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Generalization of vector bundles
coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of
Coherent_sheaf
Concept in algebraic geometry
especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties
Coherent_sheaf_cohomology
Projective analogue of the spectrum of a ring
twisting sheaf on the Proj of a ring does. Let E {\displaystyle {\mathcal {E}}} be a quasi-coherent sheaf on a scheme X {\displaystyle X} . The sheaf of symmetric
Proj_construction
Type of mathematical functions
notion of the coherent sheaf into algebraic geometry, that is, the notion of the coherent algebraic sheaf. The notion of coherent (coherent sheaf cohomology)
Function of several complex variables
Function_of_several_complex_variables
module spectrum. When the structure sheaf O X {\displaystyle {\mathcal {O}}_{X}} is not coherent, working with coherent sheaves has awkwardness (namely the
Perfect_complex
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)
Concept in algebraic geometry
bundle 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 ∗
Ample_line_bundle
Concept from algebraic geometry
In algebraic geometry, the dualizing sheaf on a proper scheme X of dimension n over a field k is a coherent sheaf ω X {\displaystyle \omega _{X}} together
Dualizing_sheaf
Sheaf consisting of modules on a ringed space; generalizing vector bundles
In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf of abelian groups F such that, for any open subset U of
Sheaf_of_modules
ν. An analytic space is coherent if its structure sheaf O {\displaystyle {\mathcal {O}}} is a coherent sheaf. A coherent sheaf of O {\displaystyle {\mathcal
Analytic_space
Theorem in algebraic geometry
geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The
Serre_duality
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
Type of ringed space
It is quasi-coherent if it is so as a module. When X is a scheme, just like a ring, one can take the global Spec of a quasi-coherent sheaf of algebras:
Sheaf_of_algebras
geometry, a quasi-coherent sheaf on an algebraic stack X {\displaystyle {\mathfrak {X}}} is a generalization of a quasi-coherent sheaf on a scheme. The
Sheaf_on_an_algebraic_stack
Coherent sheaf on a Stein manifold is spanned by sections & lacks higher cohomology
a coherent sheaf F on a Stein manifold X. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology
Cartan's_theorems_A_and_B
In algebraic geometry, the Castelnuovo–Mumford regularity of a coherent sheaf F over projective space P n {\displaystyle \mathbf {P} ^{n}} is the smallest
Castelnuovo–Mumford regularity
Castelnuovo–Mumford_regularity
mathematics, a twisted sheaf is a variant of a coherent sheaf. Precisely, it is specified by: an open covering in the étale topology Ui, coherent sheaves Fi over
Twisted_sheaf
reflexive sheaf is a coherent sheaf that is isomorphic to its second dual (as a sheaf of modules) via the canonical map. The second dual of a coherent sheaf is
Reflexive_sheaf
French mathematician (born 1926)
tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf over a finite field could not capture as much topology as singular cohomology
Jean-Pierre_Serre
Topics referred to by the same term
the degree-zero subspace of a space of characters to the whole space Coherent sheaf, a specific class of sheaves having particularly manageable properties
Coherence
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
Scheme_(mathematics)
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)
if X is a projective scheme over a Noetherian scheme S and if F is a coherent sheaf on X, then there is a scheme Quot F ( X ) {\displaystyle \operatorname
Quot_scheme
Technique from algebraic geometry
prestack of quasi-coherent sheaves over a scheme S means that, for any S-scheme X, each X-point of the prestack is a quasi-coherent sheaf on X.) Theorem—The
Faithfully_flat_descent
