Atiyah classes with values in the truncated cotangent complex
aa r X i v : . [ m a t h . AG ] N ov Atiyah classes with valuesin the truncated cotangent complex
Diploma thesis at the University of BonnFabian Langholf [email protected]
Abstract
We prove an explicit formula for the truncated Atiyah class of a bounded complex of vectorbundles. Furthermore, we show that the first truncated Chern class of such a complex onlydepends on its determinant.
Atiyah classes were first introduced in 1957.
Atiyah associates to vector bundles E over complexmanifolds X elements in H ( X, H om ( E , E ) ⊗ Ω X ). They allow him to derive a criterion for theexistence of holomorphic connections in [Ati, Thm. 2]. Furthermore, Atiyah remarks that hisclasses suggest a new way to define the already known Chern classes ([Ati, Thm. 6]).Later, it turned out that deformation theory forms an important area for applications of Atiyahclasses. If E is a vector bundle on a scheme X , and X ֒ → Y is a closed immersion with nilpotentideal sheaf, we might be interested in the question whether E can be extended to a vector bundleon Y . Illusie gives an extensive answer to questions of this type in his dissertation [Ill, Introd.of Chapt. IV, Prop. IV.3.1.8], making essential use of classes that coincide with
Atiyah’s inspecial cases. For him, the central object is the cotangent complex ˜ L X | S of a morphism X → S of schemes. The outcome shows that this complex is the correct generalization of the cotangentbundle of a smooth morphism. Consequently, his version of the Atiyah class of a vector bundle E on an S -scheme X is an element in Ext ( E , E ⊗ ˜ L X | S ). This leads to the definition of a verygeneral variant of Chern classes, whose properties are studied in [Ill, Chapt. V]. Illusie’s results impress by their overwhelming generality, but they require highly elaborate tech-niques. Already the definition of his cotangent complex is complicated. Fortunately, there is aneasier variant of this complex, introduced by
Berthelot in [SGA6, Sect. VIII.2]. It is obtained from
Illusie’s cotangent complex by truncation (see [Ill, Cor. III.1.2.9.1]), thus we call it the truncatedcotangent complex L X | S of a morphism X → S .Recently, using the easier complex, Huybrechts and
Thomas managed to prove results similar to
Illusie’s with more elementary methods that also cover the deformation theory of complexes asobjects in the derived category. Again, a version of the Atiyah class plays a key role. For a vectorbundle E on an S -scheme X , this version is an element in Ext ( E , E ⊗ L X | S ). The result motivatesus to investigate this truncated Atiyah class in more detail.In Section 2, we will explain the basic concepts. The third section is devoted to truncated Atiyahclasses of complexes E of vector bundles. Our main result (for the precise formulation see Thm.3.4) gives an explicit formula via ˇCech resolutions describing these classes:1 heorem: The truncated Atiyah class of E is given by the ˇCech cocycle (cid:18)(cid:16) M sik ( ˜ M skj · ˜ M sji − ˜ M ski ) (cid:17) ijk , (cid:16) M sij · d ˜ M sji (cid:17) ij , (1.1) (cid:16) ( − s +1 M s +1 ij ( ˜ M s +1 ji · ˜ D si − ˜ D sj · ˜ M sji ) (cid:17) ij , (cid:16) ( − s +1 d ˜ D si (cid:17) i , (cid:16) − ˜ D s +1 i · ˜ D si (cid:17) i (cid:19) . Like the other versions of Atiyah classes, the truncated one induces Chern classes. We will discussthe first one in Section 4. It turns out (see Thm. 4.5) the first truncated Chern class of a complexof vector bundles only depends on its determinant:
Theorem:
For the first truncated Chern class c ( E ) ∈ H ( X, L X ) , we have c ( E ) = c (det( E )) . The result simplifies an argument in [HT] to prove the existence of a perfect obstruction theoryfor stable pairs as used in [PT].Notation: We fix a noetherian separated scheme S . We are mostly dealing with the category of S -schemes, therefore we use the abbreviations X × Y := X × S Y for fibre products and Ω X := Ω X | S for relative cotangent sheaves of S -schemes X and Y .Rings and algebras are always assumed to be commutative and unitary. If k is a ring, A a k -algebra, and I ⊆ A ⊗ k A the ideal of the diagonal, we frequently use the isomorphism Ω A | k ∼ = I/I , da ⊗ a − a ⊗ X , we always mean sheaves of O X -modules, whose categoryis denoted by Mod( X ). We often consider a sheaf as a complex of sheaves which is concentratedin degree 0.Complexes E of sheaves on a scheme X always have increasing differentials d s : E s → E s +1 . Thecomplex E [1] has components E [1] s = E s +1 and differentials d s E [1] = − d s +1 E . If F is another complexon X , the complex E ⊗ F with components (
E ⊗ F ) u = L s + t = u E s ⊗ F t is induced by the (anti-commuting) bicomplex with differentials d s E ⊗ id : E s ⊗F t → E s +1 ⊗F t and ( − s id ⊗ d t F : E s ⊗F t →E s ⊗ F t +1 . The complex H om ( E , F ) with components ( H om ( E , F )) u = Q − s + t = u H om ( E s , F t ) isinduced by the bicomplex with differentials H om ( d s − E , F t ) : H om ( E s , F t ) → H om ( E s − , F t ) and( − − s + t +1 H om ( E s , d t F ) : H om ( E s , F t ) → H om ( E s , F t +1 ).In the literature, there are different conventions concerning the signs of Atiyah and Chern classes.Basically, there are three choices that have to be made. One of them is the convention concerningthe K¨ahler differentials we introduced above. The other ones appear in the definitions of theclasses. Our sign conventions necessarily differ from those of other authors. However, we achievetransparency by avoiding superfluous identifications and by disclosing our choices. In this section, the truncated cotangent complex L X of a (suitable) S -scheme X is to be introduced,and we want to explain how we can associate to each complex E of sheaves on X a morphism At ( E ) : E → E ⊗ L L X [1] in the derived category D (Mod( X )), called the truncated Atiyah classof E . Furthermore, we will discuss the relationship between the truncated Atiyah class and theclassical version At cl ( E ) : E → E ⊗ L Ω X [1]. Our definition of the truncated cotangent complex will not be applicable to all S -schemes, butonly to those with a smooth ambient space in the sense of the following definition. Definition:
Let X be an S -scheme. A smooth ambient space of X is a smooth, separated, andquasi-compact S -scheme U with a closed immersion X ֒ → U .2ll quasi-projective S -schemes have a smooth ambient space because there exists a closed immer-sion into an open subscheme of a projective space P NS . Furthermore, all S -schemes with a smoothambient space are separated and of finite type over S . Hence they are noetherian separatedschemes.Following Berthelot ([SGA6, Sect. VIII.2, Prop. VIII.2.2]), we will define the truncated cotangentcomplex. The following lemma will make sure that it is well-defined.
Lemma 2.1 (Berthelot):
Let X be an S -scheme. Let ι U : X ֒ → U and ι V : X ֒ → V besmooth ambient spaces, given by ideal sheaves J U and J V . Then the complexes of sheaves L UX :=( J U / J U d → Ω U | X ) and L VX := ( J V / J V d → Ω V | X ) (which are concentrated in degrees -1 and 0,and are induced by the conormal sequence) are canonically isomorphic in the derived category D (Mod( X )) . Definition:
Let X be an S -scheme with a smooth ambient space U . The truncated cotangentcomplex of X is the complex L X := L UX . It is unique up to canonical isomorphism in the derivedcategory D (Mod( X )).In the situation of Lemma 2.1, the conormal sequences of the two immersions induce morphismsof complexes L UX → Ω X and L VX → Ω X . They are compatible with the canonical isomorphismof the lemma. This follows directly from the definitions. Hence there is a canonical morphism L X → Ω X in D (Mod( X )).If X is smooth over S , then this canonical morphism is an isomorphism. We can use X as its ownambient space, and then L XX = Ω X . In the sequel, we will discuss the truncated Atiyah class, which is introduced by
Huybrechts and
Thomas in [HT]. Let X be an S -scheme with a smooth ambient space. We want to associateto complexes E of sheaves on X morphisms At ( E ) : E → E ⊗ L L X [1] in a natural way, hence anatural transformation of functors id , ( ⊗ L L X [1]) : D (Mod( X )) ⇒ D (Mod( X )). It is helpfulto consider these functors as Fourier–Mukai transforms. From this point of view, the requirednatural transformation can be defined by giving a single morphism between the Fourier–Mukaikernels ∆ X ∗ O X and ∆ X ∗ L X [1], which we will call universal truncated Atiyah class, and theconstruction of which is prepared by the following lemma. Lemma 2.2 (Huybrechts, Thomas):
