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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
Mathematical group occurring in algebraic geometry and the theory of complex manifolds
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
Picard_group
formula Exponential sheaf sequence Exponential smoothing Exponential stability Exponential sum Exponential time Sub-exponential time Exponential tree Exponential
List_of_exponential_topics
Group in algebraic geometry
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
Néron–Severi_group
(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
Lefschetz theorem on (1,1)-classes
Lefschetz_theorem_on_(1,1)-classes
Type of mathematical functions
{\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
Function of several complex variables
Function_of_several_complex_variables
Tool to track locally defined data attached to the open sets of a topological space
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
Sheaf_(mathematics)
Make a meromorphic function from local data in multiple variables
Chern class (see also exponential sheaf sequence). In terms of sheaf theory, let O ∗ {\displaystyle \mathbf {O} ^{*}} be the sheaf of holomorphic functions
Cousin_problems
connected open subset of C {\displaystyle \mathbb {C} } ). analytic sheaf analytic sheaf archimedean The archimedean property of real numbers says: given
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
Extension of the domain of an analytic function (mathematics)
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
Analytic_continuation
Homological construction in category theory
which are a special case of this: De Rham cohomology is the sheaf cohomology of the sheaf of locally constant R {\displaystyle \mathbb {R} } -valued functions
Derived_functor
{O}}_{\widetilde {X}}^{*}).} ) Taking the long exact sequence of the exponential sheaf sequence gives the degree map: deg : Pic ( X ) → H 2 ( X ;
Complete_algebraic_curve
Overview of and topical guide to category theory
Comma category Localization of a category Enriched category Bicategory Sheaf Gluing axiom Descent (category theory) Grothendieck topology Introduction
Outline_of_category_theory
Branch of mathematics
satisfies certain infinite-categorical versions of sheaf axioms (and to be algebraic, inductively a sequence of representability conditions). Quillen model
Algebraic_geometry
Vector bundle of rank 1
structures. The Chern class statements are easily proven using the exponential sequence of sheaves on the manifold. One can more generally view the classification
Line_bundle
Unsolved problem in geometry
{\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
Hodge_conjecture
Agricultural practice of changing crops
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
Crop_rotation
Belgian mathematician
Rham spectral sequence Logarithmic form Kodaira vanishing theorem Moduli of algebraic curves Motive (algebraic geometry) Perverse sheaf Riemann–Hilbert
Pierre_Deligne
Functor that preserves short exact sequences
sheaves of abelian groups on X. The covariant functor that associates to each sheaf F the group of global sections F(X) is left-exact. If R is a ring and T
Exact_functor
Type of function in mathematics
must vanish identically. Standard examples include polynomials, the exponential function, and the trigonometric functions on their domains of analyticity
Analytic_function
Mathematical functions which are smooth but not analytic
geometry and analytic geometry. In terms of sheaf theory, this difference can be stated as follows: the sheaf of differentiable functions on a differentiable
Non-analytic_smooth_function
On generating functions from counting points on algebraic varieties over finite fields
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
Weil_conjectures
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}
N-group_(category_theory)
Special objects used in (mathematical) category theory
Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge:
Initial_and_terminal_objects
History of maths
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
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Haynes (2000). "Leray in Oflag XVIIA: The origins of sheaf theory, sheaf cohomology, and spectral sequences" (ps). Archived from the original on 9 September
List of publications in mathematics
List_of_publications_in_mathematics
Mathematical group of the homotopy classes of loops in a topological space
fundamental group have a very geometric significance: any local system (i.e., a sheaf F {\displaystyle {\mathcal {F}}} on X {\displaystyle X} with the property
Fundamental_group
of morphisms by the (∞, n - 1)-category of morphisms. ∞-sheaf Another term for a homotopy sheaf. initial 1. An object A is initial if there is exactly
Glossary_of_category_theory
Algebraic structure with addition and multiplication
is that the localization allows one to view a module as a sheaf over prime ideals and a sheaf is inherently a local notion.) In category theory, a localization
Ring_(mathematics)
Surjective homomorphism
Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory. Encyclopedia of Mathematics and its Applications. Vol. 97. Cambridge
Epimorphism
Construction in group theory
the functor PSL(n, K) does not define an algebraic group, or even an fppf sheaf, and its sheafification in the fppf topology is in fact PGL(n, K). PSL and
Projective_linear_group
Special case of colimit in category theory
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
Direct_limit
Type of smooth complex surface of kodaira dimension 0
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
K3_surface
Concept in category theory
image functor f ∗ {\displaystyle f^{*}} can be described as follows: for a sheaf F {\displaystyle F} on E S {\displaystyle E_{S}} and an object p : U → T
Fibred_category
Kind of complex manifold
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
Complex_torus
category theory. Grothendieck and Serre recast algebraic geometry using sheaf theory. Large advances were made in the qualitative study of dynamical systems
20th_century_in_science
Sequence of major events in a virus pandemic
of S classes moved to remote delivery to prevent COVID-19 spread". The Sheaf. University of Saskatchewan Students' Union. Archived from the original
Timeline of the COVID-19 pandemic in Saskatchewan
Timeline_of_the_COVID-19_pandemic_in_Saskatchewan
EXPONENTIAL SHEAF-SEQUENCE
EXPONENTIAL SHEAF-SEQUENCE
Surname or Lastname
English
English : variant spelling of Shedd.Irish : reduced variant of Sheedy.
