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.
Contents
1 Definitions
2 The case of schemes
3 Properties
4 Basic constructions of coherent sheaves
5 Functoriality
6 Local behavior of coherent sheaves
7 Examples of vector bundles
7.1 Vector bundles on a hypersurface
8 Chern classes and algebraic K-theory
9 Bundle homomorphism v.s. sheaf homorphism
10 The category of quasi-coherent sheaves
11 Coherent cohomology
12 See also
13 Notes
14 References
15 External links
Definitions
A quasi-coherent sheaf on a ringed space (X,OX) is a sheaf F of OX-modules which has a local presentation, that is, every point in X has an open neighborhood U in which there is an exact sequence
- OX⊕I|U→OX⊕J|U→F|U→0_Uto mathcal O_X^oplus J
for some sets I and J (possibly infinite).
A coherent sheaf on a ringed space (X,OX) is a quasi-coherent sheaf F satisfying the following two properties:
F is of finite type over OX, 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 OX-modules, the kernel of φ is of finite type.
Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of OX-modules.
The case of schemes
When X is a scheme, the general definitions above are equivalent to more explicit ones. A sheaf F of OX-modules is quasi-coherent if and only if over each open affine subscheme U=Spec(R) the restriction F|U is isomorphic to a sheaf M~displaystyle 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 OX-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 OU-module to the sheaf ~M associated to some O(U)-module M.
- There is an open affine cover Uαdisplaystyle U_alpha of X such that for any Uα from the cover F|Uα is the sheaf associated to some O(Uα)-module.
- For each pair of open affine subschemes V ⊂ U of X, the natural homomorphism
- O(V)⊗O(U)F(U)→F(V),f⊗s↦f⋅s|Vdisplaystyle O(V)otimes _O(U)F(U)to F(V),,fotimes smapsto fcdot s
- 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)[1f]→F(f≠0)displaystyle F(U)bigg [frac 1fbigg ]to F(fneq 0)
- is an isomorphism. The homomorphism comes from the universal property of localization.
Properties
On an arbitrary ringed space quasi-coherent sheaves do not necessarily 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 OX-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 OX-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 OX-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 OXn|U → OXm|U for some natural numbers n and m. If OX is coherent, then, conversely, every sheaf of finite presentation over OX is coherent.
The sheaf of rings OX 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 OX is a coherent sheaf of rings.[5]
Basic constructions of coherent sheaves
- An OX-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 OX|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 Uα with given isomorphisms f−1(Uα) ≅ An × Uα over Uα such that the two isomorphisms over an intersection Uα ∩ Uβ 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.
- Locally free sheaves come equipped with the standard OXdisplaystyle mathcal O_X-module operations, but these give back locally free sheaves.[vague]
- 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 Ndisplaystyle mathbb N -graded ring, be a projective scheme over a Noetherian ring R0. Then each Zdisplaystyle mathbb Z -graded R-module M determines a quasi-coherent sheaf F on X such that F|f≠0_fneq 0 is the sheaf associated to the R[f−1]0displaystyle R[f^-1]_0-module M[f−1]0displaystyle M[f^-1]_0, where f is a homogeneous element of R of positive degree and f≠0=SpecR[f−1]0displaystyle fneq 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=Rn+ldisplaystyle R(n)_l=R_n+l. Then each R(n)displaystyle R(n) determines the quasi-coherent sheaf OX(n)displaystyle mathcal O_X(n) on X. If R is generated as R0-algebra by R1, then OX(n)displaystyle mathcal O_X(n) are line bundles (invertible sheaves) on X and OX(n)displaystyle mathcal O_X(n) is the n-th tensor power of OX(1)displaystyle mathcal O_X(1). In particular, OPn(−1)displaystyle 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 P2displaystyle mathbb P ^2 which is not a vector bundle is given by the cokernel in the following sequence
- O(1)→⋅(x2−yz,y3+xy2−xyz)O(3)⊕O(4)→E→0displaystyle mathcal O(1)xrightarrow cdot (x^2-yz,y^3+xy^2-xyz)mathcal O(3)oplus mathcal O(4)to mathcal Eto 0
- this is because Edisplaystyle 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 IZ/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 IZ/X is coherent.
- The structure sheaf OZ 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*OZ, where i: Z → X is the inclusion. Likewise for a closed analytic subspace of a complex analytic space. The sheaf i*OZ 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→IZ/X→OX→i∗OZ→0.displaystyle 0to I_Z/Xto O_Xto i_*O_Zto 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⊗ OXG and the sheaf of homomorphisms HomOX(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!OXdisplaystyle i_!mathcal O_X for
X=Spec(C[x,x−1])→iSpec(C[x])=Ydisplaystyle X=textSpec(mathbb C [x,x^-1])xrightarrow itextSpec(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 OX-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*OY = OX, 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*OX 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 Fp⊗OX,pk(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 Fp is a free module over the local ring OX,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 ×YX 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 ×YX. Then the sheaf of differentials Ω1X/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 ∑ fj dgj for regular functions fj and gj. If X is locally of finite type over a field k, then Ω1X/k is a coherent sheaf on X.
If X is smooth over k, then Ω1 (meaning Ω1X/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→TY→TX|Y→NY/X→0,_Yto N_Y/Xto 0,
which can be used as a definition of the normal bundle NY/X to Y in X.
