Operations on t-structures and perverse coherent
sheaves
Alexei Bondal

Introduction.
Suppose we have an equivalence D(A) ' D(B) between the derived categories of two
abelian categories A and B. How this helps to study D(A)? The virtue is that we cantransport the standard t-structure from D(B) into D(A) via the equivalence and look how
it interplays with the standard t-structure in D(A). In particular, we can try to produce
new t-structures from these two.
In this paper we consider two binary operations on t-structures in a triangulated
category, D. More precisely, there is a natural partial order on the set of t-structures andthe operations are just that of intersection and union as they are defined in the latticetheory. The basic problem is to prove the existence of intersection and union for two given
t-structures.
We define lower and upper consistent pairs of t-structures. These are some elementary
conditions which guarantee the existence of intersection or union for the pairs. Then
we take a triple of t-structures and examine various conditions of consistency in pairs oft-structures in this triple which imply consistency in the pair that comprises one of themand the intersection (or union) t-structure of the other two. This gives some rules for
iterating the operations of union and intersection.
The lattice of vector subspaces in a vector space does not satisfy distributivity law,
but rather modularity law. The partially ordered set of t-structures is somewhat similar.
We prove, in particular, some versions of modularity law under suitable conditions on
consistency in pairs. We consider abstract partially ordered sets with two binary relations subject to axioms, which upper and low consistency obey. We call them sets withconsistencies. We prove that such a set P allows a map into a universal lattice withconsistencies, U (P ).
A chain in a poset is a fully ordered set of elements. A theorem due to Birkhoff [4]claims that two chains in a modular lattice generate a distributive lattice. A pair of chains{ai } and {bj } is said to be consistent if all pairs (ai , bj ) are both upper and low consistent.
We prove a theorem that a consistent pair of chains in a set with consistencies generates
a distributive lattice L.
Our application of the developed techniques is to perverse coherent sheaves on schemesbof finite type over a field. We consider the derived category D(X) = Dcoh(X) of coherent
sheaves on such a scheme X. It has the standard t-structure. The category possesses
Grothendieck-Serre duality functor D : D(X) → D(X)opp , defined for F ∈ D(X) by· ),DF = HomX (F, ωX
1· stands for the dualizing complex on X. Since D is an anti-equivalence, it allows
where ωX
to define another, ‘dual’, t-structure on D(X) by transporting the standard t-structure
via D. We consider two chains of t-structures: one is obtained by shifts of the standard
t-structure and the other one by shifts of the dual t-structure. We prove that this pair ofchains is consistent. Therefore, they generate a distributive lattice of t-structures by theabove version of Birkhoff theorem. It is a set of perverse t-structures on coherent sheaves(those which are independent of stratification of X) considered independently by Deligne,
Bezrukavnikov [3], Kashiwara [11], Gabber [9].
The hard part in constructing a t-structure with required properties is to prove existence of the adjoint to the embedding of what-is-to-be D≤0 into D. The standard way to
produce such a functor is by using limits in the category.
According to the proposal in paper [6], derived categories of coherent sheaves can playa crucial role in the Minimal Model Program of birational algebraic geometry. We expectthat important ingredients of the Minimal Model Program, such as flips and flops, could
be constructed and understood by means of transformation of t-structures (see [7] for
3-dimensional flops).
This problem requires considering triangulated categories of ‘small’ size, like thosearising in ‘realistic’ algebraic geometry, such as the bounded derived categories of complexes of coherent sheaves on algebraic varieties as opposed to the unbounded derived
categories of quasi-coherent sheaves. There is no suitable general properties on existenceof limits in such categories. For that reason, standard constructions often lead us a priori
beyond the ‘small’ category. For example, the derived functor of cohomology with support
takes, in general, coherent complexes to quasi-coherent ones.
One of the main technical problems in studying algebraic geometry by means of triangulated categories can be formulated as to find appropriate finiteness properties onsuitable subcategories in the derived categories of coherent sheaves and keep track ofthese properties under relevant transformations of these subcategories.
We hope that this paper shed light on the interplay of t-structures, thus giving approach to understand the finiteness conditions relevant to t-structures.
It would be interesting to compare the approach of this paper with Tom Bridgeland’s
work on producing stability conditions (hence a plenty of t-structures!) via deformationargument [8].
I am indebt to Michel Van-den-Bergh for useful discussions.
This work was done more than 15 years ago and was, for the first time, reportedon Shafarevich’s seminar at Steklov Institute in late 90’s. It is my pleasure to devote
this paper to I.R. Shafarevich on the occasion of his 90-th anniversary. The influence of
Shafarevich on Russian algebraic geometry can hardly be overestimated.

1

Consistent systems of t-structures

1.1

The partially ordered set of t-structures

Let k be any field, D a k-linear triangulated category with an auto-equivalence T , called
translation (or shift) functor (see [15]).

2

For a subcategory C in D its right (resp. left) orthogonal C ⊥ (⊥ C) is the strictly fullsubcategory in D of objects X such that Hom(C, X) = 0 (resp. Hom(X, C) = 0).
We use the standard notation X[k] := T k X, k ∈ Z.
Recall (see [1]) that a t-structure in a triangulated category D is a pair of strictly fullsubcategories (D≤0 , D≥1 ), satisfying the following conditions:i) T D≤0 ⊂ D≤0 , T −1 D≥1 ⊂ D≥1 ;ii) Hom(D≤0 , D≥1 ) = 0;iii) for any X ∈ D there exist X− ∈ D≤0 , X+ ∈ D≥1 and an exact triangle
X− → X → X+ → X− [1](1)The basic example of a t-structure comes up when D is equivalent to the boundedderived category Db (A) of some abelian category A. For this case D≥1 (resp. D≤0 )comprises the complexes with cohomology in positive (resp. nonpositive) degree.
Categorically, the main virtue of the notion of t-structure is that in view of conditions(i) and (ii) the triangle (1) for a given X is unique up to unique isomorphism [1]. In otherwords, formulasτ≤0 (X) = X− , τ≥1 (X) = X+
define the truncation functors τ≤0 : D → D≤0 , τ≥1 : D → D≥1 , which are easily seen to berespectively the right and the left adjoint to the inclusion functors i≤0 : D≤0 → D, i≥1 :D≥1 → D. This suggest the following reformulation of the data defining a t-structure.
Lemma 1 To define a t-structure in a triangulated category D is equivalent to one of thefollowing data:
Data 1. A strictly full subcategory D≤0 in D, closed under the action of T and such thatthe inclusion functor i≤0 : D≤0 → D has a right adjoint τ≤0 ,Data 2. A strictly full subcategory D≥1 in D, closed under the action of T −1 and suchthat the inclusion functor i≥1 : D≥1 → D has a left adjoint τ≥1 .
Proof. As we have already constructed adjoint functors for a given t-structure, it remains
to explain how to construct a t-structure from say data 1.
Put D≥1 = (D≤0 )⊥ . As T D≤0 ⊂ D≤0 , then obviously T −1 D≥1 ⊂ D≥1 .
For any X ∈ D, the adjunction morphism i≤0 τ≤0 X → X can be inserted in a triangle:i≤0 τ≤0 X → X → X+ → i≤0 τ≤0 X[1].
X+ is easily seen to be in D≥1 . Hence the triangle is of the form (1).
This simple observation is useful in constructing t-structures.
Remark 2 Note that in general neither inclusion nor truncation functors are exact. But
one can easily see that if A → B → C is an exact triangle with A, C ∈ D≤0 (resp. D≥1 ),then B ∈ D≤0 (resp. D≥1 ). It follows by ‘rotation of triangle’ that if A and B are in D≤0(resp. B, C ∈ D≥1 ), then also C ∈ D≤0 (resp. A ∈ D≥1 ).
3

