coherent sheaf

A quasi-coherent sheaf on a ringed space $(X,\; \backslash mathcal\; O\_X)$ is a sheaf $\backslash mathcal\; F$ of $\backslash mathcal\; O\_X$-modules that has a local presentation, that is, every point in $X$ has an open neighborhood $U$ in which there is an exact sequence

:$\backslash mathcal\{O\}\_X^\{\backslash oplus\; I\}|\_\{U\}\; \backslash to\; \backslash mathcal\{O\}\_X^\{\backslash oplus\; J\}|\_\{U\}\; \backslash to\; \backslash mathcal\{F\}|\_\{U\}\; \backslash to\; 0$

for some (possibly infinite) sets $I$ and $J$.

A coherent sheaf on a ringed space $(X,\; \backslash mathcal\; O\_X)$ is a sheaf $\backslash mathcal\; F$ of $\backslash mathcal\; O\_X$-modules satisfying the following two properties:

1. $\backslash mathcal\; F$ is of finite type over $\backslash mathcal\; O\_X$, that is, every point in $X$ has an open neighborhood $U$ in $X$ such that there is a surjective morphism $\backslash mathcal\{O\}\_X^n|\_\{U\}\; \backslash to\; \backslash mathcal\{F\}|\_\{U\}$ for some natural number $n$;
2. for any open set $U\backslash subseteq\; X$, any natural number $n$, and any morphism $\backslash varphi:\; \backslash mathcal\{O\}\_X^n|\_\{U\}\; \backslash to\; \backslash mathcal\{F\}|\_\{U\}$ of $\backslash mathcal\; O\_X$-modules, the kernel of $\backslash varphi$ is of finite type.

Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of $\backslash mathcal\; O\_X$-modules.

When $X$ is a scheme, the general definitions above are equivalent to more explicit ones. A sheaf $\backslash mathcal\; F$ of $\backslash mathcal\; O\_X$-modules is quasi-coherent if and only if over each open affine subscheme $U=\backslash operatorname\{Spec\}\; A$ the restriction $\backslash mathcal\; F|\_U$ is isomorphic to the sheaf $\backslash tilde\{M\}$ associated to the module $M=\backslash Gamma(U,\; \backslash mathcal\; F)$ over $A$. When $X$ is a locally Noetherian scheme, $\backslash mathcal\; F$ 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 $U\; =\; \backslash operatorname\{Spec\}\; A$, there is an equivalence of categories from $A$-modules to quasi-coherent sheaves, taking a module $M$ to the associated sheaf $\backslash tilde\{M\}$. The inverse equivalence takes a quasi-coherent sheaf $\backslash mathcal\; F$ on $U$ to the $A$-module $\backslash mathcal\; F(U)$ of global sections of $\backslash mathcal\; F$.

Here are several further characterizations of quasi-coherent sheaves on a scheme.{{harvnb|Mumford|1999|loc=Ch. III, § 1, Theorem-Definition 3}}.

{{math_theorem|math_statement=Let $X$ be a scheme and $\backslash mathcal\; F$ an $\backslash mathcal\; O\_X$-module on it. Then the following are equivalent.

• $\backslash mathcal\; F$ is quasi-coherent.
• For each open affine subscheme $U$ of $X$, $\backslash mathcal\; F|\_U$ is isomorphic as an $\backslash mathcal\; O\_U$-module to the sheaf $\backslash tilde\; M$ associated to some $\backslash mathcal\; O(U)$-module $M$.
• There is an open affine cover $\backslash \{U\_\backslash alpha\; \backslash \}$ of $X$ such that for each $U\_\backslash alpha$ of the cover, $\backslash mathcal\; F|\_\{U\_\backslash alpha\}$ is isomorphic to the sheaf associated to some $\backslash mathcal\; O(U\_\backslash alpha)$-module.
• For each pair of open affine subschemes $V\backslash subseteq\; U$ of $X$, the natural homomorphism
• :$\backslash mathcal\; O(V)\; \backslash otimes\_\{\backslash mathcal\; O(U)\}\; \backslash mathcal\; F(U)\; \backslash to\; \backslash mathcal\; F(V),\; \backslash ,\; f\; \backslash otimes\; s\; \backslash mapsto\; f\backslash cdot\; s|\_V$

:is an isomorphism.