For a smooth scheme X over a field k and a natural number a, the vector bundle Ωa of a-forms on X is defined as the ath exterior power of the cotangent bundle, Ωa = ΛaΩ1. For a smooth variety X of dimension n over k, the canonical bundle KX 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(x1,…,xn)dx1∧⋯∧dxn,displaystyle f(x_1,ldots ,x_n);dx_1wedge 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
- π:An+1−0→Pndisplaystyle pi colon A^n+1-0to mathbf P ^n
given in coordinates by (x0,...,xn) ↦ [x0,...,xn]. (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)=ajf(x)displaystyle f(ax)=a^jf(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,...,xn 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,...,xn] as a graded ring with each xi 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→OPn→O(1)⊕n+1→TPn→0.displaystyle 0to O_mathbf P ^nto O(1)^oplus ;n+1to Tmathbf P ^nto 0.
It follows that the canonical bundle KPn (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 ddisplaystyle d hypersurface X⊂Pndisplaystyle Xsubset mathbb P ^n defined by the homogeneous polynomial fdisplaystyle f. Then, there is an exact sequence
- 0→OX(−d)→i∗ΩPn→ΩX→0displaystyle 0to mathcal O_X(-d)to i^*Omega _mathbb P ^nto Omega _Xto 0
where the second map is the pullback of differential forms, and the first map sends
- ϕ↦d(f⋅ϕ)displaystyle phi mapsto d(fcdot phi )
Note that this sequence tells us that O(−d)displaystyle mathcal O(-d) is the conormal sheaf of Xdisplaystyle X in Pndisplaystyle mathbb P ^n. Dualizing this yields the exact sequence
- 0→TX→i∗TPn→O(d)→0displaystyle 0to T_Xto i^*T_mathbb P ^nto mathcal O(d)to 0
hence O(d)displaystyle mathcal O(d) is the normal bundle of Xdisplaystyle X in Pndisplaystyle mathbb P ^n. If we use the fact that given an exact sequence
- 0→E1→E2→E3→0displaystyle 0to mathcal E_1to mathcal E_2to mathcal E_3to 0
of ranks e1,e2,e3,displaystyle e_1,e_2,e_3,, there is an isomorphism
- Λe2E2≅Λe1E1⊗Λe2E2displaystyle Lambda ^e_2mathcal E_2cong Lambda ^e_1mathcal E_1otimes Lambda ^e_2mathcal E_2
of line bundles, then we see that there is the isomorphism
- i∗ωPn≅ωX⊗OX(−d)displaystyle i^*omega _mathbb P ^ncong omega _Xotimes mathcal O_X(-d)
showing that
- ωX≅OX(d−n−1)displaystyle omega _Xcong 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, ci(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→A→B→C→0displaystyle 0to Ato Bto Cto 0
of vector bundles on X, the Chern classes of B are given by
- ci(B)=ci(A)+c1(A)ci−1(C)+⋯+ci−1(A)c1(C)+ci(C).displaystyle 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 Ki(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.
Bundle homomorphism v.s. sheaf homorphism
When vector bundles and locally free sheaves of finite constant rank are used interchangeably,
care must be given to distinguish between bundle homorphisms and sheaf homorphisms. Specifically, given vector bundles p:E→X,q:F→Xdisplaystyle p:Eto X,,q:Fto X, by definition, a bundle homomorphism φ:E→Fdisplaystyle varphi :Eto F is a scheme morphism over X (i.e., p=q∘φdisplaystyle p=qcirc varphi ) such that, for each geometric point x in X, φ:p−1(x)→q−1(x)displaystyle varphi :p^-1(x)to q^-1(x) is a linear map of rank independent of x. Thus, it induces the sheaf homorphism φ~:E~→F~displaystyle widetilde varphi :widetilde Eto widetilde F of constant rank between the corresponding locally free OXdisplaystyle mathcal O_X-modules (sheaves of dual sections). But there may be an OXdisplaystyle mathcal O_X-module homorphism that does not arise this way; namely, those not having constant rank.
In particular, a subbundle E⊂Fdisplaystyle Esubset F is a subsheaf (E,Fdisplaystyle E,F viewed as sheaves). But the converse can fail; for example, for an effective Cartier divisor D on X, OX(−D)⊂OXdisplaystyle mathcal O_X(-D)subset mathcal O_X is a subsheaf but typically not a subbundle (since any line bundle has only two subbundles).
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.
See also
- Picard group
- Divisor (algebraic geometry)
- Reflexive sheaf
- Quot scheme
- Twisted sheaf
- Essentially finite vector bundle
- Bundle of principal parts
- Gabriel–Rosenberg reconstruction theorem
- Pseudo-coherent sheaf
Notes
^ Mumford, Ch. III, § 1, Theorem-Definition 3.
^ ab Stacks Project, Tag 01LA.mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em.
^ 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
Antieau, Benjamin (2016), "A reconstruction theorem for abelian categories of twisted sheaves", Journal für die reine und angewandte Mathematik, 712: 175–188, arXiv:1305.2541, doi:10.1515/crelle-2013-0119, MR 3466552
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, doi:10.1007/978-3-642-69582-7, 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, doi:10.1007/978-1-4612-5350-1, ISBN 978-0-387-94268-1, MR 1322960
Fulton, William (1998), Intersection Theory, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-1700-8, 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. doi:10.1007/bf02684778. MR 0217083.
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. doi:10.1007/b62130. ISBN 354063293X. MR 1748380.
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, doi:10.2307/1969915, MR 0068874
External links
The Stacks Project Authors, The Stacks Project- Part V of Vakil, Ravi, The Rising Sea