Coherent sheaf

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of 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 vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank.

Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf.

Definitions

A quasicoherent sheaf on a ringed space (X,O_X) is a sheaf F of O_X-modules which has a local presentation, that is, every point in X has an open neighborhood U in which there is an exact sequence

{\mathcal {O}}_{X}^{\oplus I}|_{U}\to {\mathcal {O}}_{X}^{\oplus J}|_{U}\to {\mathcal {F}}|_{U}\to 0

for some sets I and J (possibly infinite).

A coherent sheaf on a ringed space (X,O_X) is a quasicoherent sheaf F satisfying the following two properties:

F is of finite type over O_X, that is, every point in X has an open neighborhood U in X such that there is a surjective morphism On
X|_U → F|_U for some natural number n;
for any open set U ⊂ X, any natural number n, and any morphism φ: On
X|_U → F|_U of O_X-modules, the kernel of φ is of finite type.

Morphisms between (quasi)coherent sheaves are the same as morphisms of sheaves of O_X-modules.

The case of schemes

When X is a scheme, the general definitions above are equivalent to more explicit ones. A sheaf F of O_X-modules is quasicoherent if and only if over each open affine subscheme U=Spec(R) the restriction F|_U is isomorphic to a sheaf ${\tilde {M}}$ associated to the module M=Γ(U, F) over R. When X is a locally Noetherian scheme, F is coherent if and only if it is quasi-coherent and the modules M above can be taken to be finitely generated.

On an affine scheme U = Spec A, there is an equivalence of categories from A-modules to quasi-coherent sheaves, taking a module M to the associated sheaf ~M. The inverse equivalence takes a quasi-coherent sheaf F on U to the A-module F(U) of global sections of F.

Here are several further characterizations of quasi-coherent sheaves on a scheme.^[1]

Theorem — Let X be a scheme and F an O_X-module on it. Then the following are equivalent.

F is quasi-coherent.
For each open affine subscheme U of X, F|_U is isomorphic as an O_U-module to the sheaf ~M associated to some O(U)-module M.
There is an open affine cover $\{U_{\alpha }\}$ of X such that for any U_α from the cover F|_{U_α} is the sheaf associated to some O(U_α)-module.
For each pair of open affine subschemes V ⊂ U of X, the natural homomorphism
$O(V)\otimes _{O(U)}F(U)\to F(V),\,f\otimes s\mapsto f\cdot s|_{V}$

is an isomorphism.

For each open affine subscheme U = Spec A of X and each f in A, writing {f ≠ 0} for the open subscheme of U where f is not zero, the natural homomorphism
$F(U){\bigg [}{\frac {1}{f}}{\bigg ]}\to F(\{f\neq 0\})$

is an isomorphism. The homomorphism comes from the universal property of localization.

Properties

On an arbitrary ringed space quasi-coherent sheaves don't form an abelian category. On the other hand, the quasi-coherent sheaves on any scheme form an abelian category, and they are extremely useful in that context.^[2]

On any ringed space X, the coherent sheaves form an abelian category, a full subcategory of the category of O_X-modules.^[3] (Analogously, the category of coherent modules over any ring R is a full abelian subcategory of the category of all R-modules.) So the kernel, image, and cokernel of any map of coherent sheaves are coherent. The direct sum of two coherent sheaves is coherent; more generally, an O_X-module that is an extension of two coherent sheaves is coherent.^[4]

A submodule of a coherent sheaf is coherent if it is of finite type. A coherent sheaf is always an O_X-module of finite presentation, meaning that each point x in X has an open neighborhood U such that the restriction F|_U of F to U is isomorphic to the cokernel of a morphism O_Xⁿ|_U → O_X^m|_U for some natural numbers n and m. If O_X is coherent, then, conversely, every sheaf of finite presentation over O_X is coherent.

The sheaf of rings O_X is called coherent if it is coherent considered as a sheaf of modules over itself. In particular, the Oka coherence theorem states that the sheaf of holomorphic functions on a complex analytic space X is a coherent sheaf of rings. The main part of the proof is the case X = Cⁿ. Likewise, on a locally Noetherian scheme X, the structure sheaf O_X is a coherent sheaf of rings.^[5]

Basic constructions of coherent sheaves

An O_X-module F on a ringed space X is called locally free of finite rank, or a vector bundle, if every point in X has an open neighborhood U such that the restriction F|_U is isomorphic to a finite direct sum of copies of O_X|_U. If F is free of the same rank n near every point of X, then the vector bundle F is said to be of rank n.