• For each open affine subscheme $U\; =\; \backslash operatorname\{Spec\}\; A$ of $X$ and each $f\backslash in\; A$, writing $U\_f$ for the open subscheme of $U$ where $f$ is not zero, the natural homomorphism
• :$\backslash mathcal\; F(U)\backslash bigg[\backslash frac\{1\}\{f\}\backslash bigg]\; \backslash to\; \backslash mathcal\; F(U\_f)$

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

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.{{Citation | title=Stacks Project, Tag 01LA | url=http://stacks.math.columbia.edu/tag/01LA}}.

On any ringed space $X$, the coherent sheaves form an abelian category, a full subcategory of the category of $\backslash mathcal\; O\_X$-modules.{{Citation | title=Stacks Project, Tag 01BU | url=http://stacks.math.columbia.edu/tag/01BU}}. (Analogously, the category of coherent modules over any ring $A$ is a full abelian subcategory of the category of all $A$-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 $\backslash mathcal\; O\_X$-module that is an extension of two coherent sheaves is coherent.{{harvnb|Serre|1955|loc=§13}}

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

The sheaf of rings $\backslash mathcal\; O\_X$ 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 $X\; =\; \backslash mathbf\; C^n$. Likewise, on a locally Noetherian scheme $X$, the structure sheaf $\backslash mathcal\; O\_X$ is a coherent sheaf of rings.{{harvnb|Grothendieck|Dieudonné|1960|loc=Corollaire 1.5.2}}

• An $\backslash mathcal\; O\_X$-module $\backslash mathcal\; F$ 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 $\backslash mathcal\; F|\_U$ is isomorphic to a finite direct sum of copies of $\backslash mathcal\; O\_X|\_U$. If $\backslash mathcal\; F$ is free of the same rank $n$ near every point of $X$, then the vector bundle $\backslash mathcal\; F$ 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 $\backslash pi:\; E\backslash to\; X$ and with a covering of $X$ by open sets $U\_\backslash alpha$ with given isomorphisms $\backslash pi^\{-1\}(U\_\backslash alpha)\; \backslash cong\; \backslash mathbb\; A^n\; \backslash times\; U\_\backslash alpha$ over $U\_\backslash alpha$ such that the two isomorphisms over an intersection $U\_\backslash alpha\; \backslash cap\; U\_\backslash beta$ differ by a linear automorphism.{{harvnb|Hartshorne|1977|loc=Exercise II.5.18}} (The analogous equivalence also holds for complex analytic spaces.) For example, given a vector bundle $E$ in this geometric sense, the corresponding sheaf $\backslash mathcal\; F$ is defined by: over an open set $U$ of $X$, the $\backslash mathcal\; O(U)$-module $\backslash mathcal\; F(U)$ is the set of sections of the morphism $\backslash pi^\{-1\}(U)\; \backslash to\; U$. 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 $\backslash mathcal\; O\_X$-module operations, but these give back locally free sheaves.{{vague|date=May 2018}}

• Let $X\; =\; \backslash operatorname\{Spec\}(R)$, $R$ 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 $R$.{{Citation | title=Stacks Project, Tag 00NV | url=http://stacks.math.columbia.edu/tag/00NV}}.
• Let $X\; =\; \backslash operatorname\{Proj\}(R)$, $R$ a Noetherian $\backslash N$-graded ring, be a projective scheme over a Noetherian ring $R\_0$. Then each $\backslash Z$-graded $R$-module $M$ determines a quasi-coherent sheaf $\backslash mathcal\; F$ on $X$ such that $\backslash mathcal\; F|\_\{\backslash \{\; f\; \backslash ne\; 0\; \backslash \}\}$ 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 $\backslash \{f\; \backslash ne\; 0\; \backslash \}\; =\; \backslash operatorname\{Spec\}\; R[f^\{-1\}]\_0$ is the locus where $f$ does not 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 $\backslash mathcal\; O\_X(n)$ on $X$. If $R$ is generated as $R\_0$-algebra by $R\_1$, then $\backslash mathcal\; O\_X(n)$ is a line bundle (invertible sheaf) on $X$ and $\backslash mathcal\; O\_X(n)$ is the $n$-th tensor power of $\backslash mathcal\; O\_X(1)$. In particular, $\backslash mathcal\; O\_\{\backslash mathbb\{P\}^n\}(-1)$ is called the tautological line bundle on the projective $n$-space.
• A simple example of a coherent sheaf on $\backslash mathbb\{P\}^2$ that is not a vector bundle is given by the cokernel in the following sequence