Any t-structure generates two sequences of subcategories D≤n := D≤0 [−n], D≥n :=D≥1 [1 − n]. The subcategory C = D≤0 ∩ D≥0 has a natural structure of abelian category(see [1]), it is called the heart of t-structure. The objects which lie in the heart (and also
their shifts) are called pure.
When D = Db (A), the heart of the standard t-structure comprises the complexes overA with cohomology in degree zero. Thus, the heart is equivalent to the primary abeliancategory A. We identify A with this full subcategory in Db (A).
If D is triangulated, then the opposite category Dop is also naturally triangulated.
Namely, TDop = TD−1 and triangles in Dop are defined by inverting arrows in triangles in
D. A t-structure T in D defines a t-structure T op in Dop by exchanging the roles of D≤0
and D≥1 . The heart of T op is the opposite abelian category to the heart of T .
An important special case is a t-structure satisfying T −1 D≤0 ∈ D≤0 . It follows fromthe remark 2 that for such a t-structure both D≤0 and D≥1 are triangulated subcategories.
We call such a t-structure translation invariant or, simply, T -invariant.
A strictly full triangulated subcategory A in D which has the right (resp. left) adjoint
to the inclusion i: A → D is called right (resp. left) admissible [5]. For a T -invariant
t-structure, D≤0 (resp. D≥1 ) is right (resp. left) admissible. This gives a 1-1 correspondence between T -invariant t-structures and right (or left) admissible subcategories.
For a T -invariant t-structure all the subcategories D≤k , k ∈ Z, are equal (the same for≥k
D , k ∈ Z). Hence the heart of a T -invariant t-structure is zero.
Given a triangulated category D, consider the following partial order on the set of its
t-structures (we ignore all possible set theoretical questions which arize from the fact thatthe objects of a category comprise a class not a set, as being irrelevant to our discussion).
For a pair (T1 , T2 ) of t-structures, we say T1 ⊆ T2 if D1≤0 ⊆ D2≤0 . Then also D2≥1 ⊆ D1≥1 .
The minimal element 0 in this ordered set is the t-structure with D≤0 = 0; the maximal
one, 1, has D≤0 = D.
Following the standard definition of union and intersection for partially ordered sets
we introduce:
T
Definition The (abstract) intersection
Ti of a set {Ti }i∈I of t-structures in D is thei∈Isupremum of t-structures
S T in D such that T ⊆ Ti , for all i ∈ I. If it exists, it is unique.
The (abstract) union
Ti is defined dually.
i∈I

We also denote the intersection and union operations by · and + and call them productand sum. Indeed, these are just product and coproduct in the category related to thepartially ordered set.
The intersection and the union of T -invariant t-structures, when exists, is T -invariant.
There is a natural ‘naive’ candidateTfor D∩≤0 (resp. D∪≥1 )Tfor the intersection (resp.
union) t-structure of a set {Ti }: D∩≤0 =Di≤0 , (resp. D∪≥1 =Di≥1 ).
i∈I

i∈I

There is a duality functor on the category of partially ordered sets that inverts thepartial order. It takes the union operation into the intersection and vice versa. We referoften to this duality in the present paper.
4

1.2

An example

The following example of geometric nature provides with a pair (T1 , T2 ) of t-structures
such that D1≤0 ∩ D2≤0 does not have right adjoint. This shows that the ‘naive’ intersection
does not necessarily coincides with the abstract one.
Here we assume k to be algebraically closed and of characteristic zero. Let D =bDcoh (P2 ) be the bounded derived category of coherent sheaves on the projective plane
P2 = P(V ), dim V = 3. We consider two translation invariant t-structures in D. Definecorresponding right admissible triangulated subcategories D1≤0 and D2≤0 by≤≤

D1 0 = hO(−2), O(−1)i,D2 0 = hO(1), O(2)i,i. e. the minimal strictly full triangulated subcategories generated by the indicated sheaves.
Lemma 3 The category D∩ = D1≤0 ∩ D2≤0 is not right admissible.
Proof. The category D2≤0 is equivalent as a triangulated category to Db (mod−A), thebounded derived category of representations of the path algebra of the quiver (see [5]):V· −→ ·The equivalence can be given by the functor that assigns to a complex of repreU·
W·
sentations · −→ · with the structure map ϕ: U · ⊗ V → W · , the complex of sheaves
U · ⊗ O(1) → W · ⊗ O(2) in D≤0 with the differential induced by ϕ.
2

Since A is hereditary, any indecomposable object in Db (mod−A) is pure, i. e. it isdefined by a map ϕ: U ⊗ V → W , with U , W vector spaces (up to a common shift). The
corresponding complex of sheaves in D2≤0 has the form
U ⊗ O(1) → W ⊗ O(2)(2)

with the components in degrees 0 and 1.
Let us describe indecomposables which are in D∩ . Note that an object X ∈ D is in D1≤0
iff Ext· (O, X) = 0. Applying this to (2) we get that the mapϕ:b U ⊗ V ∗ → W ⊗ S 2V ∗,obtained from ϕ by partial dualization ϕ:e U → W ⊗ V ∗ , tensoring with the identity map
in V ∗ and then by partial symmetrization, has to be an isomorphism. As dim V = 3, we
see that for X ∈ D∩ = D1≤0 ∩ D2≤0 we have dim U = 2 dim W .
If we take W to be 1-dimensional, then dim U = 2, hence the image of ϕe annihilates aline in V . It follows that the image of ϕb consists of quadrics that vanish on this line, i. e.
ϕb is not an isomorphism. Therefore, there is no indecomposable in D∩ with dim W = 1.
Yet, D∩ is not empty. Indeed, take any exceptional vector bundle on P2 , i. e. suchthat
Hom(E, E) = k,Exti (E, E) = 0, for i 6= 0.
(3)
Consider the sheaf F = End0 E of traceless local endomorphisms of E. It is non-zero,
if rk E > 1. Then (3) implies Ext· (O, F ) = H· (F ) = 0. Since D1≤0 = O⊥ in D, then5F ∈ D1≤0 . As F ∗ = F , then Ext· (F, O) = H· (F ∗ ) = 0. Since D2≤0 = ⊥ O, then F ∈ D2≤0 ,i. e. F ∈ D∩ .
Now take G = O(−1)[1]. If F ∈ D∩ is of the form (2), thenHom1 (F, G) = W ∗ ,Homi (F, G) = 0,

for i 6= 1.

(4)

Suppose that D∩ is right admissible and τ is the right adjoint to the embedding functor
D∩ ,→ D. Put G0 = τ G. Then G0 ∈ D∩ . If G0 is of the form U 0 ⊗ O(1) → W 0 ⊗ O(2),while F is of the form (2), then Hom· (F, G0 ) is the homology of the complex
U ∗ ⊗ U 0 ⊕ W ∗ ⊗ W 0 → U ∗ ⊗ W 0 ⊗ V ∗,(5)

with two non-trivial components in degrees 0 and 1. Since dim U = 2 dim W and dim U 0 =2 dim W 0 , the Euler characteristic of this complex is − dim W · dim W 0 .
From the adjunction property and from (4) we conclude:− dim W = χ(F, G) = χ(F, G0 ) = − dim W · dim W 0 .
Therefore, dim W 0 must be equal to 1, which is, as we have seen, impossible.
Generally, G0 , being an object in D∩ , has a decomposition into a sum of objectsof the form (2) and their translations. Any non-trivial translation would give a nontrivial contribution in Homi (F, G0 ), for i 6= 1, because these Hom-groups are calculatedby translations of the complex (5). But this contradicts the adjunction property for τ .
Therefore, there is no chance for G0 to exist. The lemma is proved.
One shows in the same way that the abstract intersection T1 ∩ T2 is 0.