Let X be a flat S -scheme with smooth ambient space X ֒ → U , given by the ideal sheaf J . Then the sequence / / ∆ X ∗ J / J β / / I ∆ U | X × X α / / O X × X ε / / ∆ X ∗ O X / / of sheaves on X × X is exact. Here I ∆ U denotes the ideal of the diagonal on U × U , ε is thenatural surjection, α is the restriction of the inclusion I ∆ U ֒ → O U × U , and β is defined as follows.Let Spec( A ) ⊆ U be an affine open subset in the preimage of an affine open subset Spec( k ) ⊆ S with Y := X ∩ Spec( A ) = Spec( A/J ) . Then β maps an element in ∆ X ∗ J / J ( Y × Y ) = J/J ,given by a representative j ∈ J , to the class of the element j ⊗ − ⊗ j ∈ I in I ∆ U | X × X ( Y × Y ) = I ⊗ A ⊗ k A ( A/J ⊗ k A/J ) , where I ⊆ A ⊗ k A is the ideal of the diagonal.Proof: The lemma is proved in [HT, Sect. 2.2]. (cid:3)
The flatness assumption is missing in [HT], but it is necessary as the following example shows.Consider the situation S = U = Spec( Z ), X = Spec( Z / Z ). Here, the sequence is of the form0 → Z / Z → → Z / Z id → Z / Z → efinition: Let X be a flat S -scheme with smooth ambient space X ֒ → U , given by the idealsheaf J . The universal truncated Atiyah class of X is the morphism At X : ∆ X ∗ O X → ∆ X ∗ L X [1]in the derived category D (Mod( X × X )) which is given by the diagram∆ X ∗ O X ∆ X ∗ J / J β / / id (cid:15) (cid:15) I ∆ U | X × X α / / π (cid:15) (cid:15) O X × Xε O O ∆ X ∗ J / J − d / / I ∆ U / I U | X × X (2.1)of complexes of sheaves on X × X (which are concentrated in degrees − π isthe natural projection; the other maps have already been introduced in this section (note that I ∆ U / I U | X × X ∼ = ∆ X ∗ (Ω U | X )).The commutativity of (2.1) follows immediately from the definitions of β , π , and d . Thus thediagram really describes a morphism in the derived category. It is easy to see that it does notdepend on the choice of a smooth ambient space.We already indicated how the universal class induces the natural transformation we are aimingat. This is made precise by the following definition. Definition:
Let X be a flat S -scheme with a smooth ambient space. Let p, q : X × X ⇒ X be the natural projections. The truncated Atiyah class of X is the natural transformation At := Rq ∗ ◦ ( ⊗ L At X ) ◦ Lp ∗ between the functors id ∼ = Rq ∗ ◦ ( ⊗ L ∆ X ∗ O X ) ◦ Lp ∗ , ( ⊗ L L X [1]) ∼ = Rq ∗ ◦ ( ⊗ L ∆ X ∗ L X [1]) ◦ Lp ∗ : D (Mod( X )) ⇒ D (Mod( X )).Since this definition is of central importance for us, we repeat it with other words. In the situationof the definition, we consider a complex E of sheaves on X . We are interested in its truncatedAtiyah class. Firstly, we have to apply the derived functor Lp ∗ : D (Mod( X )) → D (Mod( X × X ))to E . Here, we can make use of the flatness of X , which implies that Lp ∗ ( E ) = p ∗ E . Next, weapply the derived tensor product with the universal truncated Atiyah class. We get a morphism p ∗ E ⊗ L ∆ X ∗ O X p ∗ E⊗ L At X −−−−−−−→ p ∗ E ⊗ L ∆ X ∗ L X [1]. In the general case, it is difficult to control thismorphism. However, if E is a bounded above complex of flat sheaves, it is not necessary to derivethe tensor product, and the morphism is given by p ∗ E ⊗ ∆ X ∗ O X p ∗ E ⊗ G o o / / p ∗ E ⊗ ∆ X ∗ L X [1] , where G denotes the complex in the middle row of (2.1), and the left arrow is a quasi-isomorphism.Finally, the derived functor Rq ∗ : D (Mod( X × X )) → D (Mod( X )) has to be applied. Unlikethe last to steps, we can not avoid deriving the functor here. Making use of the general theory ofFourier–Mukai transforms, we can identify Rq ∗ ( p ∗ E ⊗ L ∆ X ∗ O X ) with E and Rq ∗ ( p ∗ E ⊗ L ∆ X ∗ L X [1])with E ⊗ L L X [1] — however, a simple and general description of the resulting morphism At ( E ) : E → E ⊗ L L X [1] appears inaccessible. At first sight, the definition of the truncated Atiyah class may appear arbitrary. We will see,however, that it lifts the classical Atiyah class.
Definition:
Let X be a separated S -scheme of finite type. The universal (classical) Atiyah class of X is the morphism At X,cl : ∆ X ∗ O X → ∆ X ∗ Ω X [1] in the derived category D (Mod( X × X ))4hich is given by the diagram ∆ X ∗ O X ∆ X ∗ Ω X / / id (cid:15) (cid:15) O X × X / I X O O ∆ X ∗ Ω X (2.2)of complexes of sheaves on X × X (which are concentrated in degrees − I ∆ X denotesthe ideal of the diagonal on X × X , and the morphisms without labelling come from the naturalshort exact sequence (note that ∆ X ∗ Ω X ∼ = I ∆ X / I X ).In other words, the universal classical Atiyah class is the morphism ∆ X ∗ O X → ∆ X ∗ Ω X [1] givenby the short exact sequence 0 → ∆ X ∗ Ω X → O X × X / I X → ∆ X ∗ O X → Definition:
Let X be a separated S -scheme of finite type. Let p, q : X × X ⇒ X be the naturalprojections. The (classical) Atiyah class of X is the natural transformation At cl := Rq ∗ ◦ ( ⊗ L At X,cl ) ◦ Lp ∗ between the functors id ∼ = Rq ∗ ◦ ( ⊗ L ∆ X ∗ O X ) ◦ Lp ∗ , ( ⊗ L Ω X [1]) ∼ = Rq ∗ ◦ ( ⊗ L ∆ X ∗ Ω X [1]) ◦ Lp ∗ : D (Mod( X )) ⇒ D (Mod( X )).Let X be a flat S -scheme with a smooth ambient space, and let E be a complex of sheaveson X . We have defined a classical Atiyah class At cl ( E ) : E → E ⊗ L Ω X [1] and a truncated one At ( E ) : E → E⊗ L L X [1]. The natural map L X → Ω X induces a morphism E⊗ L L X [1] → E⊗ L Ω X [1].The following simple lemma shows that the resulting diagram is commutative. Lemma 2.3:
Let X be a flat S -scheme with a smooth ambient space. Then the natural map L X → Ω X induces a commutative diagram of natural transformations ( ⊗ L L X [1]) (cid:15) (cid:15) id At mmmmmmmmmmmmmmm At cl ( ( QQQQQQQQQQQQQQQ ( ⊗ L Ω X [1]) between functors D (Mod( X )) → D (Mod( X )) .Proof: It suffices to prove the commutativity of the diagram∆ X ∗ L X [1] (cid:15) (cid:15) ∆ X ∗ O X At X kkkkkkkkkkkkkk At X,cl ) ) SSSSSSSSSSSSSS ∆ X ∗ Ω X [1] . To this end, we fix a smooth ambient space of X . If we denote the complexes in the middle rowsof (2.1) and (2.2) by G and G cl , then there exists a natural map G → G cl (in degree 0 the natural5rojection, in degree − I ∆ U | X × X π → I ∆ U / I U | X × X ∼ = ∆ X ∗ (Ω U | X ) → ∆ X ∗ Ω X )making the diagram G / / (cid:15) (cid:15) { { vvvvvvvvvv ∆ X ∗ L X [1] (cid:15) (cid:15) ∆ X ∗ O X G cl c c HHHHHHHHH / / ∆ X ∗ Ω X [1]obviously commutative. This implies the assertion of the lemma. (cid:3) In the next section, we will need an alternative description of the universal truncated Atiyah class.
Lemma 2.4:
Let X be a flat S -scheme with smooth ambient space X ֒ → U , given by the idealsheaf J . Then, in the commutative diagram ∆ X ∗ O X id (cid:15) (cid:15) ∆ X ∗ J / J (cid:18) − β id (cid:19) / / I ∆ U | X × X ⊕ ∆ X ∗ J / J (cid:18) − α π − d (cid:19) / / O X × X ⊕ I ∆ U / I U | X × X (cid:0) − ε (cid:1) / / ∆ X ∗ O X ∆ X ∗ J / J (cid:18) (cid:19) O O − d / / I ∆ U / I U | X × X , (cid:18) (cid:19) O O (2.3) the morphism between the complexes in the two lower rows (which are concentrated in degrees − to ) is a quasi-isomorphism. The morphism ∆ X ∗ O X → ∆ X ∗ L X [1] given by the diagram is theuniversal truncated Atiyah class of X .Proof: Once more, we denote the complex in the middle row of (2.1) by G . The complex G in themiddle row of (2.3) is the mapping cone of the morphism (cid:18) − ε (id , π ) (cid:19) : G → ∆ X ∗ O X ⊕ ∆ X ∗ L X [1]. Since G ε → ∆ X ∗ O X is a quasi-isomorphism, this holds also true for the induced map ∆ X ∗ L X [1] → G .By construction, the composition G → ∆ X ∗ O X ⊕ ∆ X ∗ L X [1] → G is the zero map, hence thediagram G ε / / (id ,π ) (cid:15) (cid:15) ∆ X ∗ O X (cid:15) (cid:15) ∆ X ∗ L X [1] / / G , is commutative, from which the assertion follows. (cid:3) In the previous section, we introduced the truncated Atiyah class At ( E ) : E → E ⊗ L L X [1] of acomplex E of sheaves on a flat S -scheme X with a smooth ambient space. Our definition via auniversal class has the advantage that it can be applied to all complexes of sheaves, and that itsnaturality is obvious. However, it is not suitable for explicit calculations. Without changing ourpoint of view, we will even fail to prove elementary results about the first truncated Chern class,which will be discussed in the next section as an application of the truncated Atiyah class.Thus we will have to give an alternative, more concrete description of the truncated Atiyah class.We will make use of ˇCech resolutions. For the increase of concreteness, we have to accept a loss6f generality. The methods of this section only allow us to handle bounded complexes of vectorbundles. First, we fix the setting of this section.
