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Object in mathematical sheaf theory
In mathematics, the constant sheaf on a topological space X {\displaystyle X} associated to a set A {\displaystyle A} is a sheaf of sets on X {\displaystyle
Constant_sheaf
Sheaf theory
In algebraic topology, a locally constant sheaf on a topological space X is a sheaf F {\displaystyle {\mathcal {F}}} on X such that for each x in X, there
Locally_constant_sheaf
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)
Tool in algebraic topology
sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology
Sheaf_cohomology
finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its origins in algebraic geometry, where in étale
Constructible_sheaf
Type of mathematical function
referred to a constant sheaf, meaning exactly sheaf of locally constant functions taking their values in the (same) group. The typical sheaf of course is
Locally_constant_function
Theorem
Poincaré lemma implies that the de Rham cohomology is the sheaf cohomology with the constant sheaf R {\displaystyle \mathbb {R} } . Thus, for abstract reason
De_Rham_theorem
Mathematical construction
In mathematics, the stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point. Sheaves are defined on open
Stalk_(sheaf)
Sheaf consisting of modules on a ringed space; generalizing vector bundles
structure sheaf. If O is the constant sheaf Z _ {\displaystyle {\underline {\mathbf {Z} }}} , then a sheaf of O-modules is the same as a sheaf of abelian
Sheaf_of_modules
Mathematical condition
particular, it implies that the de Rham complex yields a resolution of the constant sheaf R M {\displaystyle \mathbb {R} _{M}} on M. The singular cohomology of
Poincaré_lemma
Generalization of vector bundles
Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf. A quasi-coherent sheaf on a ringed
Coherent_sheaf
Cohomology with real coefficients computed using differential forms
{\displaystyle M} , let R _ {\textstyle {\underline {\mathbb {R} }}} be the constant sheaf on M {\displaystyle M} associated to the abelian group R {\textstyle
De_Rham_cohomology
Algebraic structure used in topology
Alexander–Spanier cohomology or sheaf cohomology). (Here sheaf cohomology is considered only with coefficients in a constant sheaf.) These theories give different
Cohomology
Objects of certain abelian categories associated to topological spaces
discrete valuation ring, then the constant sheaf shifted by dim X + 1 {\displaystyle \dim X+1} is an étale perverse sheaf. Let X be a disk around the origin
Perverse_sheaf
Mathematical object in sheaf cohomology
to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There is a further group of related concepts
Injective_sheaf
Locally constant sheaf of abelian groups on topological space
groups/modules...) on X is a locally constant sheaf (of abelian groups/of modules...) on X. In other words, a sheaf L {\displaystyle {\mathcal {L}}} is
Local_system
_{X\setminus X_{n-2}}} is the constant sheaf on X ∖ X n − 2 {\displaystyle X\setminus X_{n-2}} . By replacing the constant sheaf on X ∖ X n − 2 {\displaystyle
Intersection_homology
On generating functions from counting points on algebraic varieties over finite fields
_{i}\det(1-F^{*}t\mid H_{c}^{i}(F))^{(-1)^{i+1}}} The special case of the constant sheaf gives the usual zeta function. Verdier (1974), Serre (1975), Katz (1976)
Weil_conjectures
field of algebraic topology, the orientation sheaf on a manifold X of dimension n is a locally constant sheaf oX on X such that the stalk of oX at a point
Orientation_sheaf
Type of mathematical object
infinite type. More generally, by taking a locally constant sheaf of groups on S, one obtains a locally constant group scheme, for which monodromy on the base
Group_scheme
Theory in algebraic topology
{\mathcal {F}}_{A})} where F A {\displaystyle {\mathcal {F}}_{A}} is the constant sheaf on X determined by A. A variant of Čech cohomology, called numerable
Čech_cohomology
Sheaf cohomology on the étale site
sheaves) as the much finer complex topology. However, for constant sheaves such as the sheaf of integers this does not work: the cohomology groups defined
Étale_cohomology