1.3

Consistent pairs of t-structures and operations

Here we define lower and upper consistent pairs of t-structures. There are various possible
ways for generalizing these notion, which we hope to address in an appropriate place.
We mark truncation and inclusion functors of t-structures with relevant superscripts.
Definition A pair of t-structures (T1 , T2 ) is called lower consistent if
≤

≤

≥

≥

τ≤10 D2 0 ⊂ D2 0 ,and upper consistent ifτ≥21 D1 1 ⊂ D1 1 .
It is called consistent if it is simultaneously lower and upper consistent.
For consistent t-structures the intersection (union) exists and is ‘naive’.
Proposition 4 Let (T1 , T2 ) be a lower, respectively upper, consistent pair of t-structures.
Then the formulas
≤
≥
≤
≤
≤
D∩ 1 = (D∩ 0 )⊥ ,D∩ 0 = D1 0 ∩ D2 0 ,with the truncation functor τ≤∩0 := τ≤10 τ≤20 , respectively≥≥≥D∪ 1 = D1 1 ∩ D2 1 ,D∪ 0 = ⊥ (D∪ 1 )≤≥with the truncation functor τ≥∪1 := τ≥21 τ≥11 , define a new t-structure in the category D.
6Proof. For a lower consistent pair (T1 , T2 ) of t-structures the category D∩≤0 = D1≤0 ∩ D2≤0
forms the data 1 from lemma 1. Indeed, it is obviously preserved by the shift functor T .
Moreover, the functor τ≤∩0 := τ≤10 τ≤20 has its image in D∩≤0 due to lower consistency of thepair. One can easily see that it is right adjoint to the inclusion functor i∩≤0 : D∩≤0 → D.
Similarly for an upper consistent pair.
The example from the preceding subsection yields a non-consistent pair of t-structures.
Now we shall specify conditions on a sequence of t-structures under which one caniterate the operations of intersection and union.
First, intersection defines an associative operation on ordered sequences with pairwise
lower consistent elements, preserving lower consistency at intermediate steps.
Proposition 5 Let (T1 , T2 , T3 ) be a triple of t-structures with (Ti , Tj ) being lower consistent for i < j. Then the pairs (T1 ∩T2 , T3 ) and (T1 , T2 ∩T3 ) are lower consistent. Moreover,(T1 ∩ T2 ) ∩ T3 coincides with T1 ∩ (T2 ∩ T3 ).
Proof. By proposition 4 the truncation functor for the t-structure T1 ∩ T2 is the compositeof τ≤10 and τ≤20 . As both functors preserve D3≤0 , lower consistency of the pair (T1 ∩ T2 , T3 )follows.
≤0= D2≤0 ∩D3≤0 . Both categories D2≤0 and D3≤0 are preserved by τ≤10 . Hence,
Further, D2∩3≤≤
00
, i. e. the second pair (T1 , T2 ∩ T3 ) is lower consistent.
⊂ D2∩3τ≤10 D2∩3
Finally, for both t-structures (T1 ∩T2 )∩T3 and T1 ∩(T2 ∩T3 ) the categories D≤0 coincidewith D1≤0 ∩ D2≤0 ∩ D3≤0 . This gives coincidence of these t-structures.
By the dual argument, one proves
Proposition 6 Let (T1 , T2 , T3 ) be a triple of t-structures with (Ti , Tj ) being upper consistent for i < j. Then the pairs (T1 ∪T2 , T3 ) and (T1 , T2 ∪T3 ) are upper consistent. Moreover,(T1 ∪ T2 ) ∪ T3 and T1 ∪ (T2 ∪ T3 ) coincide.
It follows that for any finite sequence {Ti }i∈[1,n] of t-structures such that for all i < jtheupper)
one can T
construct a new t-structure
T pairs (Ti , Tj )Sare lower (resp.
T consistent,≤0≤0≥1
Ti (resp.
Ti ) with D∩ =Di (resp. D∪ =Di≥1 ) and the truncationi∈[1,n]i∈[1,n]i∈[1,n]i∈[1,n]functor τ≤∩0 = τ≤10 τ≤20 . . . τ≤n0 (resp. τ≤∪0 = τ≥n1 . . . τ≥11 ).
The next two propositions describe conditions under which one can alternate theintersection and union operations.
Proposition 7 Let (T1 , T2 , T3 ) be a triple of t-structures, such that the pairs (T1 , T2 )and (T1 , T3 ) are upper consistent and the pair (T2 , T3 ) is lower consistent. Then the pair
(T1 , T2 ∩ T3 ) is upper consistent.
Proof. We denote by τ≤∩0 and τ≥∩1 the truncation functors for the t-structure T2 ∩ T3 .
We know by proposition 4 that τ≤∩0 = τ≤20 τ≤30 . Then, for any X ∈ D, by axioms oftriangulated categories, the commutative triangleτ≤30 X
@



@
@
R
@
- X

τ≤20 τ≤30 X
7

can be embedded into the following ‘octahedron’ with all columns and rows being exact
triangles (for convenience we draw it as a square):τ≥21 τ≤30 X → τ≥∩1 X → τ≥31 X
↑
↑
↑o
3
3
X
→ τ≥1 X
τ≤0 X →↑↑↑∼ τ≤2 τ≤3 X → 0τ≤20 τ≤30 X →
0 0

(6)

We have to show that τ≥∩1 D1≥1 ⊂ D1≥1 . Let X ∈ D1≥1 , then, since the pair (T1 , T3 ) is
upper consistent, then τ≥31 X ∈ D1≥1 . From the middle row in (6) and remark 2 it follows
that τ≤30 X ∈ D1≥1 . By upper consistency of the pair (T1 , T2 ) we deduce that τ≥21 τ≤30 X ∈ D1≥1 .
Now from the upper row of (6) it follows that τ≥∩1 X ∈ D1≥1 . This proves the proposition.
Thus, under conditions of proposition 7 one can construct a new t-structure T1 ∪(T2 ∩ T3 ).
By duality, one proves
Proposition 8 Let (T1 , T2 , T3 ) be a triple of t-structures, such that the pairs (T1 , T3 )and (T2 , T3 ) are lower consistent and the pair (T1 , T2 ) is upper consistent. Then the pair
(T1 ∪ T2 , T3 ) is lower consistent.
Suppose we have a pair (T1 , T2 ) of t-structures with T1 ⊆ T2 . It is easily seen to belower and upper consistent. Moreover, (T2 , T1 ) is also a lower and upper consistent pair.
In particular, the pair (T1 , T1 [k]) is consistent for any k ∈ Z. Note that truncation functors
commute:τ≤10 τ≤20 = τ≤10 = τ≤20 τ≤10 , τ≥11 τ≥21 = τ≥21 = τ≥21 τ≥11
We conclude this subsection by proving a lemma, used in the sequel, on commutationof truncation functors for an ordered pair of t-structures.
Lemma 9 For an ordered pair (T1 , T2 ) of t-structures with T1 ⊆ T2 the truncation functors commute:τ≤20 τ≥11 = τ≥11 τ≤20
Proof. For any object X ∈ D we have an isomorphism: τ≥21 τ≥11 X = τ≥21 X.
Hence, by the octahedron axiom, we have the following diagram with exact rows andcolumns:τ≤20 τ≥11 X → τ≥11 X → τ≥21 τ≥11 X
↑
↑
↑o
2
2
τ≤0 X → X → τ≥1 X
↑
↑
↑
1
1
∼
τ≤0 X → τ≤0 X →0(7)

In the upper row, two terms τ≥11 X and τ≥21 τ≥11 X are in D1≥1 . Hence, the third termτ≤20 τ≥11 X is in D1≥1 .
8Then, in the left column, the lower term is in D1≤0 and the upper one is in D1≥1 .
Therefore, this column is the canonical decomposition of the middle term τ≤20 X along thefirst t-structure. Hence, we have a canonical isomorphism: τ≤20 τ≥11 X ' τ≥11 τ≤20 X.

