In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let M be a complex manifold,...

Exponential sheaf sequence Exponential smoothing Exponential stability Exponential sum Exponential time Sub-exponential time Exponential tree Exponential type...

the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group. The name is in honour...

Look up sheaf in Wiktionary, the free dictionary. In mathematics, a sheaf (pl.: sheaves) is a tool for systematically tracking data (such as sets, abelian...

essentially topological classification by intersection numbers. The exponential sheaf sequence 0 → 2 π i Z → O V → O V ∗ → 0 {\displaystyle 0\to 2\pi i\mathbb...

{\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})=0} . The exponential sheaf sequence leads to the following exact sequence: H 1 ( X , O X ) ⟶ H 1 ( X , O X ∗ ) ⟶ H 2...

Chern class (see also exponential sheaf sequence). In terms of sheaf theory, let O ∗ {\displaystyle \mathbf {O} ^{*}} be the sheaf of holomorphic functions...

(1,1)-classes. Because X is a complex manifold, it admits an exponential sheaf sequence 0 → Z _ ⟶ 2 π i O X ⟶ exp O X × → 0. {\displaystyle 0\to {\underline...

{\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})=0} . The exponential sheaf sequence leads to the following exact sequence: H 1 ( X , O X ) ⟶ H 1 ( X , O X ∗ ) ⟶ H 2...

points, and its investigation was a major reason for the development of sheaf cohomology. Suppose f is an analytic function defined on a non-empty open...

growing a series of different types of crops in the same area across a sequence of growing seasons. This practice reduces the reliance of crops on one...

Zero object Subobject Group object Magma object Natural number object Exponential object Epimorphism Monomorphism Zero morphism Normal morphism Dual (category...

structures. The Chern class statements are easily proven using the exponential sequence of sheaves on the manifold. One can more generally view the classification...

geometry and analytic geometry. In terms of sheaf theory, this difference can be stated as follows: the sheaf of differentiable functions on a differentiable...

satisfying certain infinite categorical version of a sheaf axiom (and to be algebraic, inductively a sequence of representability conditions). Quillen model...

{\displaystyle H^{2}} . A very quick proof can be given using sheaf cohomology and the exponential exact sequence. (The cohomology class of a divisor turns out to...

generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf. Suppose that X is a non-singular n-dimensional projective algebraic variety...

universal cover π : X ~ → X {\displaystyle \pi :{\tilde {X}}\to X} , and a sheaf of abelian groups F {\displaystyle {\mathcal {F}}} on X {\displaystyle X}...

function that is differentiable but not locally Lipschitz continuous. The exponential function e x {\displaystyle e^{x}} is analytic, and hence falls into...

Rham spectral sequence Logarithmic form Kodaira vanishing theorem Moduli of algebraic curves Motive (algebraic geometry) Perverse sheaf Riemann–Hilbert...

Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge:...

Haynes (2000). "Leray in Oflag XVIIA: The origins of sheaf theory, sheaf cohomology, and spectral sequences" (ps). Archived from the original on 9 September...

compatibility conditions on the overlaps. Such a structure is known as a sheaf. Let V ⊆ U {\displaystyle V\subseteq U} be open subsets of R n . {\displaystyle...

fundamental group have a very geometric significance: any local system (i.e., a sheaf F {\displaystyle {\mathcal {F}}} on X with the property that locally in...

direct system yields the ring of symmetric functions. Let F be a C-valued sheaf on a topological space X. Fix a point x in X. The open neighborhoods of...

ISSN 0271-4132. LCCN 96-37049. MR 1436913. Retrieved 2021-12-08. George Whitehead; Fifty years of homotopy theory Haynes Miller; The origin of sheaf theory...

it has a terminal object and that any two objects have a product and exponential. cartesian functor Given relative categories p : F → C , q : G → C {\displaystyle...

Next, the exponential sequence 0 → Z X → O X → O X ∗ → 0 {\displaystyle 0\to \mathbb {Z} _{X}\to O_{X}\to O_{X}^{*}\to 0} gives an exact sequence of cohomology...

isomorphism classes of line bundles on X {\displaystyle X} . From the exponential exact sequence 0 → Z → O X → O X ∗ → 0 {\displaystyle 0\to \mathbb {Z} \to {\mathcal...

[citation needed] Grothendieck and Serre recast algebraic geometry using sheaf theory.[citation needed] Large advances were made in the qualitative study...