Situation 3.1:
Let X be a flat S -scheme with smooth ambient space ι : X ֒ → U (in particular, U is separated), given by the ideal sheaf J . We denote the natural projections by p, q : X × X ⇒ X .Let E be a bounded complex of locally free coherent sheaves on X . Let U = S Γ U i be a finite coverof U by affine open subsets U i such that the components E s of E are free on all X i := U i ∩ X , andsuch that U i maps into an affine open subset S i of S . We assume that the index set Γ is strictlyordered.For non-empty subsets Λ von Γ , we introduce the notation U Λ := T Λ U i , X Λ := T Λ X i , and S Λ := T Λ S i . Since U , X , and S are separated, U Λ = Spec( A Λ ) , X Λ = Spec( A Λ /J Λ ) , and S Λ = Spec( k Λ ) are affine.Let ϕ si : E s | X i ∼ → O m si X i be trivializations and M sij := ϕ si ◦ ( ϕ sj ) − | X ij ∈ GL( m si , A ij /J ij ) thecorresponding transition maps. (Note that in the case X ij = ∅ , we always have m si = m sj .) Foreach ordered pair ( i, j ) , we choose a lift ˜ M sij ∈ Mat( m si × m si , A ij ) von M sij . Similarly, we define D si := ϕ s +1 i ◦ d s E ◦ ( ϕ si ) − ∈ Mat( m s +1 i × m si , A i /J i ) and choose a lift ˜ D si ∈ Mat( m s +1 i × m si , A i ) . Now, we want to bring ˇCech resolutions into play. Let Y be a noetherian separated scheme and Y = S Y i a finite affine open cover of Y with strictly ordered index set. For a quasi-coherentsheaf F on Y , we denote its ˇCech resolution (with respect to the chosen cover, see [Har, Lem.III.4.2]) by C ( F ). Thus for natural numbers r , we have C r ( F ) = L Λ i Λ ∗ ( F| T Λ Y i ), where the sumruns over strictly increasing sequences Λ of length r + 1, and i Λ : T Λ Y i ֒ → Y denotes the naturalinclusion.If H is a bounded complex of quasi-coherent sheaves on Y , we write C ( H ) for the total complexassociated with the (anti-commuting) bicomplex C r ( H s ) with differentials ˇ d : C r ( H s ) → C r +1 ( H s )and ( − r d H : C r ( H s ) → C r ( H s +1 ). If we consider H as a bicomplex concentrated in column 0,the natural maps H s → C ( H s ) induce a morphism of bicomplexes and thus a morphism betweentheir total complexes H → C ( H ). For the following, see e.g. [KS, Thm. 12.5.4]. Lemma 3.2:
Let H be a bounded complex of quasi-coherent sheaves on a noetherian separatedscheme Y . Let Y = S Y i be a finite affine open cover of Y with strictly ordered index set. Thenthe natural morphism H → C ( H ) is a quasi-isomorphism. Next, we will define a morphism
E → E ⊗ L X [1] in the derived category. It will turn out thatit coincides with the truncated Atiyah class of E . It follows from Lemma 3.2 that E ⊗ L X [1]is quasi-isomorphic to E ⊗ C ( L X [1]) in a natural way, where the ˇCech resolution is formed withrespect to the cover of X we fixed above. Thus, for the construction of our morphism, it sufficesto give a morphism of complexes E → E ⊗ C ( L X [1]).The sheaf ( E ⊗ C ( L X [1])) s has the summand E s ⊗ C ( J / J ) ⊕ E s ⊗ C (Ω U | X ) ⊕ E s +1 ⊗ C ( J / J ) ⊕E s +1 ⊗ C (Ω U | X ) ⊕ E s +2 ⊗ C ( J / J ). If F is a sheaf on X , then (with the notation intro-duced above) a morphism E s → E t ⊗ C r ( F ) = E t ⊗ L Λ i Λ ∗ ( F| X Λ ) ∼ = L Λ i Λ ∗ ( E t | X Λ ⊗ F| X Λ )is, by adjunction, given by maps E s | X Λ → E t | X Λ ⊗ F| X Λ . If we use the restrictions of ϕ smin (Λ) and ϕ tmin (Λ) as trivializations on X Λ (what we will always do in the sequel), then these mapscan be considered as matrices with entries in F ( X Λ ). Hence, for the definition of a morphism E s → ( E ⊗ C ( L X [1])) s (factorizing over the summand mentioned above), it suffices to give matri-ces in Mat( m si × m si , J ijk /J ijk ), Mat( m si × m si , Ω A ij | k ij ⊗ A ij A ij /J ij ), Mat( m s +1 i × m si , J ij /J ij ),Mat( m s +1 i × m si , Ω A i | k i ⊗ A i A i /J i ), and Mat( m s +2 i × m si , J i /J i ).7ith these considerations in mind, we define a morphism E s → ( E ⊗ C ( L X [1])) s by (cid:18)(cid:16) M sik ( ˜ M skj · ˜ M sji − ˜ M ski ) (cid:17) ijk , (cid:16) M sij · d ˜ M sji (cid:17) ij , (3.1) (cid:16) ( − s +1 M s +1 ij ( ˜ M s +1 ji · ˜ D si − ˜ D sj · ˜ M sji ) (cid:17) ij , (cid:16) ( − s +1 d ˜ D si (cid:17) i , (cid:16) − ˜ D s +1 i · ˜ D si (cid:17) i (cid:19) . Here and in the sequel, we drop restrictions to open subsets from the notation. We denote theuniversal derivation by d : A Λ → Ω A Λ | k Λ , which has to be applied separately to the entries of thematrices. The equations M skj · M sji = M ski , M s +1 ji · D si = D sj · M sji , and D s +1 i · D si = 0 imply that˜ M skj · ˜ M sji − ˜ M ski , ˜ M s +1 ji · ˜ D si − ˜ D sj · ˜ M sji , and ˜ D s +1 i · ˜ D si really are matrices with entries in J Λ .We want to motivate the formula (3.1) in the case of a vector bundle E with transition maps M ij .In this situation, only the first two of the five parts of the formula play a role. It is known thatthe classical Atiyah class of E is given by the ˇCech cocycle ( M ij · dM ji ) ij : E → E ⊗ C (Ω X ) —we will discuss this formula (3.5) later. In view of Lemma 2.3, it appears sensible to search for amorphism E → E ⊗ C ( L X ) lifting this cocyle and to hope that it describes the truncated class. If E extends to a vector bundle on U , then the transition maps ˜ M ij of the extension yield naturallifts of those of E (if we choose trivializations of E which are restrictions of trivializations of theextension). The resulting map ( M ij · d ˜ M ji ) ij : E → E ⊗ C (Ω U | X ) really gives a morphism ofcomplexes E → E ⊗ C ( L X [1]) solving our problem. If E , however, cannot be extended, then thechoice of lifts ˜ M ij cannot be carried out in a natural way. The term ( M ik ( ˜ M kj · ˜ M ji − ˜ M ki )) ijk in(3.1) is needed as compensation — it makes sure that a morphism of complexes E → E ⊗ C ( L X [1])arises, as our next lemma will show.After this interlude, we return to the more general situation 3.1 of a complex E . Lemma 3.3:
In situation 3.1, the morphisms (3.1) define a morphism of complexes
E → E ⊗C ( L X [1]) and thus a morphism E → E ⊗ L X [1] in the derived category D (Mod( X )) .Proof: We have to make sure that the diagrams E s / / d E (cid:15) (cid:15) E s ⊗ C ( J / J ) ⊕ E s ⊗ C (Ω U | X ) ⊕ E s +1 ⊗ C ( J / J ) ⊕E s +1 ⊗ C (Ω U | X ) ⊕ E s +2 ⊗ C ( J / J ) ξ ˇ d − ξd ξ ˇ d d E − ξ ˇ d d E − ξd − ξ ˇ d