1.4

The standard postulates for lattices

The lattice theory allows to interpret particularly nice posets, called lattices, as abstract
algebras with the union (sum) and intersection (product) operations that satisfy so-called
standard postulates [4]. This interpretation is useful for manipulations with lattices, suchas taking quotient.
Definition A lattice is a partially ordered set for which union and intersection of any
pair of elements exist.
Recall the standard postulates known for the operations on a partially ordered set. We
shall use now the ( · ,+)-notation with the usual priority of · over +. In the following
formulae, we suppose that both sides of equations are well-defined for elements T , T1 , T2 , T3
of a partially ordered set (when this is not a priori clear, as say for lattices):
T ·T =T,
T +T =T;
T1 · T 2 = T2 · T 1 ,T 1 + T2 = T2 + T 1 ;(T1 · T2 ) · T3 = T1 · (T2 · T3 ),(T1 + T2 ) + T3 = T1 + (T2 + T3 );T1 · (T1 + T2 ) = T1 + T1 · T2 = T1 .

(8)
(9)
(10)
(11)

Proposition 10 (see [4]) Lattices are identified with abstract algebras with two binary
operations · and + satisfying standard postulates (8)-(11). Given such an algebra, thepartial order on the set of its elements can be recovered from the rule:x≤y

x · y = x.

iff

(12)

If 0 and/or 1 exist, they can be added as nullary operations.
An equivalence relation ∼ in a lattice L is called congruence if p ∼ q implies p+l ∼ q+l
and p · l ∼ q · l for any element l ∈ L. Given a number of pairs (equivalences) pi ∼ qi ,i ∈ I, we define the congruence generated by them as the minimal congruence containing
all of them. It is the intersection of all congruences that contain {pi ∼ qi }i∈I . Intersectionof arbitrary number of congruencies (as subsets in L × L) is again a congruence.
For a congruence ∼ in a lattice L, regarded as an abstract algebra, the quotient algebra
0
L = L/ ∼ is well-defined. Being an abstract algebra of the same type (satisfying thesame standard postulates), it is again a lattice.
Note that, for x0 , y 0 ∈ L, x0 ≤ y 0 in L0 iff there exist x, y ∈ L such that x ∼ x0 , y ∼ y 0
and x ≤ y in L. Indeed, if x0 ≤ y 0 in L0 , then x0 ∼ x0 y 0 by (12). Once summed up with y 0 ,this gives: x0 + y 0 ∼ x0 y 0 + y 0 = y 0 . Hence, we can take required x and y to be: x = x0 y 0 ,y = x0 + y 0 .
Unfortunately, I do not know any ‘sufficiently large’ class of t-structures in a triangulated category which constitute a lattice. It would be interesting to explore this problem.
91.5

Modularity laws

For any partially ordered set, the distributivity inequalities are verified (again underassumption that both sides are well defined):(T1 + T2 ) · T3 ≥ T1 · T3 + T2 · T3 ,T1 + T2 · T3 ≤ (T1 + T2 ) · (T1 + T3 ).

(13)
(14)

The distributivity laws, i. e. when (13) and (14) are replaced by the equations withthe same left and right hand sides, are not valid for t-structures.
Example. Let T1 and T2 be the T -invariant t-structures as in the example in subsection 1.2. Define a T -invariant t-structure T3 by≤

D3 0 = hOi
Then T1 + T2 = 1, because D1≥1 ∩ D2≥1 = hO(−3)i ∩ hOi = 0. Also T1 · T3 = T2 · T3 = 0,
because D1≤0 ∩ D3≤0 = D2≤0 ∩ D3≤0 = 0. It follows that the distributivity law(T1 + T2 ) · T3 = T1 · T3 + T2 · T3
is violated, as the left hand side is T3 , while the right hand side is 0.
If T1 ≤ T3 , then (13) and (14) yield the following autodual inequality:(T1 + T2 ) · T3 ≥ T1 + T2 · T3 .

(15)

When replaced by the equation, it is called modularity law.
We prove under appropriate assumptions on consistency several statements, whichmight be viewed as modularity laws for t-structures. The first theorem is autodual.
Theorem 11 (modularity law 1) Suppose we are given a triple (T1 , T2 , T3 ) of t-structures such that T1 ≤ T3 , the pair (T1 , T2 ) is upper consistent and the pair (T2 , T3 ) is lower
consistent. Theni) the pair (T1 + T2 , T3 ) is lower consistent,ii) the pair (T1 , T2 · T3 ) is upper consistent,iii) (T1 + T2 ) · T3 = T1 + T2 · T3 .
Proof. Since T1 ≤ T3 , the pair (T1 , T3 ) is lower consistent. Hence, lower consistency ofthe pair (T1 + T2 , T3 ) follows from proposition 8. Similarly, the pair (T1 , T3 ) is upperconsistent, hence (T1 , T2 · T3 ) is upper consistent by proposition 7.
Thus, both sides of iii) are well-defined.
We have inequality (15). The opposite to (15) inequality is, by definition, equivalent
to the inclusion of subcategories:
≤

≤

0
0
D(1+2)3⊆ D1+2·3
.

10

(16)

This, in turn, is equivalent to the following statement:≤0≥1
Let X ∈ D(1+2)3
and Y ∈ D1+2·3, then Hom(X, Y ) = 0.

(∗)

Let us prove it. To this end, decompose X into an exact triangle:τ≤10 X → X → τ≥11 X.

(17)

≤0
As T1 ⊆ (T1 +T2 )T3 , the pair (T1 , (T1 +T2 )T3 ) is lower consistent. Hence τ≤10 X ∈ D(1+2)3.
≤0
1Then, by remark 2, τ≥1 X ∈ D(1+2)3 .
≥1
Now since T1 ⊆ T1 +T2 T3 , then Hom(D1≤0 , D1+2·3) = 0. It follows that Hom(τ≤10 X, Y ) = 0.
Applying the functor Hom( · , Y ) to the triangle (17), we see that it is sufficient to showthat Hom(τ≥11 X, Y ) = 0.
≥1
≤0.
∩ D1≥1 , Y ∈ D1+2·3
Therefore, we have reduced (∗) to the case when X ∈ D(1+2)3
Now let us decompose Y with respect to the third t-structure:τ≤30 Y → Y → τ≥31 Y.

(18)

≥1
. Then,Since T1+2·3 ⊆ T3 , the pair (T1+2·3 , T3 ) is upper consistent. Hence τ≥31 Y ∈ D1+2·3≤0≥1, D3≥1 ) = 0. It. Since T(1+2)3 ⊆ T3 , then Hom(D(1+2)3
again by remark 2, τ≤30 Y ∈ D1+2·3
follows that Hom(X, τ≥31 Y ) = 0. Applying Hom(X, ·) to (18), we see that it is sufficient
to show that Hom(X, τ≤30 Y ) = 0.
Therefore, we have reduced (∗) to the statement:

(∗∗)

≥1
≤0
, then Hom(X, Y ) = 0.
∩ D1≥1 and Y ∈ D3≤0 ∩ D1+2·3
if X ∈ D(1+2)3

Let us show that≤0
∩ D1≥1 ⊂ D2≤0 ,a) D(1+2)3≥1⊂ D2≥1 .
b) D3≤0 ∩ D1+2·3

2 1
Recall that, by proposition 4, since the pair (T1 , T2 ) is upper consistent, τ≥1+2
1 = τ≥1 τ≥1 .
≤0
≥1
1+2
2
2
Therefore, if X ∈ D1 , then τ≥1 X = τ≥1 X. If, in addition, X ∈ D(1+2)3 , then τ≥1 X =τ≥1+2
1 X = 0, in view of T(1+2)3 ⊆ T1+2 . This proves a).
2 3
Dually, by proposition 4, since the pair (T2 , T3 ) is lower consistent, τ≤2·3
0 = τ≤0 τ≤0 .
≤0
≥
2
1
2
2·3
Hence for Y ∈ D3 , τ≤2·3
0 Y = τ≤0 Y . If, in addition, Y ∈ D1+2·3 , then τ≤0 Y = τ≤0 Y = 0, inview of T2·3 ⊆ T1+2·3 . This proves b).
Since a) and b) obviously imply (∗∗), this finishes the proof of the theorem.