analytic if and only if the ideal sheaf of functions vanishing on A {\displaystyle A} is coherent. This ideal sheaf also gives A {\displaystyle A} the
Ideal_sheaf
In mathematics, a mapping between categories
given a sheaf F defined on a topological space X and a continuous map f: X → Y, we can define a new sheaf f∗F on Y, called the direct image sheaf or the
Direct_image_functor
Unsolved problem in geometry
classes of vector bundles on X. Hodge conjecture for Kähler varieties, coherent sheaf version. Let X be a complex Kähler manifold. Then every Hodge class
Hodge_conjecture
section. A slope of a vector bundle (or, more generally, a torsion-free coherent sheaf) E with respect to H is a rational number defined as μ ( E ) := c 1
Stable_vector_bundle
Two closely related mathematical subjects
{\displaystyle {\mathcal {O}}_{X}} is coherent. Another important statement is as follows: for any coherent sheaf F {\displaystyle {\mathcal {F}}} on an
Algebraic geometry and analytic geometry
Algebraic_geometry_and_analytic_geometry
{\displaystyle f:X\to S} be a proper morphism of noetherian schemes with a coherent sheaf F {\displaystyle {\mathcal {F}}} on X. Let S 0 {\displaystyle S_{0}}
Theorem_on_formal_functions
Algebraic structure used in topology
André–Quillen cohomology Bounded cohomology BRST cohomology Čech cohomology Coherent sheaf cohomology Crystalline cohomology Cyclic cohomology Deligne cohomology
Cohomology
Sheaf cohomology on the étale site
cohomology groups (for coherent sheaves) as the much finer complex topology. However, for constant sheaves such as the sheaf of integers this does not
Étale_cohomology
Sheaf of rings in mathematics
{\displaystyle {\mathcal {O}}_{X}} -modules. A coherent sheaf F {\displaystyle F} is a quasi-coherent sheaf that is, locally, of finite type and for every
Ringed_space
algebraic geometry, p-curvature is an invariant of a connection on a coherent sheaf for schemes of characteristic p > 0. It is a construction similar to
P-curvature
sheaves word for word. For example, the support of a coherent sheaf (or more generally, a finite type sheaf) is a closed subspace of X. If M is a module over
Support_of_a_module
Generalisation of a sheaf; a fibered category that admits effective descent
In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of
Stack_(mathematics)
y]/(xy) are Cohen–Macaulay, but is not. coherent sheaf A coherent sheaf on a Noetherian scheme X is a quasi-coherent sheaf that is finitely generated as OX-module
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Set of a ring's prime ideals
{\displaystyle \mathbf {Spec} } . For a scheme S {\displaystyle S} and a quasi-coherent sheaf of O S {\displaystyle {\mathcal {O}}_{S}} -algebras A {\displaystyle
Spectrum_of_a_ring
cotangent sheaf (Hartshorne, Ch II. Remark 8.9.2). The construction shows in particular that the cotangent sheaf is quasi-coherent. It is coherent if S is
Cotangent_sheaf
Generalizations of codimension-1 subvarieties of algebraic varieties
determines a coherent sheaf O X ( D ) {\displaystyle {\mathcal {O}}_{X}(D)} on X. Concretely it may be defined as subsheaf of the sheaf of rational functions
Divisor_(algebraic_geometry)
Scheme theory concept
morphism of schemes. If F is a quasi-coherent sheaf of finite presentation on Y (in particular, if F is coherent), and if J is the annihilator of F on
Flat_morphism
Topics referred to by the same term
curve sequence Regular tree grammar Castelnuovo–Mumford regularity of a coherent sheaf Closed regular sets in solid modeling Irregularity of a surface in algebraic
Regular
Theorem in algebra mathematics
bundle gives way to that of a coherent sheaf. Informally, Nakayama's lemma says that one can still regard a coherent sheaf as coming from a vector bundle
Nakayama's_lemma
French mathematician (1928–2014)
cohomology of a coherent sheaf on a complete variety is finite-dimensional; Grothendieck's theorem shows that the higher direct images of coherent sheaves under