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

:this is because $\backslash mathcal\{E\}$ restricted to the vanishing locus of the two polynomials has two-dimensional fibers, and has one-dimensional fibers elsewhere.

• Ideal sheaves: If $Z$ is a closed subscheme of a locally Noetherian scheme $X$, the sheaf $\backslash mathcal\; I\_\{Z/X\}$ 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 $\backslash mathcal\; I\_\{Z/X\}$ is coherent.
• The structure sheaf $\backslash mathcal\; O\_Z$ 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 $i\_*\backslash mathcal\; O\_Z$, where $i:\; Z\; \backslash to\; X$ is the inclusion. Likewise for a closed analytic subspace of a complex analytic space. The sheaf $i\_*\backslash mathcal\; O\_Z$ 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 $X$:

::$0\backslash to\; \backslash mathcal\; I\_\{Z/X\}\; \backslash to\; \backslash mathcal\; O\_X\; \backslash to\; i\_*\backslash mathcal\; O\_Z\; \backslash to\; 0.$

• Most operations of linear algebra preserve coherent sheaves. In particular, for coherent sheaves $\backslash mathcal\; F$ and $\backslash mathcal\; G$ on a ringed space $X$, the tensor product sheaf $\backslash mathcal\; F\; \backslash otimes\_\{\backslash mathcal\; O\_X\}\backslash mathcal\; G$ and the sheaf of homomorphisms $\backslash mathcal\; Hom\_\{\backslash mathcal\; O\_X\}(\backslash mathcal\; F,\; \backslash mathcal\; G)$ are coherent.{{harvnb|Serre|1955|loc=§14}}
• A simple non-example of a quasi-coherent sheaf is given by the extension by zero functor. For example, consider $i\_!\backslash mathcal\{O\}\_X$ for

::$X\; =\; \backslash operatorname\{Spec\}(\backslash Complex[x,x^\{-1\}])\; \backslash xrightarrow\{i\}\; \backslash operatorname\{Spec\}(\backslash Complex[x])=Y${{harvnb|Hartshorne|1977}}

: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.

Let $f:\; X\backslash to\; Y$ be a morphism of ringed spaces (for example, a morphism of schemes). If $\backslash mathcal\; F$ is a quasi-coherent sheaf on $Y$, then the inverse image $\backslash mathcal\; O\_X$-module (or pullback) $f^*\backslash mathcal\; F$ is quasi-coherent on $X$.{{Citation | title=Stacks Project, Tag 01BG | url=http://stacks.math.columbia.edu/tag/01BG}}. For a morphism of schemes $f:\; X\backslash to\; Y$ and a coherent sheaf $\backslash mathcal\; F$ on $Y$, the pullback $f^*\backslash mathcal\; F$ is not coherent in full generality (for example, $f^*\backslash mathcal\; O\_Y\; =\; \backslash mathcal\; O\_X$, 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 $f:\; X\backslash to\; Y$ is a quasi-compact quasi-separated morphism of schemes and $\backslash mathcal\; F$ is a quasi-coherent sheaf on $X$, then the direct image sheaf (or pushforward) $f\_*\backslash mathcal\; F$ is quasi-coherent on $Y$.

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 $f:\; X\backslash to\; \backslash operatorname\{Spec\}(k)$; then the direct image $f\_*\backslash mathcal\; O\_X$ is the sheaf on $\backslash operatorname\{Spec\}(k)$ 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.