00 0 d E ξ ˇ d d E − ξd d E (cid:15) (cid:15) E s +1 / / E s ⊗ C ( J / J ) ⊕ E s ⊗ C (Ω U | X ) ⊕ E s +1 ⊗ C ( J / J ) ⊕E s +1 ⊗ C (Ω U | X ) ⊕ E s +2 ⊗ C ( J / J ) ⊕E s +2 ⊗ C (Ω U | X ) ⊕ E s +3 ⊗ C ( J / J )commute, where the rows are given by (3.1). Here, d is induced by the universal derivation, andwe use the abbreviation ξ := ( − s .Corresponding to the decomposition of the sheaf in the right lower corner of the diagram into sevensummands, we have to compare seven pairs of morphisms E s → E t ⊗ C r ( F ) (with F = J / J or F = Ω U | X ). To this end, we fix strictly increasing sequences of indices Λ = ( i, j, k, l ), Λ = ( i, j, k ),Λ = ( i, j ) or Λ = ( i ) and prove the equality of the two maps E s | X Λ → E t | X Λ ⊗F| X Λ by a calculationwith the corresponding matrices. 8or the first summand, we get by definition of the ˇCech differential ξ ˇ d (( M sik ( ˜ M skj · ˜ M sji − ˜ M ski )) ijk ) ijkl = ξ [ M sij ( M sjl ( ˜ M slk · ˜ M skj − ˜ M slj )) M sji − M sil ( ˜ M slk · ˜ M ski − ˜ M sli )+ M sil ( ˜ M slj · ˜ M sji − ˜ M sli ) − M sik ( ˜ M skj · ˜ M sji − ˜ M ski )] = ξ [ ˜ M sil · ˜ M slk · ˜ M skj · ˜ M sji − ˜ M sil · ˜ M slj · ˜ M sji − ˜ M sil · ˜ M slk · ˜ M ski + ˜ M sil · ˜ M sli +˜ M sil · ˜ M slj · ˜ M sji − ˜ M sil · ˜ M sli − ˜ M sik · ˜ M skj · ˜ M sji + ˜ M sik · ˜ M ski ] = ξ ( ˜ M sil · ˜ M slk − ˜ M sik ) · ( ˜ M skj · ˜ M sji − ˜ M ski ) = 0 . The last equality results from the fact that the product of two matrices with entries in J ijkl hasentries in J ijkl .For the second summand, we compute − ξd ( M sik ( ˜ M skj · ˜ M sji − ˜ M ski )) + ξ ˇ d (( M sij · d ˜ M sji ) ij ) ijk = ξ [ − M sik · d ( ˜ M skj · ˜ M sji − ˜ M ski ) + M sij ( M sjk · d ˜ M skj ) M sji − M sik · d ˜ M ski + M sij · d ˜ M sji ] = ξ [ − M sik · d ˜ M skj · M sji − M sik · M skj · d ˜ M sji + M sik · d ˜ M ski + M sik · d ˜ M skj · M sji − M sik · d ˜ M ski + M sij · d ˜ M sji ] = 0 . For the third summand, we get (using D si · M sik = M s +1 ik · D sk for the second equality) d E ( M sik ( ˜ M skj · ˜ M sji − ˜ M ski )) − ξ ˇ d (( − ξM s +1 ij ( ˜ M s +1 ji · ˜ D si − ˜ D sj · ˜ M sji )) ij ) ijk = D si · M sik ( ˜ M skj · ˜ M sji − ˜ M ski ) + M s +1 ij ( M s +1 jk ( ˜ M s +1 kj · ˜ D sj − ˜ D sk · ˜ M skj )) M sji − M s +1 ik ( ˜ M s +1 ki · ˜ D si − ˜ D sk · ˜ M ski ) + M s +1 ij ( ˜ M s +1 ji · ˜ D si − ˜ D sj · ˜ M sji ) =˜ M s +1 ik · ˜ D sk · ˜ M skj · ˜ M sji − ˜ M s +1 ik · ˜ D sk · ˜ M ski + ˜ M s +1 ik · ˜ M s +1 kj · ˜ D sj · ˜ M sji − ˜ M s +1 ik · ˜ D sk · ˜ M skj · ˜ M sji − ˜ M s +1 ik · ˜ M s +1 ki · ˜ D si + ˜ M s +1 ik · ˜ D sk · ˜ M ski + ˜ M s +1 ik · ˜ M s +1 kj · ˜ M s +1 ji · ˜ D si − ˜ M s +1 ik · ˜ M s +1 kj · ˜ D sj · ˜ M sji = M s +1 ik ( ˜ M s +1 kj · ˜ M s +1 ji − ˜ M s +1 ki ) D si = ( M s +1 ik ( ˜ M s +1 kj · ˜ M s +1 ji − ˜ M s +1 ki )) d E . For the fourth summand, we get (using D si · M sij = M s +1 ij · D sj for the last but one equality) d E ( M sij · d ˜ M sji ) − ξd ( − ξM s +1 ij ( ˜ M s +1 ji · ˜ D si − ˜ D sj · ˜ M sji )) − ξ ˇ d (( − ξd ˜ D si ) i ) ij = D si · M sij · d ˜ M sji + M s +1 ij · d ( ˜ M s +1 ji · ˜ D si − ˜ D sj · ˜ M sji ) + M s +1 ij · d ˜ D sj · M sji − d ˜ D si = D si · M sij · d ˜ M sji + M s +1 ij · d ˜ M s +1 ji · D si + M s +1 ij · M s +1 ji · d ˜ D si − M s +1 ij · d ˜ D sj · M sji − M s +1 ij · D sj · d ˜ M sji + M s +1 ij · d ˜ D sj · M sji − d ˜ D si = M s +1 ij · d ˜ M s +1 ji · D si = ( M s +1 ij · d ˜ M s +1 ji ) d E . For the fifth summand, we get d E ( − ξM s +1 ij ( ˜ M s +1 ji · ˜ D si − ˜ D sj · ˜ M sji )) + ξ ˇ d (( − ˜ D s +1 i · ˜ D si ) i ) ij = ξ [ − D s +1 i · M s +1 ij ( ˜ M s +1 ji · ˜ D si − ˜ D sj · ˜ M sji ) − M s +2 ij · ˜ D s +1 j · ˜ D sj · M sji + ˜ D s +1 i · ˜ D si ] = ξ [ − M s +2 ij · D s +1 j ( ˜ M s +1 ji · ˜ D si − ˜ D sj · ˜ M sji ) − M s +2 ij · ˜ D s +1 j · ˜ D sj · M sji + ˜ D s +1 i · ˜ D si ] = ξ [ − ˜ M s +2 ij · ˜ D s +1 j · ˜ M s +1 ji · ˜ D si + M s +2 ij · M s +2 ji · ˜ D s +1 i · ˜ D si ] = ξM s +2 ij ( ˜ M s +2 ji · ˜ D s +1 i − ˜ D s +1 j · ˜ M s +1 ji ) D si = ( ξM s +2 ij ( ˜ M s +2 ji · ˜ D s +1 i − ˜ D s +1 j · ˜ M s +1 ji )) d E . For the sixth summand, we get d E ( − ξd ˜ D si ) − ξd ( − ˜ D s +1 i · ˜ D si ) = ξ [ − D s +1 i · d ˜ D si + d ˜ D s +1 i · D si + D s +1 i · d ˜ D si )] = ξd ˜ D s +1 i · D si = ( ξd ˜ D si ) d E . d E ( − ˜ D s +1 i · ˜ D si ) = − D s +2 i · ˜ D s +1 i · ˜ D si = − ˜ D s +2 i · ˜ D s +1 i · D si = ( − ˜ D s +2 i · ˜ D s +1 i ) d E . (cid:3) We have constructed two morphisms E ⇒ E ⊗ L X [1], the rather abstract truncated Atiyah classand the concrete morphism of Lemma 3.3. Our main result shows that they coincide. Theorem 3.4:
In situation 3.1, the morphism
E → E ⊗ L X [1] in the derived category D (Mod( X )) given by the formula (3.1) is the truncated Atiyah class of E .Proof: Let ( X × X ) \ ∆ X = S Γ ′ V i be a finite cover by affine open sets. For i ∈ Γ, we define V i := X i × X i ⊆ X × X . Then X × X = S Γ ⊔ Γ ′ V i is a finite cover by affine open sets (note that X i × X i = Spec( A i /J i ⊗ k i A i /J i )). We extend the strict order on Γ to a strict order on Γ ⊔ Γ ′ .For non-empty subsets Λ of Γ ⊔ Γ ′ , we define V Λ := T Λ V i , and we denote the natural inclusion by i Λ : V Λ → X × X .We write G for the complex in the middle row of (2.3), and we consider its ˇCech resolution C ( G )with respect to the chosen cover of X × X . We have C r ( G s ) = L Λ i Λ ∗ ( G s | V Λ ), where the sumruns over strictly increasing sequences Λ of length r + 1. We form a subsheaf G rs of C r ( G s ) byconsidering only those summands with Λ ∩ Γ ′ = ∅ . The totality of these subsheaves defines asubcomplex of C ( G ), denoted by G . Further, let G := C ( G ) / G be the corresponding quotient.From the condition Λ ∩ Γ ′ = ∅ , it follows that V Λ ∩ ∆ X = ∅ . Thus sheaves which appear in thecomplex G , but are concentrated on the diagonal, do not contribute to G . More explicit, G is onlycomposed of the summands G r, − = L Λ ∩ Γ ′ = ∅ i Λ ∗ ( O X × X | V Λ ) and G r, − = L Λ ∩ Γ ′ = ∅ i Λ ∗ ( I ∆ U | V Λ ).Thus if we write e := | Γ ⊔ Γ ′ | , G is given by G , − (cid:18) ˇ d − α (cid:19) −−−−→ G , − ⊕ G , − (cid:18) ˇ d α ˇ d (cid:19) −−−−−→ G , − ⊕ G , − → · · · → G r +1 , − ⊕ G r, − (cid:18) ˇ d − r α ˇ d (cid:19) −−−−−−−−−→ G r +2 , − ⊕ G r +1 , − → · · · → G e − , − ⊕ G e − , − (cid:0) ( − e − α ˇ d (cid:1) −−−−−−−−−−→ G e − , − . Apart from the diagonal, however, α is an isomorphism, and hence it induces isomorphisms G r, − → G r, − . It follows that G is an exact complex. Thus C ( G ) → G is a quasi-isomorphism.We have constructed the commutative diagram∆ X ∗ O X / / G (cid:15) (cid:15) ∆ X ∗ L X [1] o o (cid:15) (cid:15) C ( G ) (cid:15) (cid:15) C (∆ X ∗ L X [1]) o o G of complexes of sheaves on X × X . Here, all morphisms except the left one are quasi-isomorphisms,and the first row describes the truncated Atiyah class of X . If we apply the functor q ∗ ( p ∗ E ⊗ ),10e obtain the diagram E / / q ∗ ( p ∗ E ⊗ G ) (cid:15) (cid:15) E ⊗ L X [1] (cid:15) (cid:15) o o q ∗ ( p ∗ E ⊗ C ( G )) (cid:15) (cid:15) q ∗ ( p ∗ E ⊗ C (∆ X ∗ L X [1])) o o q ∗ ( p ∗ E ⊗ G ) (cid:15) (cid:15) Rq ∗ ( p ∗ E ⊗ G ) . (3.2)of complexes of sheaves on X . The map E ⊗ L X [1] → Rq ∗ ( p ∗ E ⊗ G ) is a quasi-isomorphism, andthe truncated Atiyah class of E is given by E → Rq ∗ ( p ∗ E ⊗ G ).Since ∆ X ∗ L X [1] is concentrated on the diagonal, we have q ∗ ( p ∗ E ⊗ C (∆ X ∗ L X [1])) = q ∗ ( p ∗ E ⊗ ∆ X ∗ C ( L X [1])) = E ⊗ C ( L X [1]) — note that we compare ˇCech complexes on X × X (with thecover indexed by Γ ⊔ Γ ′ ) with ˇCech complexes on X (with the cover indexed by Γ). Thus themorphism E → E ⊗ C ( L X [1]) defined in (3.1) can be inserted into our diagram, and it induces asecond morphism E → q ∗ ( p ∗ E ⊗ G ) (besides the one given by (3.2)). We will show that these twomorphisms are homotopic. Then it follows that the induced morphisms