Vector bundles in this sheaf-theoretic sense over a scheme X are equivalent to vector bundles defined in a more geometric way, as a scheme E with a morphism f: E → X and with a covering of X by open sets U_α with given isomorphisms f⁻¹(U_α) ≅ Aⁿ × U_α over U_α such that the two isomorphisms over an intersection U_α ∩ U_β differ by a linear automorphism.^[6] (The analogous equivalence also holds for complex analytic spaces.) For example, given a vector bundle E in this geometric sense, the corresponding sheaf is defined by: the group of sections E(U) over an open set U in X is the set of sections of the morphism f⁻¹(U) → U. The sheaf-theoretic interpretation of vector bundles has the advantage that vector bundles (on a locally Noetherian scheme) are included in the abelian category of coherent sheaves.

Let X = Spec(R), R a Noetherian ring. Then vector bundles on X are exactly the sheaves associated to finitely generated projective modules over R, or (equivalently) to finitely generated flat modules over R. ^[7]
Let X = Proj(R), R a Noetherian $\mathbb {N}$ -graded ring, be a projective scheme over a Noetherian ring R₀. Then each $\mathbb {Z}$ -graded R-module M determines a quasi-coherent sheaf F on X such that $F|_{\{f\neq 0\}}$ is the sheaf associated to the $R[f^{-1}]_{0}$ -module $M[f^{-1}]_{0}$ , where f is a homogeneous element of R of positive degree and $\{f\neq 0\}=\operatorname {Spec} R[f^{-1}]_{0}$ is the locus where f doesn't vanish.
For example, for each integer n, let R(n) denote the graded R-module given by $R(n)_{l}=R_{n+l}$ . Then each $R(n)$ determines the quasi-coherent sheaf ${\mathcal {O}}_{X}(n)$ on X. If R is generated as R₀-algebra by R₁, then ${\mathcal {O}}_{X}(n)$ are line bundles (invertible sheaves) on X and ${\mathcal {O}}_{X}(n)$ is the n-th tensor power of ${\mathcal {O}}_{X}(1)$ . In particular, ${\mathcal {O}}_{{\mathbb {P} }^{n}}(-1)$ is called the tautological line bundle on the projective n-space.
A simple example of a coherent sheaf on $\mathbb {P} ^{2}$ which is not a vector bundle is given by the cokernel in the following sequence

{\mathcal {O}}(1){\xrightarrow {\cdot (x^{2}-yz,y^{3}+xy^{2}-xyz)}}{\mathcal {O}}(3)\oplus {\mathcal {O}}(4)\to {\mathcal {E}}\to 0

this is because ${\mathcal {E}}$ restricted to the vanishing locus of the two polynomials is the zero object.

Ideal sheaves: If Z is a closed subscheme of a locally Noetherian scheme X, the sheaf I_Z/X of all regular functions vanishing on Z is coherent. Likewise, if Z is a closed analytic subspace of a complex analytic space X, the ideal sheaf I_Z/X is coherent.
The structure sheaf O_Z of a closed subscheme Z of a locally Noetherian scheme X can be viewed as a coherent sheaf on X. To be precise, this is the direct image sheaf i_*O_Z, where i: Z → X is the inclusion. Likewise for a closed analytic subspace of a complex analytic space. The sheaf i_*O_Z has fiber (defined below) of dimension zero at points in the open set X−Z, and fiber of dimension 1 at points in Z. There is a short exact sequence of coherent sheaves on X:

0\to I_{Z/X}\to O_{X}\to i_{*}O_{Z}\to 0.

Most operations of linear algebra preserve coherent sheaves. In particular, for coherent sheaves F and G on a ringed space X, the tensor product sheaf F⊗ _{O_X}G and the sheaf of homomorphisms Hom_{O_X}(F,G) are coherent.^[8]
A simple non-example of a quasi-coherent sheaf is given by the extension by zero functor. For example, consider $i_{!}{\mathcal {O}}_{X}$ for
$X={\text{Spec}}(\mathbb {C} [x,x^{-1}]){\xrightarrow {i}}{\text{Spec}}(\mathbb {C} [x])=Y$ ^[9]

Since this sheaf has non-trivial stalks, but zero global sections, this cannot be a quasi-coherent sheaf. This is because quasi-coherent sheaves on an affine scheme are equivalent to the category of modules over the underlying ring, and the adjunction comes from taking global sections.

Functoriality

Let ƒ: X → Y be a morphism of ringed spaces (for example, a morphism of schemes). If F is a quasi-coherent sheaf on Y, then the inverse image O_X-module (or pullback) f*F is quasi-coherent on X.^[10] For a morphism of schemes f: X → Y and a coherent sheaf F on Y, the pullback f^*F is not coherent in full generality (for example, f*O_Y = O_X, which might not be coherent), but pullbacks of coherent sheaves are coherent if X is locally Noetherian. An important special case is the pullback of a vector bundle, which is a vector bundle.