An important feature of coherent sheaves $\backslash mathcal\; F$ is that the properties of $\backslash mathcal\; F$ at a point $x$ control the behavior of $\backslash mathcal\; F$ in a neighborhood of $x$, more than would be true for an arbitrary sheaf. For example, Nakayama's lemma says (in geometric language) that if $\backslash mathcal\; F$ is a coherent sheaf on a scheme $X$, then the fiber $\backslash mathcal\; F\_x\backslash otimes\_\{\backslash mathcal\; O\_\{X,x\}\}\; k(x)$ of $F$ at a point $x$ (a vector space over the residue field $k(x)$) is zero if and only if the sheaf $\backslash mathcal\; F$ is zero on some open neighborhood of $x$. A related fact is that the dimension of the fibers of a coherent sheaf is upper-semicontinuous.{{harvnb|

Hartshorne|1977|loc=Example III.12.7.2}} 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 $\backslash mathcal\; F$ on a scheme $X$ is a vector bundle if and only if its stalk $\backslash mathcal\; F\_x$ is a free module over the local ring $\backslash mathcal\; O\_\{X,x\}$ for every point $x$ in $X$.{{harvnb|Grothendieck|Dieudonné|1960||loc=Ch. 0, 5.2.7}}

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.{{harvnb|Eisenbud|1995|loc=Exercise 20.13}}

For a morphism of schemes $X\backslash to\; Y$, let $\backslash Delta:\; X\backslash to\; X\backslash times\_Y\; X$ be the diagonal morphism, which is a closed immersion if $X$ is separated over $Y$. Let $\backslash mathcal\; I$ be the ideal sheaf of $X$ in $X\backslash times\_Y\; X$. Then the sheaf of differentials $\backslash Omega^1\_\{X/Y\}$ can be defined as the pullback $\backslash Delta^*\backslash mathcal\; I$ of $\backslash mathcal\; I$ 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 $\backslash textstyle\backslash sum\; f\_j\backslash ,\; dg\_j$ for regular functions $f\_j$ and $g\_j$. If $X$ is locally of finite type over a field $k$, then $\backslash Omega^1\_\{X/k\}$ is a coherent sheaf on $X$.

If $X$ is smooth over $k$, then $\backslash Omega^1$ (meaning $\backslash Omega^1\_\{X/k\}$) is a vector bundle over $X$, called the cotangent bundle of $X$. Then the tangent bundle $TX$ is defined to be the dual bundle $(\backslash Omega^1)^*$. 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 $Y$:

:$0\backslash to\; TY\; \backslash to\; TX|\_Y\; \backslash to\; N\_\{Y/X\}\backslash to\; 0,$

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 $i$, the vector bundle $\backslash Omega^i$ of i-forms on $X$ is defined as the $i$-th exterior power of the cotangent bundle, $\backslash Omega^i\; =\; \backslash Lambda^i\; \backslash Omega^1$. For a smooth variety $X$ of dimension $n$ over $k$, the canonical bundle $K\_X$ means the line bundle $\backslash Omega^n$. 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 $\backslash mathbb\; A^n$ over $k$

can be written as

:$f(x\_1,\backslash ldots,x\_n)\; \backslash ;\; dx\_1\; \backslash wedge\backslash cdots\backslash wedge\; dx\_n,$

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 $\backslash mathbb\; P^n$ over $R$, called $\backslash mathcal\; O(j)$. To define this, consider the morphism of $R$-schemes

:$\backslash pi:\; \backslash mathbb\; A^\{n+1\}-0\backslash to\; \backslash mathbb\; P^n$

given in coordinates by $(x\_0,\backslash ldots,x\_n)\; \backslash mapsto\; [x\_0,\backslash ldots,x\_n]$. (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 $\backslash mathcal\; O(j)$ over an open subset $U$ of $\backslash mathbb\; P^n$ is defined to be a regular function $f$ on $\backslash pi^\{-1\}(U)$ that is homogeneous of degree $j$, meaning that

:$f(ax)=a^jf(x)$

as regular functions on ($\backslash mathbb\; A^\{1\}\; -\; 0)\; \backslash times\; \backslash pi^\{-1\}(U)$. For all integers $i$ and $j$, there is an isomorphism $\backslash mathcal\; O(i)\; \backslash otimes\; \backslash mathcal\; O(j)\; \backslash cong\; \backslash mathcal\; O(i+j)$ of line bundles on $\backslash mathbb\; P^n$.

