, For sufficiently large k (usually either greater than either r(E) or dim X), the classes are zero

/. Naturality and . Functoriality, For any continuous f : Y ? X, the classes respect pullbacks along f

, Whitney sum/product formula. The class of a direct sum of bundles is the cup product of the classes of each summand: ?(E ? F) = ?(E) ?(F)

, The class of a bundle of rank 1 is forced to be something 'simple

, In a more abstract sense, stability refers to some sort of invariance under the inclusion BG(n) ? BG(n + 1)

X. and O. , the category QCoh(X) of quasicoherent sheaves of O X -modules on X is a full abelian subcategory of the category Sh(X) of sheaves of O X -modules on X, and its objects are those that correspond locally to modules over a ring. If X is further assumed to be Noetherian, then the category Coh(X) of coherent sheaves of O X -modules on X is a full abelian subcategory of QCoh(X), and its objects are those that correspond locally to finitely generated modules. By definition, we see that the full subcategory Pf(X) of D(Sh(X)) is contained inside D coh (Sh(X)), which is the full subcategory of D(Sh(X)) spanned by objects whose internal

(. E. , 1.2) Proposition. Let X be a Noetherian scheme, Then the canonical fully faithful functor D b (Coh(X)) ? D

, Coh(X)) with the full subcategory D b coh (Sh(X)) of D(Sh(X))

, Then there is a canonical equivalence of triangulated categories Pf(X) ? ? ? ? D b (Coh(X))

, This follows from

, 1.4) Morally, then, when we want to work with coherent algebraic sheaves, if X is nice enough, then working with perfect complexes, complexes of coherent sheaves, or complexes of sheaves with coherent cohomology are all equivalent. (E.1.5) A nice summary of a lot of the fundamental results of

, ) For coherent analytic sheaves, the story is more subtle: as far as the author is aware, the question of whether or not D b (Coh(X)) and D b coh (Sh(X)) are equivalent is still open

, Sh where ? 0 : St ? Sh gives the sheaf of connected components and j 0 : Sh ? St sends a sheaf of sets to the corresponding presheaf of discrete simplicial sets. Further, j 0 is fully faithful and its essential image [1] consists of 0-truncated stacks, 3.1) Lemma. There is an adjunction St

, sions i : U ? V and j : V ? W of open subsets, the morphism ? ij : (ji) * ? i * j * is only required to be an isomorphism, not necessarily the identity. We content ourselves in saying that a 1-stack in groupoids X is a fibred category X ? Grpd of groupoids that satisfies some extra descent conditions (whatever this might mean, formally). The morphisms between two 1-stacks in groupoids are just commutative diagrams of functors, and we say that two such morphisms are homotopic if there exists a natural isomorphism between them that is compatible with the projection to Grpd. We write 1?St to mean the category of 1-stacks and Ho, F.4 1-stacks-in-groupoids and stacks (F.4.1) A fibred category is a presheaf of groupoids 'up to isomorphism': for inclu

, We can replace a fibred category of groupoids by an actual presheaf of groupoids in a functorial way (the so-called strictification of fibred categories)

, ?(X) given by ?(X)(X) := Hom Top

. Lemma, Writing BG to mean the classifying simplicial set of a groupoid G, we have the composition B? : 1-St ? sPrSh ? St

, These are the stacks A such that ? n (A x ; y) = 0 for all X ? Top, x ? X, y ? A x, p.1

, We are implicitly using the equivalence between Top and Grpd

, We defined St as a localisation along weak equivalences, so taking the homotopy category changes nothing

?. Top, 3.6) Now to explain why we consider maps of the form ? a in the first place. The primary justification in the context of [TT76] is that such maps occur naturally in various situations, and so it's nice to abstractify it. The justification for us, here, is (once more) largely based on (as of yet) unfounded analogy: this looks like the local definition of a connection, in that it is a connection ? (of curvature ?) plus some operator of differential degree 0 and total ('algebraic') degree 1
URL : https://hal.archives-ouvertes.fr/hal-01230391

, The associated twisted complex (M, ? a ) is the complex given by M with the differential ? a : m ? ?m + a · m

, Triangulated categories can be understood as some sort of zero truncation of particularly nice dg-categories (as we explain in more detail in (G.4.2)), and so we can ask whether or not we can enrich a specific triangulated category with some dg structure. In particular, we can ask whether or not the (derived) category of perfect complexes on a (sufficiently nice) scheme admits such an enrichment. Formally, given some triangulated category T , and a pair (P, ), where P is a stable [1] dg-category, and : Ho P ? T is a functor of triangulated categories

, 2) We can explain the relationship between triangulated categories and other higher-categorical structures in a bit more detail. In particular, triangulated categories are the H 0 of stable dg-categories, which are semi-strictifications of A ? -categories

, Also known as an enhanced triangulated, or pre-triangulated, dg-category

, It turns out that, for both questions, A is given exactly by the category of twisted complexes over A. The embedding A ? A lets us 'pull back' the shift and cones from A to ones in A. Even nicer, it turns out that the homotopy category of A is triangulated with this shift functor and these cones. Further, if A is pre-triangulated, then this embedding A ? A is a quasiequivalence [1] of dg-categories. This means that, if A does already have some pre-triangulated structure, then we don't really change anything about it, One of the motivations behind Definition (G.6.2) is the following question: "given a dg-category A, what is the 'smallest' dg-category A , into which A embeds, such that we can define shifts and functorial cones in A ?" Or, thinking about

?. and E. , Decomposing a into a k,1?k ?? k (U , End 1?k (V )) we recover exactly the Maurer-Cartan condition for twisting cochains. Thus any holomorphic twisting cochain gives us a twisted complex in the dg-category of B-dg-modules. It is not the case, however, that by picking the 'right' dg-category A, we recover the definition of holomorphic twisting cochains from that of twisted complexes. In particular, twisting cochains are twisted complexes with only one non-zero object. Further, this object E 0 is not an arbitrary B-module: it comes from? ? (U , V ), where V is graded

, That is, it induces an equivalence on the level of homotopy categories

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