we specialize to the case of projective line P 1 k over a field k. By the results of earlier sections, classifying compactly generated tensor t-structures of D(Qcoh P 1 k ) is equivalent to classifying thick ⊗-preaisles of Perf (P 1 k ). For any smooth Noetherian scheme X the inclusion functor from Perf (X ) to the derived category of bounded complexes of coherent sheaves D b (Coh X ) is an equivalence. Therefore, we restrict our attention to D b (Coh P 1 k ). Note that we define ⊗-preaisles of D b (Coh P 1 k ) with respect to the standard preaisle It is now easy to see that for every n ≥ 0 we have O (−n) ∈ B.
As Coh P 1 k has homological dimension one, every complex of D b (Coh P 1 k ) is quasi isomorphic to the direct sum of its cohomology sheaves, see [8,Proposition 6.1]. Also, every coherent sheaf over P 1 k is the direct sum of line bundles and torsion sheaves. Since B is a preaisle, to show B = D b,≤0 (Coh P 1 k ) it is enough to show that B contains all the line bundles and torsion sheaves. For any m ≥ 0 consider the following triangle coming from the corresponding short exact sequence in Coh P 1 k ; see for instance [8,Equation 6.3], Since B is closed under extensions and positive shifts we have O (m) ∈ B.
Next, for any indecomposable torsion sheaf of degree d say T x , which is supported on a closed point x ∈ P 1 k , consider the following triangle coming from the corresponding short exact sequence in Coh P 1 k ; see [8,Equation 6.5], Again using the fact that B is closed under extensions and positive shifts we have T x ∈ B. □ Recall that for a set of objects S of T we denote the smallest cocomplete preaisle containing S by 〈S 〉 ≤0 . If T does not have coproducts, for instance D b (Coh P 1 k ), we denote 〈S 〉 ≤0 to be the smallest preaisle containing S . Similarly we denote 〈S 〉 ≥0 to be the smallest copreaisle containing S . Also recall that for any subcategory U we denote U * the full subcategory given by Example 23. For a fixed n ∈ Z we denote Recall that a Thomason filtration of X is a map φ : Z → 2 X such that φ(i ) is a Thomason subset of X and φ(i ) ⊃ φ(i + 1) for all i ∈ Z. We say φ is of type-1 if i φ(i ) ̸ = X ; and we say φ is of type-2 if i φ(i ) = X but not all φ(i ) = X . Let x ∈ P 1 k be a closed point. We denote the simple torsion sheaf supported on x by k(x). Now, we give an explicit description of the ⊗-preaisles of D b (Coh P 1 k ) in terms of simple torsion sheaves and line bundles.
Proposition 24. Any proper thick ⊗-preaisle of D b (Coh P 1 k ) is one of the following forms: where φ is a type-1 Thomason filtration of P 1 k .
where φ is a type-2 Thomason filtration of P 1 k and i 0 a fixed integer.
Proof. Suppose A is a thick ⊗-preaisle of D b (Coh P 1 k ). By Proposition 20 there is a unique Thomason filtration φ such that k has homological dimension one, every complex of D b (Coh P 1 k ) is quasi isomorphic to the direct sum of its cohomology sheaves. Therefore, we can write A in terms of coherent sheaves alone, k for all i . Every coherent sheaf over P 1 k is the direct sum of line bundles and torsion sheaves. Since the support of any line bundle is whole P 1 k . In this case, A contains only torsion sheaves. As torsion sheaves can be generated by simple torsion sheaves we have, Case 2. φ is type-2. Since i φ(i ) = X there is an integer i 0 such that φ(i 0 ) contains the generic point of P 1 k . We can take i 0 to be the largest such integer. Observe that φ(i ) = P 1 k for all i ≤ i 0 and Next, we will show which of these ⊗-preaisles of D b (Coh P 1 k ) are t-structures on D b (Coh P 1 k ). First, we prove a few lemmas. F E [1] such that E ∈ A and F ∈ A ⊥ . This contradicts the fact that A is an aisle.
□ Definition 27. We say φ is a one-step Thomason filtration of P 1 k if there is an integer i 0 and a Thomson subset Z i 0 such that

. Then, A is an aisle if and only if the corresponding Thomason filtration is a one-step filtration.
Proof. If A is a ⊗-preaisle which is also an aisle then by Lemma 26 the corresponding filtration is a one-step filtration.Conversely, suppose the filtration is one step, we will show that every complex of D b (Coh P 1 k ) can be decomposed into a triangle where the first term is in A and the third term is in A ⊥ . Without loss of generality we may assume the one step occurs at i 0 = 0, and φ(0) = Z 0 is a Thomason subset.
If Z 0 = P 1 k , then A = D b,≤0 (Coh P 1 k ) and we get the standard t-structure. Now, suppose Z 0 ̸ = P 1 k . Since the filtration is one step we only need to show sheaves at degree zero have tdecompositions, all other shifted sheaves have obvious t-decompositions. The functor Γ Z 0 (−) gives a t-decomposition of sheaves at degree zero. □ Next, we give an explicit description of ⊗-copreaisles of D b (Coh P 1 k ) in terms of simple torsion sheaves and line bundles.
Proposition 29. Any proper thick ⊗-copreaisle of D b (Coh P 1 k ) is one of the following forms: where φ is a type-1 Thomason filtration of P 1 k .
where φ is a type-2 Thomason filtration of P 1 k and i 0 a fixed integer.
Proof. By the proof of Lemma 10, we know that every ⊗-copreaisle of D b (Coh P 1 k ) is of the form A * where A is a ⊗-preaisle. Now using the description given in Proposition 24 we conclude our result.

□
The trivial ⊗-copreaisles D b (Coh P 1 k ) and 0 give rise to tensor weight structures on D b (Coh P 1 k ). In contrast to the case of t-structures (see Proposition 28), the next result shows that, there are no other tensor weight structures on D b (Coh P 1 k ).
Proposition 30. The trivial weight structures are the only tensor weight structures on D b (Coh P 1 k ).

Proof. Suppose
A is a non-zero ⊗-copreaisle of D b (Coh P 1 k ) which induces a weight structure. We first claim that A can not be a copreaisle containing only torsion sheaves. Indeed, since A is non-zero it contains some indecomposable torsion sheaf (up to shift), say T x [i ]. Without loss of generality we can assume i = 0, that is, T x ∈ A . By our assumption A contains only torsion sheaves hence L As A is closed under extensions, T x is in A . This proves A contains all the torsion sheaves and their shifts. Therefore, A must be equal to D b (Coh P 1 k ).
□ Remark 31. The above proposition says when X = P 1 k , the bounded derived category D b (Coh X ) has no non-trivial tensor weight structure. However, when X = Spec R for a regular Noetherian ring R, the derived category D b (Coh X ) has tensor weight structures. In this case, D b (mod-R) is equivalent to the homotopy category K b (proj-R), and the brutal truncations provide non-trivial tensor weight structures.