E → Rq ∗ ( p ∗ E ⊗ G ) arehomotopic as well, and this implies the assertion of the theorem.Throughout the rest of the proof, we will frequently consider sheaves on X × X that are pushedforward from X via the diagonal map. In this case, we will omit the term ∆ X ∗ in order to simplifythe notation.A morphism E → q ∗ ( p ∗ E ⊗ G ) is given by maps E s → q ∗ ( p ∗ E ⊗ G ) s . If we introduce the notation˜ C r ( F ) := L Λ i Λ ∗ ( F| V Λ ) for a sheaf F on X × X , where the sum runs only over strictly increasingsubsequences of Γ of length r + 1 (not subsequences of Γ ⊔ Γ ′ ), then we have G − = ˜ C ( J / J ) G − = ˜ C ( J / J ) ⊕ ˜ C ( I ∆ U | X × X ) ⊕ ˜ C ( J / J ) G − = ˜ C ( J / J ) ⊕ ˜ C ( I ∆ U | X × X ) ⊕ ˜ C ( J / J ) ⊕ ˜ C ( O X × X ) ⊕ ˜ C ( I ∆ U / I U | X × X ) G = ˜ C ( J / J ) ⊕ ˜ C ( I ∆ U | X × X ) ⊕ ˜ C ( J / J ) ⊕ ˜ C ( O X × X ) ⊕ ˜ C ( I ∆ U / I U | X × X ) ⊕ ˜ C ( O X ) . Thus q ∗ ( p ∗ E ⊗ G ) s has the summand q ∗ ( p ∗ E s ⊗ ˜ C ( J / J )) ⊕ q ∗ ( p ∗ E s ⊗ ˜ C ( I ∆ U | X × X )) ⊕ q ∗ ( p ∗ E s ⊗ ˜ C ( J / J )) ⊕ (3.3) q ∗ ( p ∗ E s ⊗ ˜ C ( O X × X )) ⊕ q ∗ ( p ∗ E s ⊗ ˜ C ( I ∆ U / I U | X × X )) ⊕ q ∗ ( p ∗ E s ⊗ ˜ C ( O X )) ⊕ q ∗ ( p ∗ E s +1 ⊗ ˜ C ( J / J )) ⊕ q ∗ ( p ∗ E s +1 ⊗ ˜ C ( I ∆ U | X × X )) ⊕ q ∗ ( p ∗ E s +1 ⊗ ˜ C ( J / J )) ⊕ q ∗ ( p ∗ E s +1 ⊗ ˜ C ( O X × X )) ⊕ q ∗ ( p ∗ E s +1 ⊗ ˜ C ( I ∆ U / I U | X × X )) ⊕ q ∗ ( p ∗ E s +2 ⊗ ˜ C ( J / J )) ⊕ q ∗ ( p ∗ E s +2 ⊗ ˜ C ( I ∆ U | X × X )) ⊕ q ∗ ( p ∗ E s +2 ⊗ ˜ C ( J / J )) ⊕ q ∗ ( p ∗ E s +3 ⊗ ˜ C ( J / J )) . The restrictions of J / J , I ∆ U | X × X , O X × X , I ∆ U / I U | X × X , and O X to open sets of the form V Λ (with Λ ⊆ Γ) correspond to the A Λ /J Λ ⊗ k Λ A Λ /J Λ -modules J Λ /J , I Λ ⊗ A Λ ⊗ k Λ A Λ ( A Λ /J Λ ⊗ k Λ A Λ /J Λ ), A Λ /J Λ ⊗ k Λ A Λ /J Λ , I Λ /I ⊗ A Λ ⊗ k Λ A Λ ( A Λ /J Λ ⊗ k Λ A Λ /J Λ ), and A Λ /J Λ , where I Λ ⊆ A Λ ⊗ k Λ A Λ denotes the ideal of the diagonal. A morphism E s → q ∗ ( p ∗ E t ⊗ i Λ ∗ ( F| V Λ )) with one ofthese sheaves F corresponds to a m ti × m si -matrix with entries in the modules mentioned, where i is the minimal element in Λ, and where we use the restrictions of ϕ si bzw. ϕ ti as trivializationsof E s and E t on X Λ .The diagrams (2.3) and (3.2) show that the map E s → q ∗ ( p ∗ E ⊗ G ) s induced by the truncatedAtiyah class factorizes over the summand (3.3), and that, in the above sense, it can be described11y (0 , , , , , ( ) i , , , , , , , , , , where denotes the identity matrix. Correspondingly, the morphism defined by (3.1) is given by (cid:18) , , (cid:16) M sik ( ˜ M skj · ˜ M sji − ˜ M ski ) (cid:17) ijk , , (cid:16) M sij · d ˜ M sji (cid:17) ij , , , , (cid:16) − ξM s +1 ij ( ˜ M s +1 ji · ˜ D si − ˜ D sj · ˜ M sji ) (cid:17) ij , , (cid:16) − ξd ˜ D si (cid:17) i , , , (cid:16) − ˜ D s +1 i · ˜ D si (cid:17) i , (cid:19) . Once more, we write ξ := ( − s .We have to construct a homotopy between these two morphisms E → q ∗ ( p ∗ E ⊗ G ). This requiresa map h s : E s → q ∗ ( p ∗ E ⊗ G ) s − in each degree. We define such a morphism, factorizing over thesummand q ∗ ( p ∗ E s ⊗ ˜ C ( J / J )) ⊕ q ∗ ( p ∗ E s ⊗ ˜ C ( I ∆ U | X × X )) ⊕ q ∗ ( p ∗ E s ⊗ ˜ C ( O X × X )) ⊕ (3.4) q ∗ ( p ∗ E s +1 ⊗ ˜ C ( J / J )) ⊕ q ∗ ( p ∗ E s +1 ⊗ ˜ C ( I ∆ U | X × X )) ⊕ q ∗ ( p ∗ E s +2 ⊗ ˜ C ( J / J ))of q ∗ ( p ∗ E ⊗ G ) s − , by (cid:18)(cid:16) − ξM sik ( ˜ M skj · ˜ M sji − ˜ M ski ) (cid:17) ijk , (cid:16) ξ (( ˜ M sij ⊗ ⊗ ˜ M sji ) − ( ˜ M sij · ˜ M sji ) ⊗ (cid:17) ij , ( − ξ ) i , (cid:16) M s +1 ij ( ˜ M s +1 ji · ˜ D si − ˜ D sj · ˜ M sji ) (cid:17) ij , (cid:16) ˜ D si ⊗ − ⊗ ˜ D si (cid:17) i , (cid:16) ξ ˜ D s +1 i · ˜ D si (cid:17) i (cid:19) . Here, for a matrix M with entries in A Λ , we denote by 1 ⊗ M or M ⊗ A Λ ⊗ k Λ A Λ that result from M by replacing each entry m by 1 ⊗ m or m ⊗ q ∗ ( p ∗ E ⊗ G ) s − ξ ˇ d − ξβ ξ ˇ d ξ id 0 0 0 0 00 ξα ξ ˇ d − ξπ − ξε d E − ξ ˇ d d E − ξβ − ξ ˇ d
00 0 0 ξ id 0 00 0 d E ξα
00 0 0 0 − ξπ
00 0 0 d E ξ ˇ d d E − ξβ ξ id0 0 0 0 0 d E (cid:15) (cid:15) E s h s : : uuuuuuuuuuuuuuuuuuuuu d E (cid:15) (cid:15) At ( E ) / / ( ) / / q ∗ ( p ∗ E ⊗ G ) s E s +1 , h s +1 : : tttttttttttttttttttt we only give the restriction of the differential of the complex q ∗ ( p ∗ E ⊗ G ) to the summand (3.4)of q ∗ ( p ∗ E ⊗ G ) s − which is relevant for our calculation, and which is mapped by the differentialinto the summand (3.3) of q ∗ ( p ∗ E ⊗ G ) s . Corresponding to the decomposition (3.3), we haveto execute fifteen calculations. Fortunately, we have already finished the first, seventh, twelfth,and fifteenth one in the proof of Lemma 3.3 (as first, third, fifth, and seventh calculation there;the situations only differ by the factor − ξ ). Furthermore, the calculations three, six, nine, andfourteen are trivial, and for the fifth and eleventh one, we only have to recall the identificationΩ A Λ | k Λ ∼ = I Λ /I , da ⊗ a − a ⊗ − ξβ ( − ξM sik ( ˜ M skj · ˜ M sji − ˜ M ski )) + ξ ˇ d (( ξ (( ˜ M sij ⊗ ⊗ ˜ M sji ) − ( ˜ M sij · ˜ M sji ) ⊗ ij ) ijk = − [ − ( ˜ M sik · ˜ M skj · ˜ M sji ) ⊗ ⊗ ( ˜ M sik · ˜ M skj · ˜ M sji ) + ( ˜ M sik · ˜ M ski ) ⊗ − ⊗ ( ˜ M sik · ˜ M ski )]+[( M sij ⊗ M sjk ⊗ ⊗ ˜ M skj ) − ( ˜ M sjk · ˜ M skj ) ⊗ ⊗ M sji ) − ( ˜ M sik ⊗ ⊗ ˜ M ski ) + ( ˜ M sik · ˜ M ski ) ⊗ M sij ⊗ ⊗ ˜ M sji ) − ( ˜ M sij · ˜ M sji ) ⊗
1] =( ˜ M sik · ˜ M skj · ˜ M sji ) ⊗ − ⊗ ( ˜ M sik · ˜ M skj · ˜ M sji ) + 1 ⊗ ( ˜ M sik · ˜ M ski )+(( ˜ M sij · ˜ M sjk ) ⊗ ⊗ ( ˜ M skj · ˜ M sji )) − (( ˜ M sij · ˜ M sjk · ˜ M skj ) ⊗ ⊗ ˜ M sji ) − ( ˜ M sik ⊗ ⊗ ˜ M ski ) + ( ˜ M sij ⊗ ⊗ ˜ M sji ) − ( ˜ M sij · ˜ M sji ) ⊗ M sik ⊗ − ⊗ ˜ M sik )(1 ⊗ ( M skj · M sji − M ski ))+(( M sij · M sjk · M skj − M sij ) ⊗ M sji ⊗ − ⊗ ˜ M sji )+(( M sij · M sjk − M sik ) ⊗ ⊗ ( ˜ M skj · ˜ M sji ) − ( ˜ M skj · ˜ M sji ) ⊗