If f: X → Y is a quasi-compact quasi-separated morphism of schemes and E is a quasi-coherent sheaf on X, then the direct image sheaf (or pushforward) f_*E is quasi-coherent on Y.^[2]

The direct image of a coherent sheaf is often not coherent. For example, for a field k, let X be the affine line over k, and consider the morphism f: X → Spec(k); then the direct image f_*O_X is the sheaf on Spec(k) associated to the polynomial ring k[x], which is not coherent because k[x] has infinite dimension as a k-vector space. On the other hand, the direct image of a coherent sheaf under a proper morphism is coherent, by results of Grauert and Grothendieck.

Local behavior of coherent sheaves

An important feature of coherent sheaves F is that the properties of F at a point p control the behavior of F in a neighborhood of p, more than would be true for an arbitrary sheaf. For example, Nakayama's lemma says (in geometric language) that if F is a coherent sheaf on a scheme X, then the fiber F_p⊗_{O_X,p}k(p) of F at a point p (a vector space over the residue field k(p)) is zero if and only if the sheaf F is zero on some open neighborhood of p. A related fact is that the dimension of the fibers of a coherent sheaf is upper-semicontinuous.^[11] Thus a coherent sheaf has constant rank on an open set, while the rank can jump up on a lower-dimensional closed subset.

In the same spirit: a coherent sheaf F on a scheme X is a vector bundle if and only if its stalk F_p is a free module over the local ring O_X,p for every point p in X.^[12]

On a general scheme, one cannot determine whether a coherent sheaf is a vector bundle just from its fibers (as opposed to its stalks). On a reduced locally Noetherian scheme, however, a coherent sheaf is a vector bundle if and only if its rank is locally constant.^[13]

Examples of vector bundles

For a morphism of schemes X → Y, let Δ: X → X ×_Y X be the diagonal morphism, which is a closed immersion if X is separated over Y. Let I be the ideal sheaf of X in X ×_Y X. Then the sheaf of differentials Ω¹_X/Y can be defined as the pullback Δ*(I) of I to X. Sections of this sheaf are called 1-forms on X over Y, and they can be written locally on X as finite sums ∑ f_j dg_j for regular functions f_j and g_j. If X is locally of finite type over a field k, then Ω¹_X/k is a coherent sheaf on X.

If X is smooth over k, then Ω¹ (meaning Ω¹_X/k) is a vector bundle over X, called the cotangent bundle of X. Then the tangent bundle TX is defined to be the dual bundle (Ω¹)*. For X smooth over k of dimension n everywhere, the tangent bundle has rank n.

If Y is a smooth closed subscheme of a smooth scheme X over k, then there is a short exact sequence of vector bundles on Y:

0\to TY\to TX|_{Y}\to N_{Y/X}\to 0,

which can be used as a definition of the normal bundle N_Y/X to Y in X.

For a smooth scheme X over a field k and a natural number a, the vector bundle Ω^a of a-forms on X is defined as the ath exterior power of the cotangent bundle, Ω^a = Λ^aΩ¹. For a smooth variety X of dimension n over k, the canonical bundle K_X means the line bundle Ωⁿ. Thus sections of the canonical bundle are algebro-geometric analogs of volume forms on X. For example, a section of the canonical bundle of affine space Aⁿ over k can be written as

f(x_{1},\ldots ,x_{n})\;dx_{1}\wedge \cdots \wedge dx_{n},

where f is a polynomial with coefficients in k.

Let R be a commutative ring and n a natural number. For each integer j, there is an important example of a line bundle on projective space Pⁿ over R, called O(j). To define this, consider the morphism of R-schemes

\pi \colon A^{n+1}-0\to \mathbf {P} ^{n}

given in coordinates by (x₀,...,x_n) ↦ [x₀,...,x_n]. (That is, thinking of projective space as the space of 1-dimensional linear subspaces of affine space, send a nonzero point in affine space to the line that it spans.) Then a section of O(j) over an open subset U of Pⁿ is defined to be a regular function f on π⁻¹(U) that is homogeneous of degree j, meaning that

f(ax)=a^{j}f(x)

as regular functions on (A¹ − 0) × π⁻¹(U). For all integers i and j, there is an isomorphism O(i) ⊗ O(j) ≅ O(i+j) of line bundles on Pⁿ.