In particular, every homogeneous polynomial in $x\_0,\backslash ldots,x\_n$ of degree $j$ over $R$ can be viewed as a global section of $\backslash mathcal\; O(j)$ over $\backslash mathbb\; P^n$. 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 $\backslash mathcal\; O(j)$.{{harvnb|Hartshorne|1977|loc=Corollary II.5.16}} 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 $\backslash mathbb\; P^n$ over $R$ are just the "constants" (the ring $R$), and so it is essential to work with the line bundles $\backslash mathcal\; O(j)$.

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 $S\; =\; R[x\_0,\backslash ldots,x\_n]$ as a graded ring with each $x\_i$ having degree 1. Then every finitely generated graded $S$-module $M$ has an associated coherent sheaf $\backslash tilde\; M$ on $\backslash mathbb\; P^n$ over $R$. Every coherent sheaf on $\backslash mathbb\; P^n$ arises in this way from a finitely generated graded $S$-module $M$. (For example, the line bundle $\backslash mathcal\; O(j)$ 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 $\backslash mathbb\; P^n$ 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 $\backslash mathbb\; P^n$ 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.{{Citation | title=Stacks Project, Tag 01YR | url=http://stacks.math.columbia.edu/tag/01YR}}.

The tangent bundle of projective space $\backslash mathbb\; P^n$ over a field $k$ can be described in terms of the line bundle $\backslash mathcal\; O(1)$. Namely, there is a short exact sequence, the Euler sequence:

:$0\backslash to\; \backslash mathcal\; O\_\{\backslash mathbb\; P^n\}\backslash to\; \backslash mathcal\; O(1)^\{\backslash oplus\; \backslash ;\; n+1\}\backslash to\; T\backslash mathbb\; P^n\backslash to\; 0.$

It follows that the canonical bundle $K\_\{\backslash mathbb\; P^n\}$ (the dual of the determinant line bundle of the tangent bundle) is isomorphic to $\backslash mathcal\; O(-n-1)$. 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 $\backslash mathcal\; O(1)$ 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.

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

:$0\; \backslash to\; \backslash mathcal\; O\_X(-d)\; \backslash to\; i^*\backslash Omega\_\{\backslash mathbb\{P\}^n\}\; \backslash to\; \backslash Omega\_X\; \backslash to\; 0$

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

:$\backslash phi\; \backslash mapsto\; d(f\backslash cdot\; \backslash phi)$

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

:$0\; \backslash to\; T\_X\; \backslash to\; i^*T\_\{\backslash mathbb\{P\}^n\}\; \backslash to\; \backslash mathcal\; O(d)\; \backslash to\; 0$

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

:$0\; \backslash to\; \backslash mathcal\; E\_1\; \backslash to\; \backslash mathcal\; E\_2\; \backslash to\; \backslash mathcal\; E\_3\; \backslash to\; 0$

of vector bundles with ranks $r\_1$,$r\_2$,$r\_3$, there is an isomorphism

:$\backslash Lambda^\{r\_2\}\backslash mathcal\; E\_2\; \backslash cong\; \backslash Lambda^\{r\_1\}\backslash mathcal\; E\_1\backslash otimes\; \backslash Lambda^\{r\_3\}\backslash mathcal\; E\_3$

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

:$i^*\backslash omega\_\{\backslash mathbb\; P^n\}\; \backslash cong\; \backslash omega\_X\backslash otimes\; \backslash mathcal\; O\_X(-d)$

showing that

:$\backslash omega\_X\; \backslash cong\; \backslash mathcal\; O\_X(d\; -\; n\; -1)$

