1,721,083 research outputs found
Extensions of picard 2-stacks and the cohomology groups Exti of length 3 complexes
No full textThe aim of this paper is to define and study the 3-category of extensions of Picard 2-stacks over a site S and to furnish a geometrical description of the cohomology groups Exti of length 3 complexes of abelian sheaves. More precisely, our main theorem furnishes (1) a parametrization of the equivalence classes of objects, 1-arrows, 2-arrows, and 3-arrows of the 3-category of extensions of Picard 2-stacks by the cohomology groups Exti, and (2) a geometrical description of the cohomology groups Exti of length 3 complexes of abelian sheaves via extensions of Picard 2-stacks. To this end, we use the triequivalence between the 3-category 2Picard(S) of Picard 2-stacks and the tricategory T[−2,0] (S) of length 3 complexes of abelian sheaves over S introduced by the second author in [12], and we define the notion of extension in this tricategory T[−2,0] (S), getting a pure algebraic analog of the 3-category of extensions of Picard 2-stacks. The calculus of fractions that we use to define extensions in the tricategory T[−2,0] (S) plays a central role in the proof of our main theorem
Multilinear morphisms between 1-motives
Full text linkWe introduce the notion of biextensions of 1-motives over an arbitrary scheme S and we define bilinear morphisms between 1-motives as isomorphism classes of such biextensions. If S is the spectrum of a field of characteristic 0, we check that these biextensions define bilinear morphisms between the realizations of 1-motives. Generalizing we obtain the notion of multilinear morphisms between 1-motives. © 2009 Walter de Gruyter Berlin - New York
G-fonctions et cohomologie des hypersurfaces singulières
Full text linkOur object of study is the arithmetic of the differential modules W(l) (l ∈ N - {0}), associated by Dwork's theory to a homogeneous polynomial f(λ, X) with coefficients in a number field. Our main result is that W(l) is a differential module of type G, c'est-à-dire, a module whose solutions are G-functions. For the proof we distinguish two cases: the regular one and the non regular one. Our method gives us an effective upper bound for the global radius of W(l), which doesn't depend on "l" but only on the polynomial f(λ, X). This upper bound is interesting because it gives an explicit estimate for the coefficients of the solutions of W(l). In the regular case we know there is an isomorphism of differential modules between W(1) and a certain De Rham cohomology group, endowed with the GaussManin connection, c'est-à-dire, our module "comes from geometry". Therefore our main result is a particular case of André's theorem which assert that at least in the regular case, all modules coming from geometry are of type G
Périodes de 1-motifs et transcendance
Full text linkAbstractThe generalized Grothendieck's conjecture of periods (CPG)K predicts that if M is a 1-motive defined over an algebraically closed subfield K of C, then deg.transcQK(périodes(M))⩾dimQMT(MC). In this article we propose a conjecture of transcendance that we call the elliptico-toric conjecture (CET). Our main result is that (CET) is equivalent to (CPG)K applied to 1-motives defined over K of the kind M=[Zr∏j=1nEj×Gms]. (CET) implies some classical conjectures, as the Schanuel's conjecture or its elliptic analogue, but it implies new conjectures as well. All these conjectures following from (CET) are equivalent to (CPG)K applied to well chosed 1-motives: for example the Schanuel's conjecture is equivalent to (CPG)K applied to 1-motives of the kind M=[ZrGms]
Biextensions of Picard stacks and their homological interpretation
Full text linkAbstractLet S be a site. We introduce the 2-category of biextensions of strictly commutative Picard S-stacks. We define the pull-back, the push-down, and the sum of such biextensions and we compute their homological interpretation: if P,Q and G are strictly commutative Picard S-stacks, the equivalence classes of biextensions of (P,Q) by G are parametrized by the cohomology group Ext1([P]⊗L[Q],[G]), the isomorphism classes of arrows from such a biextension to itself are parametrized by the cohomology group Ext0([P]⊗L[Q],[G]) and the automorphisms of an arrow from such a biextension to itself are parametrized by the cohomology group Ext−1([P]⊗L[Q],[G]), where [P],[Q] and [G] are the complexes associated to P,Q and G respectively
Third kind elliptic integrals and 1-motives
Full text linkIn [4] we have showed that the Generalized Grothendieck's Period Conjecture applied to 1-motives, whose underlying semi-abelian variety is a product of elliptic curves and of tori, is equivalent to a transcendental conjecture involving elliptic integrals of the first and second kind, and logarithms of complex numbers. In this paper we investigate the Generalized Grothendieck's Period Conjecture in the case of a 1-motive M whose underlying semi-abelian variety is a non trivial extension of a product of elliptic curves by a torus. This will imply the introduction of elliptic integrals of the third kind for the computation of the periods of M and therefore the Generalized Grothendieck's Period Conjecture applied to M will be equivalent to a transcendental conjecture involving elliptic integrals of the first, second and third kind
Homological interpretation of extensions and biextensions of 1-motives
Full text linkAbstractLet k be a separably closed field. Let Ki=[Ai→uiBi] (for i=1,2,3) be three 1-motives defined over k. We define the geometrical notions of extension of K1 by K3 and of biextension of (K1,K2) by K3. We then compute the homological interpretation of these new geometrical notions: namely, the group Biext0(K1,K2;K3) of automorphisms of any biextension of (K1,K2) by K3 is canonically isomorphic to the group Ext0(K1⊗LK2,K3), and the group Biext1(K1,K2;K3) of isomorphism classes of biextensions of (K1,K2) by K3 is canonically isomorphic to the group Ext1(K1⊗LK2,K3)
Brauer groups of 1-motives
Full text linkOver a normal base scheme, we prove the generalized Theorem of the Cube for 1-motives and that a torsion class of the group He ́t2(M,Gm,M) of a 1-motive M, whose pull-back via the unit section ε:S→M is zero, comes from an Azumaya algebra. In particular, we deduce that over an algebraically closed field of characteristic zero, all classes of He ́t2(M,Gm,M) come from Azumaya algebras
Extensions and biextensions of locally constant group schemes, tori and abelian schemes
Full text linkLet S be a scheme. We compute explicitly the group of homomorphisms, the S-sheaf of homomorphisms, the group of extensions, and the S-sheaf of extensions involving locally constant S-group schemes, abelian S-schemes, and S-tori. Using the obtained results, we study the categories of biextensions involving these geometrical objects. In particular, we prove that if G i (for i = 1, 2, 3) is an extension of an abelian S-scheme A i by an S-torus T i , the category of biextensions of (G 1, G 2) by G 3 is equivalent to the category of biextensions of the underlying abelian S-schemes (A 1, A 2) by the underlying S-torus T 3. © 2008 Springer-Verlag
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