We shall also need another incarnation of the modularity law.
Theorem 12 (modularity law 2) Suppose we are given a triple (T1 , T2 , T3 ) of t-structures such that T1 ≤ T3 , the pair (T1 , T2 ) is upper consistent and the pair (T3 , T2 ) is lower
consistent. Theni) the pair (T1 , T2 · T3 ) is upper consistent,ii) the pair (T3 , T1 + T2 ) is lower consistent,
11iii) (T1 + T2 ) · T3 = T1 + T2 · T3 .
Note that in the contrary to the theorem 11, lower consistency of the pair (T3 , T1 + T2 )is not a formal consequence of the propositions from the previous subsection.
Proof. As T1 ≤ T3 , the pair (T1 , T3 ) is upper consistent. Hence, we can apply proposition
7 to the triple (T1 , T3 , T2 ) to get upper consistency of the pair (T1 , T2 · T3 ). This proves i)and the fact that T1+2·3 exists.
3 2
We know by proposition 4 that τ≤2·3
0 = τ≤0 τ≤0 .
Then, for any X ∈ D, by the octahedron axiom, we have the diagram with exact rows
and columns:
2τ≥31 τ≤20 X → τ≥2·3
1 X → τ≥1 X
↑
↑
↑o
2
2
τ≤0 X →X→ τ≥1 X(19)↑↑↑
∼ τ≤3 τ≤2 X → 0τ≤30 τ≤20 X →
0 0
1+2·3
1+2
1
1
2 1
By proposition 4, τ≥1 = τ≥1 τ≥1 and τ≥1 = τ≥2·3
1 τ≥1 . Hence, substitution X = τ≥1 Y in
the upper row of the octahedron yields:

Y → τ≥1+2
τ≥31 τ≤20 τ≥11 Y → τ≥2·3+1
1 Y.
1Now, substitution Y = τ≤1+2
0 Z gives a natural isomorphism:∼ 1+2·3 τ≤1+2 Z.
τ≥31 τ≤20 τ≥11 τ≤1+2
0 Z → τ≥1
0
It follows that, for any Z ∈ D,τ≤30 τ≥1+2·3
τ≤1+2
1
0 Z = 0.
Observe an obvious inequality: T1+2·3 ≤ T3 . Hence, by lemma 9, τ≤30 and τ≥1+2·3
1commute. This implies:τ≥1+2·3τ≤30 τ≤1+2
1
0 = 0.
≤0
Equivalently, the image of τ≤30 τ≤1+2
0 is in D1+2·3 .
≤0
≤0
is in
⊂ D1+2∩ D3≤0 . Therefore, the image of τ≤30 τ≤1+2
On the other hand, D1+2·3
0
≤0
≤0
≤0
≤0
3
D1+2 ∩ D3 . Then τ≤0 D1+2 ⊂ D1+2 , hence lower consistency of (T3 , T1 + T2 ).
By proposition 4, functor τ≤30 τ≤1+2
0 is the truncation functor for T(1+2)·3 . Its image is≤0≤0≤0
D(1+2)·3= D1+2∩ D3≤0 . It was shown that the image is in D1+2·3. Together with the
opposite inclusion (15) this yields iii).

The dual statement to theorem 12 isTheorem 13 (modularity law 20 ) Suppose we are given a triple (T1 , T2 , T3 ) of t-structures such that T1 ≤ T3 , pair (T2 , T1 ) is upper consistent and pair (T2 , T3 ) is lower consistent. Theni) pair (T2 · T3 , T1 ) is upper consistent,ii) pair (T1 + T2 , T3 ) is lower consistent,iii) (T1 + T2 ) · T3 = T1 + T2 · T3 .
12

2

Sets with consistencies and pairs of chains

For the sake of clear and logical exposition, we formalize the properties of consistency
proved in the previous subsections and present a convenient pictorial description. Then
we prove that a consistent pair of chains generates a distributive lattice.

2.1

Sets with consistencies

By a set with consistencies we call a partially ordered set containing 0 and 1 with twobinary (non-reflexive) relations, called lower and upper consistencies, such that for any
lower, respectively upper, consistent pair the abstract intersection, respectively union,
exists and satisfies the following axioms ( prime means the dual axiom):SC1. if a ≤ b, then both (a, b) and (b, a) are lower and upper consistent,
SC2. implication of lower consistency under intersection as in proposition 5,
SC20 . implication of upper consistency under union as in proposition 6,
SC3. implication of upper consistency under intersection as in proposition 7,
SC30 . implication of lower consistency under union as in proposition 8,
SC4. modularity law as in theorem 11,
SC5. modularity law as in theorem 12,
SC50 . modularity law as in theorem 13.
In this formalism, the words ‘of t-structures’ must be removed from all the quoted
statements. In the sequel, we shall sometimes directly refer to the corresponding statements, meaning the above axioms for general sets with consistencies.
Note that the standard postulates (8),(9),(10),(11) are automatically satisfied for setswith consistencies. It follows from SC1 that 0 and 1 constitute a lower and upper consistent pair with any element.
Contrary to a lattice, a set with consistencies is not an abstract algebra.

2.2

Diagrams

It is instructive to use the following pictorial description for consistencies and operations
with them.
Suppose we are given a set with consistencies V . Consider a graph Γ(V ) with vertices
labelled by elements of V .
We draw a full arrow:a −→ bif (a, b) is a lower consistent pair in V , and a dashed arrow
a 99K b13

if (a, b) is an upper consistent pair.
Now the operations of union and intersection are interpreted as contractions of fulland dashed arrows. For the intersection, this is:
a −→ b=⇒a·b

a 99K b=⇒a+b

For the union, this is:
The duality on t-structures described in subsection 1.1 exchanges full arrows withdashed ones and reverses their directions.
We shall actually consider subgraphs of Γ(V ) which are significant to ensure iterated
contractions.
Axioms SC2, SC20 , SC3, SC30 of the definition of a set with consistencies provides
us with patterns which allow to connect by arrows a new vertex obtained as result ofcontractions of an arrow with other (old) vertices of the graph. Axiom SC2 (proposition
5) reads as possibility of contractions of two edges in a triangle with full arrows:ba −→ b · c

⇐=



@
@

=⇒

a · b −→ c

@
R
@
- c

a

Axiom SC20 reads as the same picture with dashed arrows.
Axiom SC3 (proposition 7) reads:
b


@
@
@
R
@
- c

a

=⇒

a 99K b · c=⇒

a + b −→ cAxiom SC30 reads:
b


a

@
@
@
R
@
- c

Note that we don’t have in general the implications which are obtained from the lasttwo pictures by changing full arrows for dashed ones and vice versa (while preserving theirdirections).
Sometimes it is useful to put the vertex marked with the intersection (resp. union)
t-structure at the middle point of the corresponding full (resp. dashed) arrow, if it exists.