One useful technique for constructing rank 2 vector bundles is the Serre construction{{Cite journal|last=Serre|first=Jean-Pierre|date=1960–1961|title=Sur les modules projectifs|url=http://www.numdam.org/item/SD_1960-1961__14_1_A2_0/|journal=Séminaire Dubreil. Algèbre et théorie des nombres|language=fr|volume=14|issue=1|pages=1–16}}{{Cite journal|last=Gulbrandsen|first=Martin G.|date=2013-05-20|title=Vector Bundles and Monads On Abelian Threefolds|url=https://www.ux.uis.no/~martingg/papers/abmonad.pdf|journal=Communications in Algebra|volume=41|issue=5|pages=1964–1988|doi=10.1080/00927872.2011.645977|issn=0092-7872|arxiv=0907.3597}}^{pg 3} which establishes a correspondence between rank 2 vector bundles $\backslash mathcal\{E\}$ on a smooth projective variety $X$ and codimension 2 subvarieties $Y$ using a certain $\backslash text\{Ext\}^1$-group calculated on $X$. This is given by a cohomological condition on the line bundle $\backslash wedge^2\backslash mathcal\{E\}$ (see below).

The correspondence in one direction is given as follows: for a section $s\; \backslash in\; \backslash Gamma(X,\backslash mathcal\{E\})$ we can associated the vanishing locus $V(s)\; \backslash subseteq\; X$. If $V(s)$ is a codimension 2 subvariety, then

1. It is a local complete intersection, meaning if we take an affine chart $U\_i\; \backslash subseteq\; X$ then $s|\_\{U\_i\}\; \backslash in\; \backslash Gamma(U\_i,\backslash mathcal\{E\})$ can be represented as a function $s\_i:U\_i\; \backslash to\; \backslash mathbb\{A\}^2$, where $s\_i(p)\; =\; (s\_i^1(p),\; s\_i^2(p))$ and $V(s)\backslash cap\; U\_i\; =\; V(s\_i^1,s\_i^2)$
2. The line bundle $\backslash omega\_X\backslash otimes\; \backslash wedge^2\backslash mathcal\{E\}|\_\{V(s)\}$ is isomorphic to the canonical bundle $\backslash omega\_\{V(s)\}$ on $V(s)$

In the other direction,{{Cite journal|last=Hartshorne|first=Robin|year=1978|title=Stable Vector Bundles of Rank 2 on P3|url=https://eudml.org/doc/163199|journal=Mathematische Annalen|volume=238|issue=3 |pages=229–280|doi=10.1007/BF01420250 }} for a codimension 2 subvariety $Y\; \backslash subseteq\; X$ and a line bundle $\backslash mathcal\{L\}\; \backslash to\; X$ such that

1. $H^1(X,\backslash mathcal\{L\})\; =\; H^2(X,\backslash mathcal\{L\})\; =\; 0$
2. $\backslash omega\_Y\; \backslash cong\; (\backslash omega\_X\backslash otimes\backslash mathcal\{L\})|\_Y$

there is a canonical isomorphism

$\backslash text\{Hom\}((\backslash omega\_X\backslash otimes\backslash mathcal\{L\})|\_Y,\backslash omega\_Y)\; \backslash cong\; \backslash text\{Ext\}^1(\backslash mathcal\{I\}\_Y\backslash otimes\backslash mathcal\{L\},\; \backslash mathcal\{O\}\_X)$,

$0\; \backslash to\; \backslash mathcal\{O\}\_X\; \backslash to\; \backslash mathcal\{E\}\; \backslash to\; \backslash mathcal\{I\}\_Y\backslash otimes\backslash mathcal\{L\}\; \backslash to\; 0$

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 $CH^i(X)$ for $i\backslash geq\; 0$.{{harvnb|Fulton|1998|loc=§3.2 and Example 8.3.3}} These satisfy the same formal properties as Chern classes in topology. For example, for any short exact sequence

:$0\backslash to\; A\; \backslash to\; B\; \backslash to\; C\; \backslash to\; 0$

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

:$c\_i(B)\; =\; c\_i(A)+c\_1(A)c\_\{i-1\}(C)+\backslash cdots+c\_\{i-1\}(A)c\_1(C)+c\_i(C).$

It follows that the Chern classes of a vector bundle $E$ depend only on the class of $E$ in the Grothendieck group $K\_0(X)$. By definition, for a scheme $X$, $K\_0(X)$ 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 $K\_0(X)$ 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 $i>0$.