Type of mathematical functions
\xrightarrow {\exp } \mathbf {O} ^{*}\to 0} where the leftmost sheaf is the locally constant sheaf with fiber 2 π i Z {\displaystyle 2\pi i\mathbb {Z} } . The
Function of several complex variables
Function_of_several_complex_variables
Mathematical theory
if F = k _ {\displaystyle {\mathcal {F}}={\underline {k}}} is the constant sheaf and Y {\displaystyle Y} is a smooth submanifold, then we get Ext k n
Alexander_duality
the sheaf of rings on G/B formed by taking the *-pushforward of DG/U along the T-bundle G/U → G/B, a sheaf of rings whose center is the constant sheaf of
Beilinson–Bernstein localization
Beilinson–Bernstein_localization
Category where every morphism is invertible; generalization of a group
[\sigma ]\in {\check {H}}^{k}({\mathcal {U}},{\underline {A}})} for some constant sheaf of abelian groups can be represented as a function σ : ∐ U i 1 ⋯ i k
Groupoid
Generalization of a complex manifold that allows the use of singularities
variety is a zero locus of a set of a polynomial function. Denote the constant sheaf on a topological space with value C {\displaystyle \mathbb {C} } by
Complex_analytic_variety
Duality for sheaves of k-modules over a locally compact space
machinery of sheaf cohomology. Suppose X is a compact orientable n-dimensional manifold, k is a field and k X {\displaystyle k_{X}} is the constant sheaf on X
Verdier_duality
Mathematical sequence
generalized by instead considering sheaves of modules over a locally constant sheaf of rings A _ {\displaystyle {\underline {A}}} for a fixed commutative
Leray_spectral_sequence
Exact sequence used to describe the structure of an object
F(E_{*}).} This situation applies in many situations. For example, for the constant sheaf R on a differentiable manifold M can be resolved by the sheaves C ∗
Resolution_(algebra)
Expression that may be integrated over a region
is the space of locally constant functions on M {\displaystyle M} . Therefore, the complex is a resolution of the constant sheaf R _ {\displaystyle {\underline
Differential_form
Mathematical group of the homotopy classes of loops in a topological space
on X {\displaystyle X} , the restriction of F {\displaystyle F} is a constant sheaf of the form F | U = Q n {\displaystyle {\mathcal {F}}|_{U}=\mathbb {Q}
Fundamental_group
Mathematics glossary
locally constant sheaf A locally constant sheaf on a space X is a sheaf such that each point of X has an open neighborhood on which the sheaf is constant. loop
Glossary of algebraic topology
Glossary_of_algebraic_topology
Invariant of algebraic varieties and of more general schemes
is the constant sheaf Z, and Z(1) is isomorphic in the derived category of X to Gm[−1]. Here Gm (the multiplicative group) denotes the sheaf of invertible
Motivic_cohomology
General concept and operation in mathematics
singular cohomology with C-coefficients (equivalently, sheaf cohomology of the constant sheaf C) Hi(X) ⊗ H2n−i(X) → C, where n is the (complex) dimension
Duality_(mathematics)
Algebraic structure
structure of weight n on a complex manifold X consists of a locally constant sheaf S of finitely generated abelian groups on X, together with a decreasing
Hodge_structure
Concept in mathematics
group scheme G on a scheme X over a base scheme S, an equivariant sheaf F on X is a sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules together with
Equivariant_sheaf
In algebraic geometry, an ℓ-adic sheaf on a Noetherian scheme X is an inverse system consisting of Z / ℓ n {\displaystyle \mathbb {Z} /\ell ^{n}} -modules
ℓ-adic_sheaf
Type of Grothendieck topology on the category of schemes
Suppose that X is a Noetherian scheme. An abelian étale sheaf F on X is called finite locally constant if it is a representable functor which can be represented
Étale_topology
Sheaf of rings in mathematics
Precisely, it is a topological space equipped with a sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of continuous
Ringed_space
In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most
Exceptional inverse image functor
Exceptional_inverse_image_functor
Its kernel is the sheaf 2πiZ of locally constant functions on M taking the values 2πin, with n an integer. The exponential sheaf sequence is therefore