In particular, every homogeneous polynomial in x₀,...,x_n of degree j over R can be viewed as a global section of O(j) over Pⁿ. Note that every closed subscheme of projective space can be defined as the zero set of some collection of homogeneous polynomials, hence as the zero set of some sections of the line bundles O(j).^[14] This contrasts with the simpler case of affine space, where a closed subscheme is simply the zero set of some collection of regular functions. The regular functions on projective space Pⁿ over R are just the "constants" (the ring R), and so it is essential to work with the line bundles O(j).

Serre gave an algebraic description of all coherent sheaves on projective space, more subtle than what happens for affine space. Namely, let R be a Noetherian ring (for example, a field), and consider the polynomial ring S = R[x₀,...,x_n] as a graded ring with each x_i having degree 1. Then every finitely generated graded S-module M has an associated coherent sheaf ~M on Pⁿ over R. Every coherent sheaf on Pⁿ arises in this way from a finitely generated graded S-module M. (For example, the line bundle O(j) is the sheaf associated to the S-module S with its grading lowered by j.) But the S-module M that yields a given coherent sheaf on Pⁿ is not unique; it is only unique up to changing M by graded modules that are nonzero in only finitely many degrees. More precisely, the abelian category of coherent sheaves on Pⁿ is the quotient of the category of finitely generated graded S-modules by the Serre subcategory of modules that are nonzero in only finitely many degrees.^[15]

The tangent bundle of projective space Pⁿ over a field k can be described in terms of the line bundle O(1). Namely, there is a short exact sequence, the Euler sequence:

0\to O_{\mathbf {P} ^{n}}\to O(1)^{\oplus \;n+1}\to T\mathbf {P} ^{n}\to 0.

It follows that the canonical bundle K_Pⁿ (the dual of the determinant line bundle of the tangent bundle) is isomorphic to O(−n−1). This is a fundamental calculation for algebraic geometry. For example, the fact that the canonical bundle is a negative multiple of the ample line bundle O(1) means that projective space is a Fano variety. Over the complex numbers, this means that projective space has a Kähler metric with positive Ricci curvature.

Vector Bundles on a Hypersurface

Consider a smooth degree $d$ hypersurface $X\subset \mathbb {P} ^{n}$ defined by the homogeneous polynomial $f$ . Then, there is an exact sequence

0\to {\mathcal {O}}_{X}(-d)\to i^{*}\Omega _{\mathbb {P} ^{n}}\to \Omega _{X}\to 0

where the second map is the pullback of differential forms, and the first map sends

\phi \mapsto d(f\cdot \phi )

Note that this sequence tells us that ${\mathcal {O}}(-d)$ is the conormal sheaf of $X$ in $\mathbb {P} ^{n}$ . Dualizing this yields the exact sequence

0\to T_{X}\to i^{*}T_{\mathbb {P} ^{n}}\to {\mathcal {O}}(d)\to 0

hence ${\mathcal {O}}(d)$ is the normal bundle of $X$ in $\mathbb {P} ^{n}$ . If we use the fact that given an exact sequence

0\to {\mathcal {E}}_{1}\to {\mathcal {E}}_{2}\to {\mathcal {E}}_{3}\to 0

of ranks $e_{1},e_{2},e_{3},$ , there is an isomorphism

\Lambda ^{e_{2}}{\mathcal {E}}_{2}\cong \Lambda ^{e_{1}}{\mathcal {E}}_{1}\otimes \Lambda ^{e_{2}}{\mathcal {E}}_{2}

of line bundles, then we see that there is the isomorphism

i^{*}\omega _{\mathbb {P} ^{n}}\cong \omega _{X}\otimes {\mathcal {O}}_{X}(-d)

showing that

\omega _{X}\cong {\mathcal {O}}_{X}(d-n-1)

Chern classes and algebraic K-theory

A vector bundle E on a smooth variety X over a field has Chern classes in the Chow ring of X, c_i(E) in CHⁱ(X) for i ≥ 0.^[16] These satisfy the same formal properties as Chern classes in topology. For example, for any short exact sequence

0\to A\to B\to C\to 0

of vector bundles on X, the Chern classes of B are given by

c_{i}(B)=c_{i}(A)+c_{1}(A)c_{i-1}(C)+\cdots +c_{i-1}(A)c_{1}(C)+c_{i}(C).

It follows that the Chern classes of a vector bundle E depend only on the class of E in the Grothendieck group K₀(X). By definition, for a scheme X, K₀(X) is the quotient of the free abelian group on the set of isomorphism classes of vector bundles on X by the relation that [B] = [A] + [C] for any short exact sequence as above. Although K₀(X) is hard to compute in general, algebraic K-theory provides many tools for studying it, including a sequence of related groups K_i(X) for integers i.