A variant is the group $G\_0(X)$ (or $K\_0\text{'}(X)$), 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 $K\_0(X)\backslash to\; G\_0(X)$ 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.

{{harvnb|Fulton|1998|loc=B.8.3}} 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.

Since the resolution property states that a coherent sheaf $\backslash mathcal\; E$ on a Noetherian scheme is quasi-isomorphic in the derived category to the complex of vector bundles :$\backslash mathcal\; E\_k\; \backslash to\; \backslash cdots\; \backslash to\; \backslash mathcal\; E\_1\; \backslash to\; \backslash mathcal\; E\_0$

we can compute the total Chern class of $\backslash mathcal\; E$ with

:$c(\backslash mathcal\; E)\; =\; c(\backslash mathcal\; E\_0)c(\backslash mathcal\; E\_1)^\{-1\}\; \backslash cdots\; c(\backslash mathcal\; E\_k)^\{(-1)^k\}$

For example, this formula is useful for finding the Chern classes of the sheaf representing a subscheme of $X$. If we take the projective scheme $Z$ associated to the ideal $(xy,xz)\; \backslash subseteq\; \backslash mathbb\; C[x,y,z,w]$, then

:$c(\backslash mathcal\; O\_Z)\; =\; \backslash frac\{c(\backslash mathcal\; O)c(\backslash mathcal\; O(-3))\}\{c(\backslash mathcal\; O(-2)\backslash oplus\; \backslash mathcal\; O(-2))\}$

since there is the resolution

:$0\; \backslash to\; \backslash mathcal\; O(-3)\; \backslash to\; \backslash mathcal\; O(-2)\backslash oplus\backslash mathcal\; O(-2)\; \backslash to\; \backslash mathcal\; O\; \backslash to\; \backslash mathcal\; O\_Z\; \backslash to\; 0$

over $\backslash mathbb\{CP\}^3$.

When vector bundles and locally free sheaves of finite constant rank are used interchangeably,

care must be given to distinguish between bundle homomorphisms and sheaf homomorphisms. Specifically, given vector bundles $p:\; E\; \backslash to\; X,\; \backslash ,\; q:\; F\; \backslash to\; X$, by definition, a bundle homomorphism $\backslash varphi:\; E\; \backslash to\; F$ is a scheme morphism over $X$ (i.e., $p\; =\; q\; \backslash circ\; \backslash varphi$) such that, for each geometric point $x$ in $X$, $\backslash varphi\_x:\; p^\{-1\}(x)\; \backslash to\; q^\{-1\}(x)$ is a linear map of rank independent of $x$. Thus, it induces the sheaf homomorphism $\backslash widetilde\{\backslash varphi\}:\; \backslash mathcal\; E\; \backslash to\; \backslash mathcal\; F$ of constant rank between the corresponding locally free $\backslash mathcal\; O\_X$-modules (sheaves of dual sections). But there may be an $\backslash mathcal\; O\_X$-module homomorphism that does not arise this way; namely, those not having constant rank.

In particular, a subbundle $E\; \backslash subseteq\; F$ is a subsheaf (i.e., $\backslash mathcal\; E$ is a subsheaf of $\backslash mathcal\; F$). But the converse can fail; for example, for an effective Cartier divisor $D$ on $X$, $\backslash mathcal\; O\_X(-D)\; \backslash subseteq\; \backslash mathcal\; O\_X$ is a subsheaf but typically not a subbundle (since any line bundle has only two subbundles).

The quasi-coherent sheaves on any fixed 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.{{Citation | title=Stacks Project, Tag 077K | url=http://stacks.math.columbia.edu/tag/077K}}. 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.{{harvnb|Antieau|2016|loc=Corollary 4.2}}

{{main|Coherent sheaf 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.

• 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
• Quasi-coherent sheaf on an algebraic stack

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Category:Algebraic geometry

Category:Sheaf theory

Category:Vector bundles

Category:Topological methods of algebraic geometry

Category:Complex manifolds