Surname or Lastname
Irish
Irish : reduced form of Sheehan.English : nickname for an attractive person, from Middle English schene ‘fair’, ‘comely’, ‘handsome’.English : habitational name from Sheen in Surrey and Staffordshire, both named in Old English with the plural of scēo ‘shed’, ‘shelter’.
Girl/Female
Indian
Shefa, Healthiness
Girl/Female
Biblical
Roundness of a sheaf.
Boy/Male
Muslim
One who gives health
Biblical
sheaf of corn
Surname or Lastname
English (Kent)
English (Kent) : from Middle English shefe ‘sheaf’, ‘bundle’ (Old English scēaf), hence possibly a metonymic occupational name for a harvest worker, or for someone who paid or collected tithes, from the same term in the sense ‘tenth’ (or other proportion of produce paid as a tithe).Jacob Sheafe (d. 1658) was one of the founds of Boston MA. He is buried in the King’s Chapel Burying Ground there.
Male
English
Irish surname transferred to unisex forename use, from an Anglicized form of Gaelic Ó Séaghdha ("descendant of Séaghdha"), possibly SHEA means "hawk-like."Â
Boy/Male
Arabic, Muslim
One who Gives Health
Boy/Male
British, English
Shear Man
Boy/Male
Arabic, Australian, Biblical, Christian, French, Hebrew, Jewish
Sheaf of Corn; King of Israel; Servant of Jehovah; My Sheaf
Boy/Male
Biblical
Sheaf of corn.
Boy/Male
Muslim/Islamic
One who gives health
Biblical
roundness of a sheaf
Boy/Male
Irish American
Majestic.
Surname or Lastname
English
English : nickname for a beautiful or radiant person, or one with fair hair, from Middle English scher, schir ‘bright’, ‘fair’.
Boy/Male
Anglo, British, English
Shear Man
Boy/Male
Arabic, Muslim
The Healing
Boy/Male
British, English
Shear Man
Girl/Female
Muslim
Shefa, Healthiness
EXPONENTIAL SHEAF-SEQUENCE
EXPONENTIAL SHEAF-SEQUENCE
Female
Slovene
(Ðленка) Slovene form of Greek HelénÄ“, possibly ALENKA means "torch."
Boy/Male
Hindu
The knower of hymns
Boy/Male
American, Anglo, Australian, British, Christian, English, Hawaiian, Hebrew, Indian
Father
Boy/Male
Tamil
Boy/Male
Indian, Sanskrit
Truth
Boy/Male
Muslim
Gold
Male
Slovene
Slovene form of Latin Demetrius, DIMITRIJ means "loves the earth" or "follower of Demeter."
Boy/Male
English
Phonetic spelling of Schuyler.
Boy/Male
American, Australian, British, Danish, English, German, Indian, Latin, Tamil
Power; Strong; Healthy
Boy/Male
Anglo Saxon Hebrew Irish
Name of an abbot.
EXPONENTIAL SHEAF-SEQUENCE
EXPONENTIAL SHEAF-SEQUENCE
EXPONENTIAL SHEAF-SEQUENCE
EXPONENTIAL SHEAF-SEQUENCE
EXPONENTIAL SHEAF-SEQUENCE
v. t.
To separate or sever with shears or a similar instrument; to cut off; to clip (something) from a surface; as, to shear a fleece.
v. t.
To gather and bind into a sheaf; to make into sheaves; as, to sheaf wheat.
pl.
of Sheaf
a.
Pertaining to, or consisting of, a sheaf or sheaves; resembling a sheaf.
v. t.
To shear.
p. pr. & vb. n.
of Shear
v. t.
To put under a sheal or shelter.
v. t.
To produce a change of shape in by a shear. See Shear, n., 4.
v. i.
To collect and bind cut grain, or the like; to make sheaves.
n.
Shears See Shear.
imp.
of Shear
v. t.
To shear.
n.
A sheaf of grain (wheat, unless otherwise specified).
pl.
of Shelf
p. p.
of Shear
v. t.
To cut, clip, or sever anything from with shears or a like instrument; as, to shear sheep; to shear cloth.
a.
Pertaining to exponents; involving variable exponents; as, an exponential expression; exponential calculus; an exponential function.