Alexander_Grothendieck
Homological construction
Unbounded derived categories were introduced by Spaltenstein in 1988. In coherent sheaf theory, pushing to the limit of what could be done with Serre duality
Derived_category
Branch of mathematics
from the fact every vector bundle can be equivalently described as a coherent sheaf. This is done using the Grothendieck group of the Singularity category
K-theory
Coherent sheaf Invertible sheaf Sheaf cohomology Coherent sheaf cohomology Hirzebruch–Riemann–Roch theorem Grothendieck–Riemann–Roch theorem Coherent
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Objects of certain abelian categories associated to topological spaces
individual D-modules (and not more general complexes thereof); a perverse sheaf is in general represented by a complex of sheaves. The concept of perverse
Perverse_sheaf
Tool in mathematical dimension theory
generated graded modules correspond to coherent sheaves. If F {\displaystyle {\mathcal {F}}} is a coherent sheaf over a projective scheme X, we define
Hilbert series and Hilbert polynomial
Hilbert_series_and_Hilbert_polynomial
Concept in mathematics
this free module. If f is étale and if X, Y are complete, then for any coherent sheaf F on Y, writing χ for the Euler characteristic, χ ( f ∗ F ) = deg
Morphism of algebraic varieties
Morphism_of_algebraic_varieties
quotient stack if and only if it has the resolution property; i.e., every coherent sheaf is a quotient of a vector bundle. Earlier, Robert Wayne Thomason proved
Quotient_stack
Mathematical manifold theory
The piece Hp,q(X) of the Hodge decomposition can be identified with a coherent sheaf cohomology group, which depends only on X as a complex manifold (not
Hodge_theory
is the quasi-coherent sheaf of ideals cutting out Z, then the direct image i ∗ {\displaystyle i_{*}} from the category of quasi-coherent sheaves over
Closed_immersion
Exact sequence used to describe the structure of an object
{\displaystyle X=\mathbb {P} _{S}^{n}} is projective space, any coherent sheaf M {\displaystyle {\mathcal {M}}} on X {\displaystyle X} has a presentation
Resolution_(algebra)
Mathematical space
representable functor. If E {\displaystyle {\mathcal {E}}} is a quasi-coherent sheaf on a scheme S {\displaystyle S} for a positive integer k {\displaystyle
Grassmannian
Type of geometric transformation
generality, let X be a scheme, and let I {\displaystyle {\mathcal {I}}} be a coherent sheaf of ideals on X. The blow-up of X with respect to I {\displaystyle {\mathcal
Blowing_up
Generalisations of Serre duality in mathematics
In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex
Coherent_duality
Concept in algebraic geometry
property. In particular, given a fixed coherent sheaf F {\displaystyle {\mathcal {F}}} and a sub-coherent sheaf F ′ {\displaystyle {\mathcal {F}}'} , showing
Noetherian_scheme
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
Meromorphic differential form
strengthening of Alexander Grothendieck's algebraic de Rham theorem, relating coherent sheaf cohomology with singular cohomology. Namely, for any smooth scheme X
Logarithmic_form
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
British mathematician (1923–2010)
Jean-Pierre Serre's coherent sheaf proof of the Riemann–Roch theorem for curves, and found his first research direction in sheaf methods for ruled surfaces
Peter_Hilton
"Smallest" commutative algebra that contains a vector space
exterior algebra by a quadratic form Proj construction § Proj of a quasi-coherent sheaf, an application of symmetric algebras in algebraic geometry Bourbaki
Symmetric_algebra
Mathematics study in geometry
then every bounded complex of coherent sheaves is perfect. Hence the singularity category is trivial Any coherent sheaf F {\displaystyle {\mathcal {F}}}
Derived noncommutative algebraic geometry
Derived_noncommutative_algebraic_geometry