1) = 0 . For the fourth summand, we get ξα ( ξ (( ˜ M sij ⊗ ⊗ ˜ M sji ) − ( ˜ M sij · ˜ M sji ) ⊗ ξ ˇ d (( − ξ ) i ) ij =(( M sij ⊗ ⊗ M sji ) − ( M sij · M sji ) ⊗
1) + ( − ( M sij ⊗ ⊗ M sji ) + ) = 0 . For the eighth summand, we get d E ( ξ (( ˜ M sij ⊗ ⊗ ˜ M sji ) − ( ˜ M sij · ˜ M sji ) ⊗ − ξβ ( M s +1 ij ( ˜ M s +1 ji · ˜ D si − ˜ D sj · ˜ M sji )) − ξ ˇ d (( ˜ D si ⊗ − ⊗ ˜ D si ) i ) ij + ( − ξ (( ˜ M s +1 ij ⊗ ⊗ ˜ M s +1 ji ) − ( ˜ M s +1 ij · ˜ M s +1 ji ) ⊗ d E = ξ [( D si ⊗ M sij ⊗ ⊗ ˜ M sji ) − ( ˜ M sij · ˜ M sji ) ⊗ − ( ˜ M s +1 ij · ˜ M s +1 ji · ˜ D si ) ⊗ ⊗ ( ˜ M s +1 ij · ˜ M s +1 ji · ˜ D si ) + ( ˜ M s +1 ij · ˜ D sj · ˜ M sji ) ⊗ − ⊗ ( ˜ M s +1 ij · ˜ D sj · ˜ M sji ) − ( M s +1 ij ⊗ D sj ⊗ − ⊗ ˜ D sj )(1 ⊗ M sji )+˜ D si ⊗ − ⊗ ˜ D si − (( ˜ M s +1 ij ⊗ ⊗ ˜ M s +1 ji ) − ( ˜ M s +1 ij · ˜ M s +1 ji ) ⊗ ⊗ D si )] = ξ [(( ˜ D si · ˜ M sij ) ⊗ ⊗ ˜ M sji ) − ( ˜ D si · ˜ M sij · ˜ M sji ) ⊗ − ( ˜ M s +1 ij · ˜ M s +1 ji · ˜ D si ) ⊗ ⊗ ( ˜ M s +1 ij · ˜ M s +1 ji · ˜ D si ) + ( ˜ M s +1 ij · ˜ D sj · ˜ M sji ) ⊗ − ⊗ ( ˜ M s +1 ij · ˜ D sj · ˜ M sji ) − (( ˜ M s +1 ij · ˜ D sj ) ⊗ ⊗ ˜ M sji ) + ( ˜ M s +1 ij ⊗ ⊗ ( ˜ D sj · ˜ M sji )) + ˜ D si ⊗ − ⊗ ˜ D si − ( ˜ M s +1 ij ⊗ ⊗ ( ˜ M s +1 ji · ˜ D si )) + (( ˜ M s +1 ij · ˜ M s +1 ji ) ⊗ ⊗ ˜ D si )] = ξ [(( − M s +1 ij · M s +1 ji ) ⊗ D si ⊗ − ⊗ ˜ D si )+( ˜ M s +1 ij ⊗ − ⊗ ˜ M s +1 ij )(1 ⊗ ( D sj · M sji − M s +1 ji · D si ))+(( D si · M sij − M s +1 ij · D sj ) ⊗ ⊗ ˜ M sji − ˜ M sji ⊗ . For the tenth summand, we compute d E ( − ξ ) + ξα ( ˜ D si ⊗ − ⊗ ˜ D si ) + ( ξ ) d E = ξ [ − ( D si ⊗ + D si ⊗ − ⊗ D si + (1 ⊗ D si )] = 0 . Finally, we get d E ( ˜ D si ⊗ − ⊗ ˜ D si ) − ξβ ( ξ ˜ D s +1 i · ˜ D si ) + ( ˜ D s +1 i ⊗ − ⊗ ˜ D s +1 i ) d E =( D s +1 i ⊗ D si ⊗ − ⊗ ˜ D si ) − ( ˜ D s +1 i · ˜ D si ) ⊗ ⊗ ( ˜ D s +1 i · ˜ D si )+( ˜ D s +1 i ⊗ − ⊗ ˜ D s +1 i )(1 ⊗ D si ) = 0for the thirteenth summand. (cid:3) .3 The classical Atiyah class of a complex of vector bundles In the last subsection, we found a description of the truncated Atiyah class. There is a corre-sponding description of the classical class, found by
Ang´eniol and
Lejeune-Jalabert in [AL, Prop.II.1.5.2.1]. Since the truncated class lifts the classical one, our formula allows us to recover thealready known one.As in the truncated case, morphisms E s → ( E ⊗ C (Ω X [1])) s that factorize over the summand E s ⊗ C (Ω X ) ⊕ E s +1 ⊗ C (Ω X ) of ( E ⊗ C (Ω X [1])) s can be constructed by giving matrices (cid:16)(cid:0) M sij · dM sji (cid:1) ij , (cid:0) ( − s +1 dD si (cid:1) i (cid:17) . (3.5) Corollary 3.5 (Ang´eniol, Lejeune-Jalabert):
In situation 3.1, the maps (3.5) define a mor-phism of complexes
E → E ⊗ C (Ω X [1]) . The induced morphism E → E ⊗ Ω X [1] in the derivedcategory is the classical Atiyah class of E .Proof: The diagram
E ⊗ C ( L X [1]) (cid:15) (cid:15) E ⊗ L X [1] (cid:15) (cid:15) o o E ( ) sssssssssss ( ) % % KKKKKKKKKKK
E ⊗ C (Ω X [1]) E ⊗ Ω X [1] o o is commutative by definition of the morphism L X → Ω X and the maps (3.1) and (3.5). ThusLemma 2.3 gives the assertion of the corollary. (cid:3) The formula given in [AL, Prop. II.1.5.2.1] and our result (3.5) differ by signs. Apart from thedifferent conventions (that we discussed at the end of the introduction), there is another reasonfor this. Whereas we calculate with the complex
E ⊗ C (Ω X [1]) the complex C (Ω X ⊗ E )[1] is usedin [AL]. There is a natural isomorphism between these complexes bringing further signs into play. Via a trace map, the classical Atiyah class of a perfect complex gives rise to an element in the firstcohomology group of the cotangent sheaf, called the first Chern class. Similarly, the truncatedAtiyah class induces an element in the first cohomology group of truncated cotangent complex,the first truncated Chern class. This class will be investigated in this section. We will find outthat the basic properties of the classical Chern class can also be proven for the truncated version.At the end of the section, we will discuss an application.
First, we have to recall the trace map, introduced by
Illusie in [SGA6, Sect. I.8]. In [Ill, Sect.V.3], he generalizes his construction in order to define Chern classes. We need the notion of aperfect complex. Recall that a complex E of sheaves on a scheme X is said to be perfect if everypoint of X is contained in an open set Y ⊆ X on which there exist a bounded complex F of finitefree sheaves and a quasi-isomorphism F → E| Y of complexes on Y .Let X be a scheme, let E be a perfect complex, and let G be a bounded complex of sheaves on X . In the derived category D (Mod( X )), there are a natural isomorphism R H om ( E , E ⊗ L G ) ∼ ←−E ∨ ⊗ L E ⊗ L G (see [SGA6, Cor. I.7.7]) and a map E ∨ ⊗ L E ⊗ L G → G which is induced by thecontraction morphism E ∨ ⊗ L E → O X (see [SGA6, Lem. I.7.5]), where we use the abbreviation E ∨ = R H om ( E , O X ). The composition R H om ( E , E ⊗ L G ) → G gives the trace maptr : Hom( E , E ⊗ L G ) → Hom( O X , G )14fter application of the functor Hom( O X , ) and in view of the isomorphism Hom( E , E ⊗ L G ) ∼ −→ Hom( O X , R H om ( E , E ⊗ L G )) given by adjunction (see [SGA6, Lem. I.7.4]).This definition of the trace map is complicated. The following lemma collects all its propertiesthat are relevant for us. Lemma 4.1 (Illusie):
Let X be a scheme. Let E and F be perfect complexes, and let G and H be bounded complexes of sheaves on X . Let µ : E → E ⊗ L G be a morphism in D (Mod( X )) .(i) If E is a bounded complex of locally free coherent sheaves, and µ is a morphism of complexes,given by maps µ st : E s → E t ⊗G s − t of sheaves, then tr( µ ) is a morphism of complexes as well,given by P s ( − s tr( µ ss ) : O X → G . The term tr( µ ss ) is to be understood in the followingway: µ ss corresponds to a map O X → ( E s ) ∨ ⊗ E s ⊗ G . This map has to be composed withthe morphism ( E s ) ∨ ⊗ E s ⊗ G → G induced by the usual trace map ( E s ) ∨ ⊗ E s → O X .(ii) If χ : E → F is an isomorphism in D (Mod( X )) , and ν := ( χ ⊗ L id) ◦ µ ◦ χ − : F → F ⊗ L G is the morphism induced by µ , then tr( µ ) = tr( ν ) .(iii) If τ : G → H is a morphism in D (Mod( X )) , then the diagram G τ (cid:15) (cid:15) O X tr( µ ) ppppppppppppp tr(( E⊗ L τ ) ◦ µ ) ' ' NNNNNNNNNNNNN H commutes.Proof: Statement (i) is proved in [SGA6, I.8.1.2] in slightly weaker generality. Statements (ii) and(iii) are discussed in [Ill, V.3.8.11.1] and [Ill, V.3.7.4.1]. (cid:3)
We are interested in the trace of the truncated Atiyah class.