14

2.3

The universal lattice with consistencies

A morphism of sets with consistencies is a morphism of the corresponding posets whichtakes any upper (lower) consistent pair to an upper (lower) consistent pair. A set withconsistencies which is a lattice is called a lattice with consistencies. Morphisms of lattices
with consistencies are defined in the obvious way.
A morphism φ : P → L, where P is a poset and L a lattice, is an order preserving
map such that φ(x + y) = φ(x) + φ(y), φ(xy) = φ(x)φ(y), whenever x + y or xy exists.
Consider a poset P on which two arbitrary binary relations are fixed. We are interested
in morphisms P → L of P into a lattice L with consistencies, such that the images ofany pair of elements in P which is in the first, respectively, the second, relation, is lower,
respectively upper, consistent in L. Such a morphism P → U is called universal if for any
other morphism P → L of this sort, there exists a unique morphism of lattices U → L
with consistencies which makes commutative the following diagram (of morphisms):
U


@
@
@
R
@
- L

P

In this case, we call U = U (P ) the universal lattice with consistencies generated by P .
When consistency relations are omitted in the above definitions, the notion of theuniversal lattice L(P ) generated by a poset P takes place. The universal lattice L(P )classically known to exist [4]. Indeed, remember that the notion of a lattice is equivalent
to that of abstract algebra satisfying the standard postulates. Thus, we can consider afree abstract algebra F (P ), generated by P as a disjoint set, and take its quotient by the
congruence generated by relations x + y = ”x + y” and x · y = ”x · y” for all x and y
for which the union ”x + y” (resp. intersection ”x · y”) in poset P exists (cf. 1.4). Thisquotient is L(P ).
A lattice is said to be distributive if for any x, y, z:x(y + z) = xy + xz,
x + yz = (x + y)(x + z).
These postulates are mutually dual and each one implies the other.
We can similarly define the universal distributive lattice D(P ) generated by poset P .
D(P ) is the quotient of L(P ) by the congruence generated by equivalences (x + y)z ∼xz + yz for all x, y, z ∈ L(P ).
Proposition 14 Let P be a partially ordered set with two binary relations. Then a universal lattice with consistencies generated by P exists.
Proof. Consider the universal lattice L(P ) generated by the poset P . Endow L(P ) withtwo binary relations, called respectively upper and lower consistency: first, demand theimages in L(P ) of all pairs in P which belong to the first (resp. second) given binary
relation to be upper (resp. lower) consistent; second, extend the two binary relations of15

consistency by the minimal set of pairs that satisfy axioms SC1,. . . , SC30 and those partsof axioms SC5 and SC50 which imply consistencies (recall that not all of them formallyfollow from the other axioms).
Note that this procedure does not make L(P ) a set with consistencies, because themodularity equations of axioms SC4, SC5, SC50 are not satisfied yet.
To meet this constraints, consider the quotient lattice L1 (P ) of L(P ) by the congruence
generated by equivalences (t1 + t2 )t3 ∼ t1 + t2 t3 , for all the triples (t1 , t2 , t3 ) ∈ L(P ) whichmeet conditions of one of the modularity laws SC4, SC5, SC50 . Endow L1 (P ) withconsistency relations inherited from L(P ). The axiom SC1 obviously follows from thefact (proved in 1.4) that x0 ≤ y 0 in L1 (P ) iff there exists x, y ∈ L(P ) such that x ∼ x0 ,y ∼ y 0 and x ≤ y in L(P ).
But it is not so good in what concerns the other axioms. For this reason, subject
L1 (P ) (with its consistency relations) to the same procedure as we did with L(P ), i.e.
extend appropriately the binary relations and take the similar quotient, to get L2 (P ) fromL1 (P ). By iterating this process, we get a sequence of lattices Li (P ) together with maps
P → Li (P ), compatible with the quotient maps. Then U (P ) = colim Li (P ) with two
colimit binary relations is the universal lattice with consistencies.
Indeed, consider any morphism P → L, where L is a lattice with consistencies andthe morphism takes the pairs in L from the first, respectively, the second binary relations
on P into lower, respectively, upper consistent pairs in L. The universality of P → L(P )implies existence of the lattice homomorphism L(P ) → L, which makes the corresponding
diagram commute. This homomorphism clearly descends to a homomorphism Li (P ) → L,
which, when passing to the limit, gives a homomorphism U (P ) → L. It is easy to seethat this homomorphism is unique.
We can consider a distributive lattice as a lattice with consistencies for which all pairsof elements are both upper and lower consistent. Since distributivity implies modularity,
we indeed get a correctly defined lattice with consistencies. A canonical homomorphism
ψ : U (P ) → D(P )follows from the universality of U (P ). It is clearly an epimorphism, because congruencies
used to construct U (P ) as a quotient of the lattice L(P ) in proposition 14 are among
those which were used to construct D(P ) as the quotient of L(P ).

2.4

Consistent pairs of chains

A chain in a partially ordered set is an ordered sequence of elements. A pair of chainsa1 ≥ . . . ≥ an and b1 ≤ . . . ≤ bm in a set with consistencies such that (ai , bj ) is upper andlower consistent for any pair (i, j), is called consistent pair of chains. Our purpose in thissubsection will be to prove that a consistent pair of chains generates (under operations ofunion and intersection) a distributive lattice.
There is a theorem due to Birkhoff [4] that two chains in a modular lattice generate a distributive lattice. When working with sets with consistencies we have two new
difficulties against the hypothesis of Birkhoff theorem: first, intersection and union do
not a priori exist, second, the modularity laws are verified only under some consistency
conditions.
16

The notion of sum and intersection of any number of elements in a partially ordered
set is intrinsically defined. Say, the sum of a given set S of elements is the element whichis greater than any element from S but less than any other element with this property.
It may not exist. Recall the following trivial observation, which slightly strengthensassociativity postulates (10).
Lemma 15 Let (a, b, c) be a triple of elements in a partially ordered set. If a + b exists,then existence of one side of the following equation implies existence of the other and theequation itself:(a + b) + c = a + b + c.

(20)

In the dual statement, + is replaced by ·.
Proof. Obvious.
Remark. Note that in the constructions of this subsection, this lemma is the only
tool applied to deduce existence of union, respectively intersection, for pairs of elements((a + b, c) here) which do not a priori constitute an upper, respectively lower, consistentpair. The typical situation is the following. Suppose we have a triple of elements (sayt-structures) (a1 , a2 , a3 ) with ai → aj , for i < j. Then, by proposition 8 and the lemma,(a1 + a2 ) + a3 = a1 + a2 + a3 exists. Hence, again by the lemma, (a1 + a3 ) + a2 exists,though neither (a1 + a3 , a2 ) nor (a2 , a1 + a3 ) is a priori lower consistent.
For the rest of the subsection, we assume to be given a consistent pair of chainsa1 ≥ . . . ≥ an and b1 ≤ . . . ≤ bm . Denote:uij = ai bj ,vij = ai + bj .