Mathematical conjecture
one can build up more complicated examples of coherent sheaves, analogous to building a coherent sheaf using the torsion filtration. As a simple example
SYZ_conjecture
Term in algebraic geometry
preserve coherent sheaves, in the sense that the higher direct images Rif∗(F) (in particular the direct image f∗(F)) of a coherent sheaf F are coherent (EGA
Proper_morphism
{\mathcal {N}}} denote the sheaf of Nash function germs on a Nash manifold M, and I {\displaystyle {\mathcal {I}}} be a coherent sheaf of N {\displaystyle {\mathcal
Nash_function
Relate the direct image and the pull-back of sheaves
B ′ ) {\displaystyle X'=\operatorname {Spec} (B')} , and the quasi-coherent sheaf F := M ~ {\displaystyle {\mathcal {F}}:={\tilde {M}}} associated to
Base_change_theorems
Topological invariant in mathematics
Euler characteristic used in algebraic geometry is as follows. For any coherent sheaf F {\displaystyle {\mathcal {F}}} on a proper scheme X, one defines its
Euler_characteristic
{O}}_{X}} is coherent, so we may define Fitt 0 ( f ∗ O X ) {\displaystyle \operatorname {Fitt} _{0}(f_{*}{\mathcal {O}}_{X})} as a coherent sheaf of O Y {\displaystyle
Fitting_ideal
Type of smooth complex surface of kodaira dimension 0
(the dimension h 1 ( X , O X ) {\displaystyle h^{1}(X,O_{X})} of the coherent sheaf cohomology group H 1 ( X , O X ) {\displaystyle H^{1}(X,O_{X})} ) is
K3_surface
Algebraic variety defined within an affine space
{\displaystyle H^{i}(X,F)=0} for any i > 0 {\displaystyle i>0} and any quasi-coherent sheaf F on X. (cf. Cartan's theorem B.) This makes the cohomological study
Affine_variety
Manifold with Riemannian, complex and symplectic structure
{\displaystyle 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
Relation between genus, degree, and dimension of function spaces over surfaces
route to generalization. Consequently, the Euler characteristic of a coherent sheaf is reasonably computable. For just one summand within the alternating
Riemann–Roch_theorem
Theory in algebraic topology
scheme, Čech and sheaf cohomology agree for any quasi-coherent sheaf. For the étale topology, the two cohomologies agree for any étale sheaf on X, provided
Čech_cohomology
Category mapping
which is a contravariant pseudofunctor since, for example for a quasi-coherent sheaf F {\displaystyle {\mathcal {F}}} , we only have: ( g ∘ f ) ∗ F ≃ f ∗
Pseudo-functor
Vector bundles theorem
analytic coherent sheaf over ( X , ω ) {\displaystyle (X,\omega )} . Namely in the algebraic setting the rank and degree of a coherent sheaf are encoded
Kobayashi–Hitchin correspondence
Kobayashi–Hitchin_correspondence
Concept in mathematics
equivariant sheaf to be an equivariant object in the category of, say, coherent sheaves. A structure of an equivariant sheaf on an invertible sheaf or a line
Equivariant_sheaf
Topics referred to by the same term
Cartan's theorems A and B, c.1931 results by Henri Cartan concerning a coherent sheaf on a Stein manifold Cartan's lemma, several results by Élie or Henri
Cartan's_theorem
On the Euler characteristic of a holomorphic vector bundle on a compact complex manifold
m^{n}+O(m^{n-1}).} More generally, if F {\displaystyle {\mathcal {F}}} is any coherent sheaf on X then χ ( X , F ⊗ O X ( m D ) ) = rank ( F ) ( D n ) n ! . m n
Hirzebruch–Riemann–Roch theorem
Hirzebruch–Riemann–Roch_theorem
Curve defined as zeros of polynomials
{\displaystyle {\frac {(k-1)(k-2)}{2}}} which can be computed using coherent sheaf cohomology. Here's a brief summary of the curves' genera relative to
Algebraic_curve
whose kernel (as a sheaf) has a vanishing first cohomology group. The dimension of a vector space of sections of a coherent sheaf is finite, in projective
List of publications in mathematics
List_of_publications_in_mathematics
Characteristic class in algebraic topology