Exponential_sheaf_sequence
conditions, a local system can be equivalently described as a locally constant sheaf. The fundamental groupoid of the singleton space is the trivial groupoid
Fundamental_groupoid
Make a meromorphic function from local data in multiple variables
{\xrightarrow {\exp }}\mathbf {O} ^{*}\to 0} where the leftmost sheaf is the locally constant sheaf with fiber 2 π i Z {\displaystyle 2\pi i\mathbb {Z} } . The
Cousin_problems
Specific algebraic group
H^{1}(S,GL_{n}(\mathbb {Z} ))} , where the coefficient group forms a constant sheaf. In particular, twisted forms of a split torus T over a field K are
Algebraic_torus
Expresses the number of points of a variety over a finite field
{tr} (F_{E}\,\,|\,\,H_{c}^{i}(X_{\bar {k}},{\mathcal {F}}))} For a constant sheaf Q l {\displaystyle \mathbb {Q} _{l}} (viewed as ( lim ← Z / l n Z
Grothendieck_trace_formula
R^{i}p_{!}{\mathcal {F}}.} Taking for F {\displaystyle {\mathcal {F}}} the constant sheaf with coefficients in a ring R {\displaystyle R} recovers the previous
Cohomology with compact support
Cohomology_with_compact_support
Manifold upon which it is possible to perform calculus
charts and atlases). Third, the sheaf OM is not manifestly a sheaf of functions at all. Rather, it emerges as a sheaf of functions as a consequence of
Differentiable_manifold
localization of the global sections of X, and consequently KX will be the constant sheaf whose value is the fraction field of the global sections of X. If X
Function field (scheme theory)
Function_field_(scheme_theory)
differential forms Local system, the more general notion of a locally constant sheaf. Orientation character, a characteristic form related to the orientation
Flat_vector_bundle
{\displaystyle {\mathsf {Sch}}_{S}} to the category of groups that is a Zariski sheaf (i.e., satisfying the gluing axiom for the Zariski topology). For example
Group_functor
Concept in algebraic geometry
p : E → Y {\displaystyle p\colon E\to Y} (or more generally a coherent sheaf on Y {\displaystyle Y} ) has a pullback to X {\displaystyle X} , f ∗ E =
Ample_line_bundle
n\}\mapsto F(H_{n})} . Any sheaf F on the site can be considered as a stack by viewing F ( X ) {\displaystyle F(X)} as a constant simplicial set; this way
Simplicial_presheaf
Construction in homological algebra
{\displaystyle X} , and Z X {\displaystyle \mathbb {Z} _{X}} is the sheaf of locally constant Z {\displaystyle \mathbb {Z} } -valued functions. Instead of Z
Ext_functor
Concept in homological algebra
F_{Z}} is an ordinary sheaf, U {\displaystyle U} is a smooth open subscheme and F U {\displaystyle F_{U}} is a locally constant sheaf on U. Note the presence
T-structure
Concept in mathematics
} {\displaystyle Y=\lbrace p\rbrace } is a point, then the ideal sheaf is the sheaf of smooth germs vanishing at p {\displaystyle p} and the isomorphism
Normal_bundle
Two closely related mathematical subjects
if F {\displaystyle {\mathcal {F}}} is a sheaf on X {\displaystyle X} , then there is a corresponding sheaf F an {\displaystyle {\mathcal {F}}^{\text{an}}}
Algebraic geometry and analytic geometry
Algebraic_geometry_and_analytic_geometry
Type of topological space
space. On a stratified space, a constructible sheaf can be defined as a sheaf that is locally constant on each stratum. Among the several ideals, Grothendieck's
Stratified_space
Mathematical group occurring in algebraic geometry and the theory of complex manifolds
group Arakelov class group Group-stack Picard category Sheaf cohomology#Sheaf cohomology with constant coefficients Kleiman 2005, Definition 9.2.2. Grothendieck
Picard_group
Mathematical theorem
versions of it were found by Max Noether (1886) and Enriques (1894). The sheaf-theoretic version is due to Hirzebruch. One form of the Riemann–Roch theorem
Riemann–Roch theorem for surfaces
Riemann–Roch_theorem_for_surfaces
Homological construction in category theory
a special case of this: De Rham cohomology is the sheaf cohomology of the sheaf of locally constant R {\displaystyle \mathbb {R} } -valued functions on
Derived_functor