Lemma 16 Let j ≤ l. Then, the pair (ukl , uij ) is upper consistent. Dually, the pair(vij , vkl ) is lower consistent for i ≤ k.
Proof. First, we may suppose that i ≤ k, because, if i > k, then uij ≤ ukl and upperconsistency for (ukl , uij ) follows.
Applying SC3 to (ak , ai , bj ), we obtain upper consistency for the pair (ak , ai bj ).
Hence, SC50 (the modularity law 20 ) is applicable to the triple (ai bj , ak , bl ). It follows,
the pair (ukl , uij ) = (ak bl , ai bj ) is upper consistent. The rest follows from the duality.
Recall that a decomposition x = x1 + . . . + xn is called irreducible, if x 6= x1 + . . . +xi−1 + xi+1 + . . . + xn , for all i.
Proposition 17 Any finite sum of elements uij exists and has a decomposition of theform:u = ui1 j1 + . . . + uik jk , with i1 < . . . < ik ; j1 < . . . < jk .
(21)
Dually, any finite product of vij exists and has the formv = vi1 j1 · . . . · vik jk , with i1 < . . . < ik ; j1 < . . . < jk .
17

(22)

Proof. First, any sum of the form (21) exists. Indeed, summation can be done step bystep from the left to the right in view of lemma 16 and SC20 .
The rest of the proof is done by induction on number of summands uij .
Let us add to a sum u of the form (21) some element uij . In fact, it is sufficient toprove only existence of
u0 = u + uij .
(23)
Indeed, if it exists, then we can reduce, if necessary, the decomposition of u0 to get an
irreducible one. Any irreducible decomposition is automatically of the form (21), because
otherwise there would exist summands upq and urs with upq ≤ urs .
If i ≤ it and j ≥ jt for some t, then uij ≥ uit jt . Similarly, if i ≥ it and j ≤ jt forsome t, then uij ≤ uit jt . In both cases, one of two elements in the sum for u0 , either uij
or uit jt can be absorbed by the rule (11). Then, in view of lemma 15 and the induction
hypothesis the sum u0 exists.
The alternative to the above inequalities is the possibility to order all summands inthe decomposition (23) of u0 to be of the form (21). It has already been shown that any
such sum exists.
A chain is called extended if it contains 0 and 1 as extreme elements.
Proposition 18 Let a1 ≥ . . . ≥ an and b1 ≤ . . . ≤ bm be a consistent pair of extended
chains. Choose two ordered sets of indices I = {i1 ≤ . . . ≤ ik }, il ∈ [1, n], and J = {j1 ≤. . . ≤ jk }, jl ∈ [1, m]. Denote:rIJ = ai1 (bj1 + ai2 ) . . . (bjk−1 + aik )bjk ,sIJ = ai1 bj1 + ai2 bj2 + . . . + aik bjk .
Then:i) rIJ = sIJ ,ii) ai 99K sIJ −→ bj , for all (i, j) ∈ [1, n] × [1, m].
Remark. If we extend the chains of {ai }i∈I and {bj }j∈J by 0, then the formula for rIJ
is simply the product of suitable vij ’s. We allow the indices for {ai }’s and {bj }’s coincide
for the reason of simplifying the write-down of the proof by induction.
Proof. Both rIJ and sIJ exist due to proposition 17.
We shall prove both statements of the proposition by simultaneous induction on k =|I| = |J|. For k = 1, rIJ = ai1 bj1 = sIJ and ii) follows from SC2 and SC30 .
Now, take the case |I| = |J| = k − 1 for granted.
Denote I 0 = I \ ik , J 0 = J \ jk . Evidently, sI 0 J 0 ≤ bjk . By the induction hypothesis
aij 99K sI 0 J 0 . Then, we can apply SC50 (theorem 13) to the triple (sI 0 J 0 , aik , bjk ) to get(rI 0 J 0 + aik )bjk = (sI 0 J 0 + aik )bjk = sI 0 J 0 + aik bik = sIJ .
Let us show that the left hand side is rIJ .
Decompose rI 0 J 0 = qbjk−1 , whereq = ai1 (bj1 + ai2 ) . . . (bjk−2 + aik−1 ).
18

(24)

Since the second chain is extended, the last element bm in it is 1. Then, q = sI 0 J 00 , where
J 00 = {j1 , . . . jk−2 , m}. Hence, applying the induction hypothesis to the subsets I 0 and J 00 ,we have: q −→ bjk−1 . Since, in addition, aik ≤ q and aik 99K bjk−1 , we can apply SC5 to
the triple (aik , bjk−1 , q):rI 0 J 0 + aik = qbjk−1 + aik = q(bjk−1 + aik ).
It follows that the left hand side of (24) is rIJ . Thus, we have proven i).
In view of aik 99K sI 0 J 0 and the induction hypothesis, we can apply axioms SC20 and
SC30 to get:ai 99K (sI 0 J 0 + aik ) −→ bj
for all (i, j). Then, in view of sI 0 J 0 + aik −→ bjk we can apply axioms SC2 and SC3 to
get:ai 99K (sI 0 J 0 + aik )bjk −→ bj .
By (24) the middle entry coincides with sIJ . This proves ii) and completes the proof ofthe proposition.
Theorem 19 A consistent pair of chains in a set with consistencies generates in this set
a distributive lattice with all elements being of the form (21).
Proof. By adding, if necessary, 0 and 1 as extreme elements of the chains, we may
assume chains to be extended. Then all ai and bj are among uij . By proposition 17, any
sum of uij exists and is of the form (21).
Let us show that any product of elements of the form (21) exists and is of the sameform. By proposition 18, any sum of the form (21) can be recast as a product of vij (inthe formula for rIJ elements ai1 and bjk can be interpreted as vi1 m and v1jk respectively).
Products of these elements exist again by proposition 17 and are of the form (22), which
in turn, by proposition 18, can be recast as a sum of the form (21). This proves that theset of elements of the form (21) is a lattice, L. It is the lattice generated by the pair ofchains.
Consider the poset P = σ ∪ τ , which is a disjoint union of two chains σ = {a1 ≥ . . . ≥an } and τ = {b1 ≤ . . . ≤ bm }, as an abstract ordered set. Let D(P ) be the universal
distributive lattice generated by P .
Endow P with two coinciding binary relations, which consist of all pairs (ai , bj ). Let
U (P ) be the universal lattice with consistencies generated by P , which exists by proposition 14, and ψ : U (P ) → D(P ) the canonical epimorphism. Let us show that it is an
isomorphism.
Since U (P ) is the quotient of the universal lattice L(P ), it has only elements of theform (21). Hence, to show that ψ is an isomorphism it is sufficient to present a distributive
lattice generated by two chains such that all elements of the form (21) are taken by ψ
into distinct elements.
Such a lattice can be realized as a sublattice of subsets in the finite set of integer
nodes in the rectangle {(x, y) ∈ [1, m] × [−1, −n]}. Elements ai are presented by thesubsets {y ≤ −i} and bj by the subsets {x ≤ j}. Any element of the lattice is the setof integer points under a ladder descending from the point (1, −1) to the point (m, −n).
19Being a sublattice of the lattice of subsets, it is distributive and images under ψ of distinct
elements of the form (21) are clearly distinct. Hence ψ is injective. Therefore, U (P ) is
distributive and, by universality property, L is distributive too.
Remark. An element x in a lattice is called indecomposable if x = a + b implies
x = a or x = b, and decomposable otherwise. By another theorem of Birkhoff [4], anyelement in a finite distributive lattice can be uniquely decomposed into an irreducible
sum of indecomposable elements. Clearly, all indecomposable elements are among uij . It
is easy to see that an element uij is decomposable in L iff uij = ui+1,j + ui,j−1 .