{\displaystyle \mathbb {P} ^{2}} restricted to C {\displaystyle C} . For any coherent sheaf F on a smooth compact complex manifold M, one has χ ( F ) = ∫ M ch
Todd_class
Mathematical classification of surfaces
the classification can be given in terms of the dimensions of various coherent sheaf cohomology groups. The basic ones are the plurigenera and the Hodge
Enriques–Kodaira classification
Enriques–Kodaira_classification
In algebra, module with a finite generating set
element Artin–Rees lemma Countably generated module Finite algebra Coherent sheaf, a generalization used in algebraic geometry For example, Matsumura
Finitely_generated_module
Moduli scheme of subschemes of a scheme, represents the flat-family-of-subschemes functor
using the equality of the Hilbert polynomial of a coherent sheaf with the Euler-characteristic of its sheaf cohomology groups. Pick a sufficiently large value
Hilbert_scheme
Concept in algebraic geometry
scheme X that is locally of finite type over a field k, there is a coherent sheaf Ω1 of differentials on X. The scheme X is smooth over k if and only
Smooth_scheme
Completion of the usual space with "points at infinity"
would like to be able to associate a projective space to every quasi-coherent sheaf E over a scheme Y, not just the locally free ones.[clarification needed]
Projective_space
_{*}\omega _{g}} . ELSV formula Here, "vector bundle" in the sense of quasi-coherent sheaf on an algebraic stack van der Geer, Gerard (2008), "Siegel modular forms
Hodge_bundle
Technique in mathematical group theory
Borel–Weil–Bott construction of representations of algebraic groups using coherent sheaf cohomology is also similar. For real semisimple groups there is an analogue
Deligne–Lusztig_theory
Theorem in complex analysis about the sheaf of holomorphic functions
subsequently the sheaf O X {\displaystyle {\mathcal {O}}_{X}} of holomorphic functions on a complex manifold X {\displaystyle X} ) is coherent. Cartan's theorems
Oka_coherence_theorem
Generalization of a vector bundle
{\displaystyle \mathbb {P} (E)} . Let F {\displaystyle {\mathcal {F}}} be a coherent sheaf on a Deligne–Mumford stack X. Then let C ( F ) := Spec X ( Sym (
Cone_(algebraic_geometry)
convex spaces are still convex. This follows from the Künneth theorem in coherent sheaf cohomology. One more non-trivial class of examples of convex varieties
Convexity (algebraic geometry)
Convexity_(algebraic_geometry)
Stability conditions for triangulated cateogires
every coherent sheaf on the curve admits a filtration by semistable vector bundles and skyscraper sheaves. It follows that every complex of coherent sheaves
Bridgeland stability condition
Bridgeland_stability_condition
instance, this happens if F {\displaystyle {\mathcal {F}}} is a quasi-coherent sheaf on a scheme and each element of U {\displaystyle {\mathfrak {U}}} is
Čech-to-derived functor spectral sequence
Čech-to-derived_functor_spectral_sequence
is a smooth projective variety of positive dimension such that the coherent sheaf cohomology groups H i ( X , O X ) {\displaystyle H^{i}(X,O_{X})} are
Semiorthogonal_decomposition
Module over a sheaf of differential operators
are local and parallel the situation of coherent sheaves. This builds on the fact that DX is a locally free sheaf of OX-modules, albeit of infinite rank
D-module
Theorem relating to algebraic topology
over M U ∗ {\displaystyle {\mathcal {}}MU_{*}} is the same as a quasi-coherent sheaf F {\displaystyle {\mathcal {F}}} over Spec L {\displaystyle {\text{Spec
Landweber exact functor theorem
Landweber_exact_functor_theorem
) {\displaystyle (X,F)} of a scheme X {\displaystyle X} and a quasi-coherent sheaf F {\displaystyle F} on it, a morphism f ¯ : ( X , F ) → ( Y , G ) {\displaystyle
Cartesian_fibration
Locally constant sheaf of abelian groups on topological space
between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient
Local_system
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