Vector bundle of cotangent spaces at every point in a manifold
smooth manifolds, such as complex manifolds, or (in the form of cotangent sheaf) algebraic varieties or schemes. In the smooth case, any Riemannian metric
Cotangent_bundle
Weil cohomology theory for schemes X over a base field k
closed subscheme of a scheme T {\displaystyle T} defined by a nilpotent sheaf of ideals on T {\displaystyle T} ; for example, Spec ( k ) → Spec (
Crystalline_cohomology
Concept in mathematics
field analogue of complex multiplication theory. A shtuka (also called F-sheaf or chtouca) is a sort of generalization of a Drinfeld module, consisting
Drinfeld_module
Relate the direct image and the pull-back of sheaves
square of topological spaces and F {\displaystyle {\mathcal {F}}} is a sheaf on X. Such theorems exist in different branches of geometry: for (essentially
Base_change_theorems
Mathematical object in category theory
→ Ω(U) defined by ηU(*)=U for every open set U of X. Given a sheaf F on X and a sub-sheaf j: G → F, the classifying morphism χ j : F → Ω is given by the
Subobject_classifier
Generalization of algebraic variety
quasi-coherent sheaf on a scheme X means an OX-module that is the sheaf associated to a module on each affine open subset of X. Finally, a coherent sheaf (on a
Scheme_(mathematics)
Relates the geometric vector bundles to algebraic projective modules
variety with structure sheaf O X , {\displaystyle {\mathcal {O}}_{X},} and F {\displaystyle {\mathcal {F}}} a coherent sheaf of O X {\displaystyle {\mathcal
Serre–Swan_theorem
Application of homotopy to algebraic varieties
considering A 1 {\displaystyle \mathbb {A} ^{1}} as a sheaf via the Yoneda embedding, and the constant simplicial object functor S h v ( S m S ) N i s → Δ
A¹_homotopy_theory
Type of mathematical expression
in Nine Sections, c. 200 BCE, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou." We would
Polynomial
Function with a smaller domain
collection of all such objects is called a sheaf. If only the first two properties are satisfied, it is a pre-sheaf. More generally, the restriction (or domain
Restriction_(mathematics)
Affine space over the complex numbers
holomorphic functions on a complex affine space A forms a sheaf of rings on it. By definition, such a sheaf associates to each (analytic) open subset U of A the
Complex_affine_space
Vector bundle of rank 1
of spheres to spheres. In algebraic geometry, an invertible sheaf (i.e., locally free sheaf of rank one) is often called a line bundle. Every line bundle
Line_bundle
Mathematical parametrization of vector spaces by another space
OX denotes the structure sheaf of continuous real-valued functions on X, then F becomes a sheaf of OX-modules. Not every sheaf of OX-modules arises in
Vector_bundle
Topics referred to by the same term
degree-zero subspace of a space of characters to the whole space Coherent sheaf, a specific class of sheaves having particularly manageable properties closely
Coherence
Suburb of Greater London, England
2022. Sheaf 2015, pp. 80–86, 100–108. Sheaf & Howe 1995, pp. 91–93. Heath 2000, pp. 9–10. Orton 1965, pp. 48–49, 63. Sheaf 1997, pp. 12–13. Sheaf 2015
Hampton,_London
Field of algebraic geometry
be formulated in sheaf cohomology terms, as the non-vanishing of the H1 cohomology of the sheaf of sections of the invertible sheaf or line bundle associated
Brill–Noether_theory
Relation between genus, degree, and dimension of function spaces over surfaces
equations, and the GAGA principle says that sheaf cohomology of an algebraic variety is the same as the sheaf cohomology of the analytic variety defined
Riemann–Roch_theorem
Branch of mathematics studying functions of a complex variable
variables makes use of additional techniques such as Banach algebras and sheaf theory. It is often concerned with questions of interest in algebraic geometry
Complex_analysis
mere scheme: a sheaf in graded modules over the structure sheaf is defined in the process. The homogeneous components of this graded sheaf are denoted O
Algebraic geometry of projective spaces
Algebraic_geometry_of_projective_spaces
Equivalence class of objects sharing local properties at a point in a topological space