3

Perverse coherent sheaves

3.1

Grothendieck duality and the dual t-structure

Let k be a field of characteristic zero. In this section we assume, for simplicity, all schemes
to be of finite type over k. For such a scheme X we consider the bounded derived categoryb
D(X) := Dcoh(X) of coherent sheaves. We shall construct some perverse t-structures in
D(X) using the machinery developed in the preceding section.
Let T be the standard t-structure in D(X) with D≤0 (resp. D≥0 ) consisting of complexes of coherent sheaves on X with trivial positive (resp. negative) degree cohomology.
By Hi (F) we denote the i-th cohomology sheaf of a complex F ∈ D(X). OX is
the structure sheaf of rings on X. We write f∗ , Hom, etc. for derived push-forward,
derived local Hom functor and other functors between derived categories. For a functor
Φ : D(X) → D(Y ) we use notation Rk Φ := Hk · Φ for the corresponding cohomological
functors D(X) → Coh(Y ). Here Φ is usually the derived functor of a left exact functor
Coh(X) → Coh(Y ). If Φ is the derived one of a right exact functor Coh(X) → Coh(Y ),then, in compliance with the tradition, we write Lk Φ := H−k · Φ for the cohomology
functors.
Note that for a morphism f : X → Y there is the twisted inverse image functor++
f ! : Dqcoh(Y ) → Dqcoh(X) between the bounded below derived categories of quasi-coherent
sheaves (see [14] and [12]). It is defined by:f ! (−) = (−) ⊗ f ! (OY ),where f ! (OY ) is defined by applying the right adjoint functor for f∗ to the structure sheaf
OY . When f is proper and of finite Tor-dimension, then f ! coincides with the right adjoint
to f∗ and f ! takes D(Y ) to D(X). In the particular case when f is a closed embedding,this property gives an obvious construction for f ! at least locally over the base. Whenf is smooth of relative dimension n, f ! is defined by twisting with the relative canonicalclass ωX/Y :f ! (−) = (−) ⊗ ωX/Y [n].
If f is projective, it can be decomposed into f = pi, where i is a closed embedding and p
is smooth. Then f ! = p! i! . Note also that f is automatically of finite Tor-dimension if Y
is a smooth variety.
20

Also, for a proper f , there is a duality isomorphism for local Hom’s:f∗ HomX (F, f ! G) = HomY (f∗ F, G),

(25)

+
+
natural in F ∈ Dqcoh(X), G ∈ Dqcoh(Y ).
·· = π ! O , where π :Denote by ωX the dualizing complex on X. By definition ωXX
X pt·
X → pt is the projection to a point pt. For a smooth variety X of dimension n, ωX =
ωX [n], where ωX = ΩnX is the canonical sheaf of differential forms of the highest degree.
· is known to be in D(X).
For general X, ωX
For a closed embedding i : X → Y of X into a smooth Y we have:· = i! ω · = i! ω [dimY ].
ωX
Y
Y

(26)

The category D(X) possesses a contravariant involutive exact functor D = DX . For
an F ∈ D(X), it is defined by:· ).
DF = DX F = HomX (F, ωX(27)

D is compatible with triangulated structures, i.e. DT is naturally isomorphic to T −1 D,
and D takes an exact triangle A → B → C to the exact triangle DC → DB → DA.
If j : U → X is an open embedding, then the dualizing complex over U is obtainedby restricting of the one on X:·.
ωU· = j ∗ ωX
(28)
It follows that D is local, meaning that, given an open embedding as above, we have:j ∗ DX = DU j ∗ .

(29)

We use notation D(i) F := Hi (DF) for the cohomology sheaves of DF. D preserves
D(X) and D · D is naturally isomorphic to the identity functor ([10]).
Using D, we can define another t-structure T̃ = (D̃≤0 , D̃≥1 ) on D(X) by requiring:F ∈ D̃ 0 ⇐⇒ DF ∈ D≥0 ,(30)

F ∈ D̃≥0 ⇐⇒ DF ∈ D 0 .

(31)

≤

≤

As D is an anti-equivalence, T̃ is a t-structure. We call it the dual t-structure. Note thatthe dualizing complex is a pure object for the dual t-structure.

3.2

A consistent pair of chains and perverse sheaves

We denote by τ≤k , τ≥k (resp. τ̃≤k , τ̃≥k ) the truncation functors for T (resp. T̃ ).
The definition of T̃ implies:τ̃≤r = Dτ≥−r D, τ̃≥r = Dτ≤−r D.

(32)

Proposition 20 Let X be a scheme of finite type over k. Suppose that G ∈ D≥0 (X),then, for any r ∈ Z, D(τ≤r DG) ∈ D≥0 (X).
21

Proof. In view of (29), the statement of the proposition is local. Therefore, we may
assume that X is affine and (being of finite type) embeddable into a smooth variety overk. Fix such a closed embedding i : X → Y of X into a smooth variety Y of dimension l.
In view of (25) and (26), for any G ∈ D(X), we have:· ) = i Hom (G, i! ω · ) = Hom (i G, ω [l]).
i∗ DG = i∗ HomX (G, ωX
Y ∗
Y
∗
X
Y
Since i∗ is an exact functor with respect to the standard t-structure, theni∗ D(k) G = Rk i∗ DG = Rk HomY (i∗ G, ωY [l]).

(33)

Let G ∈ D≥0 (X). Then Rk i∗ G are zero for k < 0.
Consider the spectral sequence with E2mj = Extm+l(R−j i∗ G, ωY ) that converges toY
Rm+j HomY (i∗ G, ωY [l]).
Since ωY is locally free, then, for any coherent sheaf F on Y , ExtkY (F, ωY ) has supportof codimension ≥ k in Y (cf. [13], p.142). By convention, the support of the zero sheaf,
which is the empty set, is of infinite codimension.
As E2mj = 0, for j > 0, it follows from the spectral sequence that the support ofRs HomY (i∗ G, ωY [l]) is of codimension at least s + l. Now, formula (33) implies that thesame restriction on codimension of support is verified for i∗ D(s) G.
As the cohomology sheaves of the truncation τ≤r i∗ DG either coincide with those ofi∗ DG or equal zero, the codimension of the support of Hs (τ≤r i∗ DG) in Y is at least s + l
for any r ∈ Z.
Now consider the spectral sequence with E2js = Dj H−s (τ≤r DG), which converges tothe cohomology sheaves Hj+s (D(τ≤r DG)).
By (33) and in view of exactness of i∗ we obtain:−si∗ Dj H−s (τ≤r DG) = Rj HomY (i∗ H−s (τ≤r DG), ωY [l]) = Extj+l
Y (H (τ≤r i∗ DG), ωY ).

Using the fact that ExtkY (F, E) = 0 for any locally free sheaf E and for any coherent sheaf
F with codimension of the support greater than k, we find that the right hand side istrivial for j < −s.
Then the spectral sequence impliesi∗ Hj (D(τ≤r DG)) = 0.
The statement of the proposition follows.
Now consider the set of t-structures in D(X) as a set with consistencies where consistency binary relations are defined as in section 1.3. The sequences T [r] and T̃ [r] canbe viewed as chains in this partially ordered set. An ordered pair of t-structures is said
to be T -consistent if the pair of chains obtained from these two t-structures by applying
translation functor is consistent.
Theorem 21 The pair (T , T̃ ) is a T -consistent pair of t-structures.
Proof. We have to show that, for any r ∈ Z, the pair (T [r], T̃ ) is lower and upperconsistent.
22

If, for F ∈ D̃≤0 , we denote G = DF , then G ∈ D≥0 . By the proposition Dτ≤r DG =Dτ≤r F belongs to D≥0 . Hence τ≤r D̃≤0 ⊂ D̃≤0 , which is the condition of lower consistency.
Further, in view of (32), we have:τ̃≥r D≥0 = Dτ≤−r DD≥0 ⊂ D≥0 ,which yields the upper consistency.
Due to theorem 21, pair of chains T [r] and T̃ [r] is consistent. Hence, by theorem 19
we obtain a distributive lattice of t-structures. The t-structures that occur in this lattice
are in fact coherent versions of perverse t-structures (cf. [3]).
Now recall that for t-structures there is the naive intersection (resp. union) definedby intersecting the subcategories D≤0 (resp. D≥1 ).
Proposition 22 Any (multiple) intersection and union of t-structures in the obtained
lattice is naive.
Proof. Intersection or union in consistent pairs is manifestly naive. As follows from theremark after lemma 15, the situation described in that lemma is the only place where weshould check naivety of the union and intersection. So the fact comes with the following
reformulation of lemma 15 for t-structures: if a + b exists and is naive, then the existence
and naivety for one side of the equation (20) implies existence and naivety for the otherside and the equation itself. This statement is again obvious.

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