A variant is the group G₀(X) (or K₀'(X)), the Grothendieck group of coherent sheaves on X. (In topological terms, G-theory has the formal properties of a Borel–Moore homology theory for schemes, while K-theory is the corresponding cohomology theory.) The natural homomorphism K₀(X) → G₀(X) is an isomorphism if X is a regular separated Noetherian scheme, using that every coherent sheaf has a finite resolution by vector bundles in that case.^[17] For example, that gives a definition of the Chern classes of a coherent sheaf on a smooth variety over a field.

More generally, a Noetherian scheme X is said to have the resolution property if every coherent sheaf on X has a surjection from some vector bundle on X. For example, every quasi-projective scheme over a Noetherian ring has the resolution property.

The category of quasi-coherent sheaves

Quasi-coherent sheaves on any scheme form an abelian category. Gabber showed that, in fact, the quasi-coherent sheaves on any scheme form a particularly well-behaved abelian category, a Grothendieck category.^[18] A quasi-compact quasi-separated scheme X (such as an algebraic variety over a field) is determined up to isomorphism by the abelian category of quasi-coherent sheaves on X, by Rosenberg, generalizing a result of Gabriel.^[19]

Coherent cohomology

The fundamental technical tool in algebraic geometry is the cohomology theory of coherent sheaves. Although it was introduced only in the 1950s, many earlier techniques of algebraic geometry are clarified by the language of sheaf cohomology applied to coherent sheaves. Broadly speaking, coherent sheaf cohomology can be viewed as a tool for producing functions with specified properties; sections of line bundles or of more general sheaves can be viewed as generalized functions. In complex analytic geometry, coherent sheaf cohomology also plays a foundational role.

Among the core results of coherent sheaf cohomology are results on finite-dimensionality of cohomology, results on the vanishing of cohomology in various cases, duality theorems such as Serre duality, relations between topology and algebraic geometry such as Hodge theory, and formulas for Euler characteristics of coherent sheaves such as the Riemann–Roch theorem.

Notes

↑ Mumford, Ch. III, § 1, Theorem-Definition 3.
1 2 Stacks Project, Tag 01LA .
↑ Stacks Project, Tag 01BU .
↑ Serre (1955), section 13.
↑ Grothendieck, EGA I, Corollaire 1.5.2.
↑ Hartshorne (1977), Exercise II.5.18.
↑ Stacks Project, Tag 00NV .
↑ Serre (1955), section 14.
↑ Hartshorne, Robin. Algebraic Geometry.
↑ Stacks Project, Tag 01BG .
↑ Hartshorne (1977), Example III.12.7.2.
↑ Grothendieck, EGA I, Ch. 0, 5.2.7.
↑ Eisenbud (1995), Exercise 20.13.
↑ Hartshorne (1977), Corollary II.5.16.
↑ Stacks Project, Tag 01YR .
↑ Fulton (1998), section 3.2 and Example 8.3.3.
↑ Fulton (1998), B.8.3.
↑ Stacks Project, Tag 077K .
↑ Antieau (2016), Corollary 4.2.

References

Pedagogical

Gathman, Andreas, Algebraic Geometry (PDF)

Reference

Antieau, Benjamin (2016), "A reconstruction theorem for abelian categories of twisted sheaves", Journal für die reine und angewandte Mathematik, 712: 175–188, MR 3466552, doi:10.1515/crelle-2013-0119
Danilov, V. I. (2001) [1994], "Coherent algebraic sheaf", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Grauert, Hans; Remmert, Reinhold (1984), Coherent Analytic Sheaves, Springer-Verlag, ISBN 3-540-13178-7, MR 0755331
Eisenbud, David (1995), Commutative Algebra with a View toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1, MR 1322960
Fulton, William (1998), Intersection Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98549-7, MR 1644323
Sections 0.5.3 and 0.5.4 of Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. MR 0217083. doi:10.1007/bf02684778.
Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Mumford, David (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians (2nd ed.). Springer-Verlag. ISBN 354063293X. MR 1748380. doi:10.1007/b62130.
Onishchik, A.L. (2001) [1994], "Coherent analytic sheaf", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Onishchik, A.L. (2001) [1994], "Coherent sheaf", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Serre, Jean-Pierre (1955), "Faisceaux algébriques cohérents", Annals of Mathematics, 61: 197–278, MR 0068874, doi:10.2307/1969915

External links

The Stacks Project Authors, The Stacks Project
Part V of Vakil, Ravi, The Rising Sea

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