meaning. The name is derived from cereal germ in a continuation of the sheaf metaphor, as a germ is (locally) the "heart" of a function, as it is for
Germ_(mathematics)
Manifold with Riemannian, complex and symplectic structure
X} with complex coefficients splits as a direct sum of certain coherent sheaf cohomology groups: H r ( X , C ) ≅ ⨁ p + q = r H q ( X , Ω p ) . {\displaystyle
Kähler_manifold
Algebraic geometry analog of a principal bundle in algebraic topology
{\displaystyle V} . Given V {\displaystyle V} one can take the (representable) sheaf of local isomorphisms I s o m ( V , O X ⊕ n ) {\displaystyle Isom(V,{\mathcal
Torsor_(algebraic_geometry)
Algebraic variety in a projective space
salient feature of projective varieties are the finiteness constraints on sheaf cohomology. For smooth projective varieties, Serre duality can be viewed
Projective_variety
Concept in mathematics
the structure sheaf) is a fundamental object in affine algebraic geometry. The only regular function on a projective variety is constant (this can be viewed
Morphism of algebraic varieties
Morphism_of_algebraic_varieties
Mathematical theorem
invertible on the base scheme, F {\displaystyle {\mathcal {F}}} a locally constant étale sheaf with finite stalks and values in Z / n Z {\displaystyle \mathbb {Z}
Theorem_of_absolute_purity
Branch of mathematics
algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its
Algebraic_geometry
Objects extending the notion of functions
D(R) (functions of vanishing moments up to order q). If (E,P) is a (pre-)sheaf of semi normed algebras on some topological space X, then Gs(E, P) will
Generalized_function
Central business district in South Yorkshire, England
The Sheaf Valley Area, surrounding the River Sheaf, is dominated by Sheaf Square/Sheffield Station and the Cultural Industries Quarter (CIQ). Sheaf Square
Sheffield_city_centre
Topological space that locally resembles Euclidean space
locally constant), each connected component has a fixed dimension. Sheaf-theoretically, a manifold is a locally ringed space, whose structure sheaf is locally
Manifold
Street in Sheffield, England
almost constant incline to the beginning of Woodseats. Between the old Meersbrook Quarry and Heeley Retail Park are views to the west of the Sheaf Valley
Chesterfield_Road_(Sheffield)
and H is a hyperplane section, then a vector bundle (or a torsion-free sheaf) W is called stable (or sometimes Gieseker stable) if χ ( V ( n H ) ) rank
Stable_vector_bundle
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
Touchard polynomials (combinatorics) Exponential response formula Exponential sheaf sequence Exponential smoothing Exponential stability Exponential sum Exponential
List_of_exponential_topics
Subjective experience of victimization by extraterrestrials
syndrome), sleep paralysis, deception, and psychopathology. Skeptic Robert Sheaffer sees similarity between some of the aliens described by abductees and those
Alien_abduction
Correspondsnce between Higgs bundles and fundamental group representations
{\mathcal {F}})/B^{k}(X,{\mathcal {F}})} of sheaf cocycles by sheaf coboundaries is always well-defined. When the sheaf F {\displaystyle {\mathcal {F}}} is not
Nonabelian Hodge correspondence
Nonabelian_Hodge_correspondence
Panlo is the District 9 male tribute in the 10th Hunger Games. He and Sheaf die from injuries sustained in the bombing attack. In the movie, he was
List of The Hunger Games characters
List_of_The_Hunger_Games_characters
Writing implement designed to make lines of constant width
the caliper principle and made the points easily interchangeable. The Sheaffer company produced an expensive drafting set which included such pens for
Technical_pen
flat over R and (2) the ideal sheaf I ( D ) {\displaystyle I(D)} of D is locally free of rank one (i.e., invertible sheaf). Equivalently, a closed subscheme
Relative effective Cartier divisor
Relative_effective_Cartier_divisor
Dutch painter (1853–1890)
2017. Retrieved 20 October 2016. Sund, Judy (1988). "The Sower and the Sheaf: Biblical Metaphor in the Art of Vincent van Gogh". The Art Bulletin. 70
Vincent_van_Gogh
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