Definition:
Let X be a separated S -scheme of finite type, and let E be a perfect complex on X .Then c ,cl ( E ) := tr( At cl ( E )) ∈ H ( X, Ω X ) is said to be the first (classical) Chern class of E . If X is flat with a smooth ambient space (over S ), then c ( E ) := tr( At ( E )) ∈ H ( X, L X ) is said to bethe first truncated Chern class of E .From Lemma 2.3 and part (iii) of Lemma 4.1, it follows that in the situation of the definition (withflat X with smooth ambient space), the first classical Chern class is the composition c ,cl ( E ) : O X c ( E ) −−−→ L X [1] → Ω X [1]. Thus via the natural map H ( X, L X ) → H ( X, Ω X ), the truncatedclass lifts the classical one. For bounded complexes of vector bundles, we can find an explicit formula for their first truncatedChern class. It can be derived easily from the results of the last section. Therefore, for thissubsection, we return to the situation 3.1.In particular, we consider a flat S -scheme X with a smooth ambient space X ֒ → U and a boundedcomplex E of locally free coherent sheaves, for which we have chosen a suitable cover U = S Γ U i and trivializations of all components with transition maps M sij ∈ GL( m si , A ij /J ij ) and lifts ˜ M sij ∈ M at ( m si × m si , A ij ). For the local differentials D si ∈ M at ( m s +1 i × m si , A i /J i ), we have also chosenlifts ˜ D si ∈ M at ( m s +1 i × m si , A i ). 15 heorem 4.2: In situation 3.1, the first truncated Chern class of E is given by the ˇCech cocycle X s ( − s tr( M sik ( ˜ M skj · ˜ M sji − ˜ M ski )) ! ijk , X s ( − s tr( M sij · d ˜ M sji ) ! ij (4.1) ∈ H ( X, C ( L X [1])) = H ( X, C ( J / J )) ⊕ H ( X, C (Ω U | X )) . More precisely, the formula (4.1)defines a morphism O X → C ( L X [1]) , and the resulting diagram O X → C ( L X [1]) ← L X [1] describesthe first truncated Chern class of E .Proof: The truncated Atiyah class of E is given by the diagram E µ −→ E ⊗ C ( L X [1]) ← E ⊗ L X [1],where µ is the map (3.1). Part (i) of Lemma 4.1 shows that the formula (4.1) describes the traceof µ . The assertion of the theorem follows from part (iii) of the same lemma. (cid:3) As in the case of the Atiyah class, we can recover the already known formula for the classicalChern class from (4.1). The statement follows from Theorem 4.2 because the truncated class liftsthe classical one.
Corollary 4.3 (Ang´eniol, Lejeune-Jalabert):
In situation 3.1, the first classical Chern classof E is given by the ˇCech cocycle X s ( − s tr( M sij · dM sji ) ! ij ∈ H ( X, C (Ω X [1])) = H ( X, C (Ω X )) . Many formulas are known that relate classical Chern classes of associated vector bundles. Here,we want to derive similar formulas for truncated Chern classes.
Theorem 4.4:
Let X be a flat S -scheme with a smooth ambient space. Then the first truncatedChern class gives rise to a group homomorphism c : Pic( X ) → H ( X, L X ) .Proof: Let M and N be line bundles on X . By choosing a smooth ambient space, a suitable cover,and trivializations of the line bundles, we can achieve a situation as in the previous section, andwe stick to the notation used there. We denote the transition maps of M and N by M ij and N ij and the chosen lifts by ˜ M ij and ˜ N ij . With respect to the induced trivialization of M ⊗ N , thetransition maps are given by M ij · N ij , and we can choose the lifts ˜ M ij · ˜ N ij .By (4.1), the first truncated Chern classes of M and N are given by (cid:18)(cid:16) M ik ( ˜ M kj · ˜ M ji − ˜ M ki ) (cid:17) ijk , (cid:16) M ij · d ˜ M ji (cid:17) ij (cid:19) and (cid:18)(cid:16) N ik ( ˜ N kj · ˜ N ji − ˜ N ki ) (cid:17) ijk , (cid:16) N ij · d ˜ N ji (cid:17) ij (cid:19) and those of M ⊗ N by (cid:18)(cid:16) M ik · N ik ( ˜ M kj · ˜ N kj · ˜ M ji · ˜ N ji − ˜ M ki · ˜ N ki ) (cid:17) ijk , (cid:16) M ij · N ij · d ( ˜ M ji · ˜ N ji ) (cid:17) ij (cid:19) . To prove the theorem, we will show that the latter cocycle is the sum of the other two. We have M ik ( ˜ M kj · ˜ M ji − ˜ M ki ) + N ik ( ˜ N kj · ˜ N ji − ˜ N ki ) − M ik · N ik ( ˜ M kj · ˜ N kj · ˜ M ji · ˜ N ji − ˜ M ki · ˜ N ki ) =( ˜ M ik · ˜ M ki − N ik · ˜ N ki − − ( ˜ M ik · ˜ M kj · ˜ M ji − N ik · ˜ N kj · ˜ N ji −
1) = 016because the product of two elements of J ijk lies in J ijk ) and M ij · d ˜ M ji + N ij · d ˜ N ji = M ij · N ij · N ji · d ˜ M ji + M ij · N ij · M ji · d ˜ N ji = M ij · N ij · d ( ˜ M ji · ˜ N ji ) . (cid:3) We can associate to every bounded complex E of locally free coherent sheaves on a scheme X aline bundle, called its determinant det( E ) := N s det( E s ) ( − s . The following main theorem of thissection asserts that the first truncated Chern class of E only depends on its determinant. Theorem 4.5:
Let X be a flat S -scheme with a smooth ambient space. Let E be a bounded complexof locally free coherent sheaves on X . Then for the first truncated Chern class, we have c ( E ) = c (det( E )) . Proof:
By choosing a smooth ambient space, a suitable cover, and trivializations, we can assumethat we are in situation 3.1. The formula (4.1) shows that c ( E ) = P s ( − s c ( E s ), and becauseof the additivity of the first truncated Chern class on the Picard group, we have c (det( E )) = c ( N s det( E s ) ( − s ) = P s ( − s c (det( E s )). Thus we may assume that E is a single locally freecoherent sheaf.For the transition maps M ij , we have chosen lifts ˜ M ij . For the determinant of E , there exist triv-ializations, with respect to which the transition maps are det( M ij ). We choose the lifts det( ˜ M ij ).By (4.1), the first Chern classes of E and det( E ) are given by (cid:16) tr( M ik ( ˜ M kj · ˜ M ji − ˜ M ki )) (cid:17) ijk , X u,v ( M ij ) uv · d ( ˜ M ji ) vu ! ij and (cid:18)(cid:16) det( M ik )(det( ˜ M kj ) · det( ˜ M ji ) − det( ˜ M ki )) (cid:17) ijk , (cid:16) det( M ij ) · d (det( ˜ M ji )) (cid:17) ij (cid:19) . We will prove that these cocycles coincide. To this end, we need the following simple result. Let A be a ring with ideal J and M ∈ Mat( m × m, A ) a matrix for which the entries of M − all liein J . Then in A/J , we havedet( M ) = X σ ∈ S m sgn ( σ ) Y u M u,σ ( u ) = Y u M uu = Y u (1 + ( M uu − X u ( M uu −
1) = tr( M ) − ( m − . We use this result to prove the first of the required calculations:tr( M ik ( ˜ M kj · ˜ M ji − ˜ M ki )) = tr( ˜ M ik · ˜ M kj · ˜ M ji ) − tr( ˜ M ik · ˜ M ki ) =det( ˜ M ik · ˜ M kj · ˜ M ji ) − det( ˜ M ik · ˜ M ki ) = det( M ik )(det( ˜ M kj ) · det( ˜ M ji ) − det( ˜ M ki )) . For an algebra A over a ring k and a matrix M ∈ Mat( m × m, A ), we denote by M uv the matrixthat results from M by replacing the entry ( u, v ) by 1 and the other entries in row u or column v by 0. It suffices to prove the formula X u,v det( M vu ) · dM vu = d (det( M )) (4.2)17n Ω A | k in this situation, for in our case M = ˜ M ji , this formula implies the required equality whenwe use Cramer’s rule: X u,v ( M ij ) uv · d ( ˜ M ji ) vu = X u,v det( M ij ) · det(( M ji ) vu ) · d ( ˜ M ji ) vu = det( M ij ) · d (det( ˜ M ji )) . We prove (4.2) by induction with respect to m , where the statement is clear for m = 1. For m > d (det( M )) = d ( X n M n · det( M n )) = X n det( M n ) · d ( M n ) + X n M n · d (det( M n )) = X n det( M n ) · d ( M n ) + X n M n X u = n,v =1 det(( M n ) vu ) · d ( M vu ) = X u det( M u ) · d ( M u ) + X v =1 ,u ( X n = u M n · det(( M vu ) n )) d ( M vu ) = X u,v det( M vu ) · dM vu , where we use Laplace’s formula for the first and the last equality. To obtain the third equality,the induction hypothesis is applied to matrices that result from M by deleting one row and onecolumn. (cid:3) To explain the following theorem, we have to discuss the rank of a bounded complex E of finitelocally free sheaves on a scheme X . For a point x ∈ X , the number rk x ( E ) := P s ( − s rk x E s is called the rank of E in x . If this number is the same for all points of X , we say that E is ofconstant rank. If X is connected, then E obviously is of constant rank. Theorem 4.6:
Let X be a flat S -scheme with smooth ambient space. Let E , F , and G be boundedcomplexes of locally free coherent sheaves on X . Then for the first truncated Chern classes, wehave c ( E ∨ ) = − c ( E ) (4.3) c ( E ⊕ F ) = c ( E ) + c ( F ) . (4.4) If there exists a short exact sequence → E → G → F → , then c ( G ) = c ( E ) + c ( F ) . (4.5) If E and F are of constant rank, then c ( E ⊗ F ) = rk( F ) · c ( E ) + rk( E ) · c ( F ) (4.6) c ( H om ( E , F )) = − rk( F ) · c ( E ) + rk( E ) · c ( F ) . (4.7) Proof:
To prove (4.3), we use the previous two theorems and find c ( E ∨ ) = c (det( E ∨ )) = c (det( E ) ∨ ) = − c (det( E )) = − c ( E ).The equation (4.5) follows analogously from the calculation c ( G ) = c (det( G )) = c (det( E ) ⊗ det( F )) = c (det( E )) + c (det( F )) = c ( E ) + c ( F ), and it implies (4.4) as a special case.Again using the previous theorems, we see that it suffices to provedet( E ⊗ F ) = det( E ) ⊗ rk( F ) ⊗ det( F ) ⊗ rk( E ) (4.8)to obtain (4.6). Since it is more difficult to prove this formula than its analogues above, werecall its proof. To this end, we can restrict to the connected components of X . Thus we canassume that the components of E and F are globally of constant rank. To prove (4.8) for vector18undles E and F , we fix trivializations of these sheaves on a suitable cover, and we compare theinduced transition maps of the line bundles on both sides of the equation. For arbitrary squarematrices M and N of size m and n with entries in a ring, however, we really have det( M ⊗ N ) =det(( M ⊗ )( ⊗ N )) = det( M ⊗ ) · det( ⊗ N ) = det( M ) n · det( N ) m .From this, we can deduce the statement in the general case of complexes:det( E ⊗ F ) = O u det(( E ⊗ F ) u ) ( − u = O u det( M s + t = u E s ⊗ F t ) ( − u = O s,t det( E s ⊗ F t ) ( − s + t = O s,t (det( E s ) ⊗ rk( F t ) ⊗ det( F t ) ⊗ rk( E s ) ) ( − s + t = O s det( E s ) ⊗ ( P t ( − s + t rk( F t )) ⊗ O t det( F t ) ⊗ ( P s ( − s + t rk( E s )) = O s det( E s ) ⊗ ( − s rk( F ) ⊗ O t det( F t ) ⊗ ( − t rk( E ) = det( E ) ⊗ rk( F ) ⊗ det( F ) ⊗ rk( E ) . Finally, the statement (4.7) follows from (4.3) and (4.6) if we use the isomorphism H om ( E , F ) ∼ = E ∨ ⊗ F . (cid:3) We defined the first truncated Chern class for all perfect complexes, but only considered it inthe case of bounded complexes of vector bundles up to now. The reason is that only for thissmaller class of complexes, we have the formula (4.1) at our disposal. Although perfect complexesare locally isomorphic to complexes of this form, we cannot control their (globally defined) firsttruncated Chern classes with our methods. In order to derive statements as in the previoussubsection for arbitrary perfect complexes, technically more complicated methods as in [Ill, Chapt.V] would probably be necessary.Nevertheless, many schemes have the property that every perfect complex is even globally iso-morphic to a bounded complex of vector bundles. In this case, our results can immediately beextended to perfect complexes. We will formulate these slightly more general statements in thissubsection.In this context, we will speak about determinants of perfect complexes, see [KM]. We will considerthe determinant of a perfect complex only as an element in the Picard group. The following lemmacollects the most important properties (see [KM, Def. 4, Thm. 2, Sect. before Prop. 6]).
Lemma 4.7 (Knudsen, Mumford):
Let X be a scheme, E , F , and G be perfect complexes on X . If E is a bounded complex of locally free coherent sheaves, then det( E ) = N s det( E s ) ( − s . If E and F are isomorphic in D (Mod( X )) , then det( E ) = det( F ) . If there exists a distinguishedtriangle E → F → G → E [1] , then det( F ) = det( E ) ⊗ det( G ) . For bounded complexes of finite locally free sheaves, we introduced the rank in a point. Perfectcomplexes are locally isomorphic to such complexes. Thus the rank in a point is also defined forthem. (To see that this notion is well-defined, we have to understand that two bounded complexes E and F of free finite modules which are isomorphic in the derived category of a local ring (thelocal ring in the point considered) have the same rank. After reduction to the case of a quasi-isomorphism E → F , we consider its mapping cone G . By construction of the cone, it suffices toprove rk( G ) = 0. Since G is an exact complex of free finite modules, the images of all differentialsare projective and thus free (the ring is local). Hence G is split and exact.) All perfect complexeson connected schemes are of constant rank.Now, we can easily generalize the results of the previous subsection (use (4.9) for the proof of(4.10)). Corollary 4.8:
Let X be a flat S -scheme with smooth ambient space. Let E , F , and G be perfectcomplexes on X which are isomorphic to bounded complexes of locally free coherent sheaves in (Mod( X )) . Then for the first truncated Chern classes, we have c ( E ) = c (det( E )) (4.9) c ( E ∨ ) = − c ( E ) c ( E ⊕ F ) = c ( E ) + c ( F ) . If there exists a distinguished triangle
E → F → G → E [1] , then c ( F ) = c ( E ) + c ( G ) . (4.10) If E and F are of constant rank, then c ( E ⊗ L F ) = rk( F ) · c ( E ) + rk( E ) · c ( F ) c ( R H om ( E , F )) = − rk( F ) · c ( E ) + rk( E ) · c ( F ) . If all perfect complexes on our scheme X are isomorphic to bounded complexes of vector bundles,then the assumption of Corollary 4.8 is trivially satisfied. Therefore, we are interested in situationswhere this is the case. We will cite an interesting theorem due to Illusie in this context. It statesthat the large class of divisorial schemes has the property mentioned. Recall that a separatedquasi-compact scheme X is said to be divisorial if the open sets X s := { x ∈ X : s ( x ) = 0 } forarbitrary global sections s of arbitrary line bundles on X form a basis of the topology.Let A be a noetherian ring and X a quasi-projective scheme over Spec( A ). We show that in thissituation, X is divisorial. Let x ∈ X , and let Y ⊆ X be an open neighbourhood of x . If we denotethe ideal sheaf of X \ Y (with the induced reduced scheme structure) by I , and if M is an ampleline bundle, then there exists a natural number N for which I ⊗ M ⊗ N is globally generated. Inparticular, there is a global section s with s ( x ) = 0. Via the inclusion I ⊆ O X , we can consider itals a global section of the line bundle M ⊗ N . This section has the required property x ∈ X s ⊆ Y . Theorem 4.9 (Illusie):
Let X be a separated, quasi-compact, and divisorial scheme (e.g. aquasi-projective scheme over a noetherian ring). Then every perfect complex on X is isomorphicto a bounded complex of locally free coherent sheaves in D (Mod( X )) .Proof: A proof can be found in [TT, Prop. 2.3.1]. Originally, the theorem was proven in [SGA6,Chapt. II]. (cid:3) If S is quasi-projective over an affine noetherian scheme, and X is flat and quasi-projective over S , then the theorem shows that X is divisorial, and our Corollary 4.8 holds true for all perfectcomplexes. In this situation, the commutative diagram K (Perf( X )) det / / c ' ' OOOOOOOOOOO
Pic( X ) c x x rrrrrrrrrr H ( X, L X )of abelian groups summarizes the main results of this section. Here, Perf( X ) denotes the trian-gulated (see [SGA6, Prop. I.4.10]) category of perfect complexes on X , and K (Perf( X )) is itsGrothendieck group. In this final subsection, we want to discuss a consequence of our Theorem 4.5 concerning the paper[HT].We consider a closed immersion i : X ֒ → Y of flat S -schemes with smooth ambient spaces, and weassume that the ideal sheaf I of this immersion satisfies I = 0. We are interested in the question20hether given perfect complexes E on X extend to perfect complexes F on Y in the sense that Li ∗ ( F ) ∼ = E . We call such extensions deformations of E .It turns out that the truncated Atiyah class of E can be used to give an answer to this question.Furthermore, the truncated Kodaira–Spencer class κ ( i ) : L X → I [1] (see [HT, Def. 2.7]) is useful— note that I can be considered as sheaf on X because of I = 0. Theorem 4.10 (Huybrechts, Thomas):
Let X and Y be flat S -schemes with smooth ambientspaces and i : X ֒ → Y a closed immersion whose ideal sheaf I satisfies I = 0 . Let E be a perfectcomplex on X . Then the following statements are equivalent:(i) There exists a perfect complex F on Y with Li ∗ ( F ) ∼ = E .(ii) The map ω ( E ) := ( E ⊗ L κ ( i )[1]) ◦ At ( E ) : E → E ⊗ L I [2] is the zero morphism.Proof: The theorem is proven in [HT, Cor. 3.4]. (cid:3)
For a perfect complex E on X , the morphism ω ( E ) is called the obstruction class of E . We canuse our results to relate the obstruction class of a perfect class to the obstruction class of itsdeterminant. In this way, we give a more natural proof for a statement that Huybrechts and
Thomas use in [HT].
Theorem 4.11:
Let X and Y be flat S -schemes with smooth ambient spaces and i : X ֒ → Y aclosed immersion whose ideal sheaf I satisfies I = 0 . Let E be a perfect complex on X that isisomorphic to a bounded complex of locally free coherent sheaves in D (Mod( X )) (e.g. an arbitraryperfect complex if X is divisorial). Then ω (det( E )) = tr( ω ( E )) . Proof:
For the equality, we make use of the natural isomorphism Hom(det( E ) , det( E ) ⊗ L I [2]) ∼ =Hom( O X , I [2]). The precise formulation of the formula of the theorem istr( ω (det( E ))) = tr( ω ( E )) . We have tr( ω ( E )) = tr(( E ⊗ L κ ( i )[1]) ◦ At ( E )) = κ ( i )[1] ◦ tr( At ( E )) = κ ( i )[1] ◦ c ( E ), where weuse part (iii) of Lemma 4.1 for the second equality. Correspondingly, we have tr( ω (det( E ))) = κ ( i )[1] ◦ c (det( E )). An application of formula (4.9) finishes the proof. (cid:3) References [AL] B. ANG´ENIOL, M. LEJEUNE-JALABERT:
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