View source for Sheaf theory

A special mathematical tool which provides a unified approach to establishing connections between local and global properties of topological spaces (in particular geometric objects) and which is a powerful method for studying many problems in contemporary algebra, geometry, topology, and analysis.

A pre-sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s0848401.png" /> on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s0848402.png" /> assigns to each open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s0848403.png" /> an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s0848404.png" /> (a ring, a module over a ring, etc.) and to every pair of open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s0848405.png" /> a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s0848406.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s0848407.png" /> is the identity isomorphism and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s0848408.png" /> for every triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s0848409.png" />. In other words, a pre-sheaf is a contravariant functor from the category of open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484010.png" /> and their inclusions into the category of groups (rings, etc.) and their homomorphisms. The mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484011.png" /> are called restriction homomorphisms (for example, if the elements of the stalk <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484012.png" /> are functions of some type or other defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484014.png" /> is the restriction of these to the smaller subset). A topology on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484016.png" /> is, by definition, the direct limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484017.png" />, is defined in the following way: For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484018.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484019.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484020.png" /> consisting of those points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484022.png" />, which are images of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484023.png" /> in the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484024.png" /> is declared to be open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484025.png" />. In this topology the stalks <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484026.png" /> are discrete, the stalk-wise algebraic operations defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484027.png" /> by taking direct limits are continuous and the natural projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484029.png" />, is a local homeomorphism. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484030.png" /> together with the stalk-wise algebraic operations and the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484031.png" /> is called the sheaf of Abelian groups (rings, etc.) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484032.png" /> associated with the pre-sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484033.png" />.

Every continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484034.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484035.png" /> is called a section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484036.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484037.png" />. The section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484038.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484039.png" /> defined by the zeros in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484040.png" /> is called the zero section. If a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484041.png" /> is zero at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484042.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484043.png" /> coincides with the zero section in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484044.png" />; therefore the set of points at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484045.png" /> is not zero (the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484046.png" />) is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484047.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484048.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484050.png" /> is a certain family of closed sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484051.png" />; in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484052.png" />) be the group (ring, module, etc.) of all sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484053.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484054.png" /> (respectively, all sections over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484055.png" /> with supports in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484056.png" />; in particular, sections with compact support). The assignment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484057.png" /> is a pre-sheaf over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484058.png" />, called the pre-sheaf of sections of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484059.png" />. The assignment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484060.png" /> used in defining the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484061.png" /> also defines homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484062.png" /> which commute with the restrictions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484063.png" />, that is, it defines a homomorphism of pre-sheaves. This homomorphism is an isomorphism provided that the original pre-sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484064.png" /> satisfies the requirements: a) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484066.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484067.png" /> if the restrictions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484069.png" /> to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484070.png" /> are equal; and b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484072.png" /> is a collection of elements such that the restrictions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484074.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484075.png" /> coincide, then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484076.png" /> which has restriction to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484077.png" /> coinciding with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484078.png" />. The concept of a pre-sheaf satisfying these requirements is equivalent to the concept of the sheaf associated with it, therefore such pre-sheaves are quite often called sheaves also.

A sheaf of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484079.png" /> (with the evident projection to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484080.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484081.png" /> is a fixed group (ring, etc.), is called a constant sheaf, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484082.png" />. A sheaf which is constant in sufficiently small neighbourhoods of every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484083.png" /> is called locally constant. The topology of such sheaves is separated (i.e. Hausdorff) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484084.png" /> is a separated space. In more typical situations the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484085.png" /> can be non-separated even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484086.png" /> is separated (such is the case, for example, for the sheaf of germs of continuous (or differentiable) functions which is generated by the pre-sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484087.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484088.png" /> is the set of continuous (differentiable) functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484089.png" />; however, the sheaf of germs of analytic functions on a manifold is separated).

Every homomorphism of pre-sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484090.png" /> induces a mapping of the associated sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484091.png" />, which is a local homeomorphism and maps stalks homomorphically to stalks; such a mapping of sheaves is called a sheaf homomorphism. Mono- and epimorphisms are defined in the standard way. For any sheaf homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484092.png" /> the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484093.png" /> is an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484094.png" />, closed with respect to the stalk-wise algebraic operations. Every subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484095.png" /> which satisfies these requirements is called a subsheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484096.png" />. The quotient sheaf of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484097.png" /> by a subsheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484098.png" /> is defined as the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s08484099.png" /> associated with the pre-sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840100.png" />; moreover, there is an epimorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840101.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840102.png" />. For every open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840103.png" /> there is a subsheaf in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840104.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840105.png" />, which is the union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840106.png" /> with the zero section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840107.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840108.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840109.png" /> denotes the corresponding quotient sheaf (whose restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840110.png" /> coincides with the restriction thereto of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840111.png" />).

Since it is possible to interpret such ordinary terms as homomorphism, kernel, image, subsheaf, quotient sheaf, etc. for sheaves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840112.png" /> in such a way that these concepts have essentially the same meaning as in algebra, one can consider them from a categorical point of view and apply to sheaf theory the constructions of [[Homological algebra|homological algebra]]. The resulting category of sheaves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840113.png" /> has the same classical properties as the category of Abelian groups or the category of modules; in particular, one can define for sheaves direct sums, infinite direct products, inductive limits, and other concepts.

The apparatus of sheaf theory has penetrated into various fields of mathematics thanks to the fact that there is a natural definition of the cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840114.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840115.png" /> with coefficients in a sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840116.png" />, and this without any kind of restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840117.png" /> (this is essential, for example, in algebraic geometry, where the spaces arising are, as a rule, non-separated) and to the fact that other cohomologies (under certain specific conditions) reduce to a sheaf cohomology, at least in those situations where their application is justified.

To define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840118.png" /> one first constructs the canonical resolution

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840119.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840120.png" /> is the sheaf defined by the pre-sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840121.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840122.png" /> is the group of all (possibly discontinuous) sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840123.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840124.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840125.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840126.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840127.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840128.png" />. By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840129.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840130.png" /> is obtained by replacing the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840131.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840132.png" />). The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840133.png" /> itself can be obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840134.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840135.png" /> (in classical cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840136.png" /> is the group of locally constant functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840137.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840138.png" />). The resolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840139.png" /> is an exact covariant functor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840140.png" />: there is a short exact sequence of resolutions corresponding to a short exact  "coefficient"  sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840141.png" />. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840142.png" /> turns out to be exact on the terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840144.png" />, of the resolution, therefore there is an exact cohomology sequence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840145.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840146.png" /></td> </tr></table>

corresponding to the indicated coefficient sequence, beginning with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840147.png" />. The cohomology sequence of a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840148.png" /> corresponds to the short exact sequence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840149.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840150.png" /> is a closed set).

The cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840151.png" /> have the following  "universality"  property, which casts light on their meaning: For any other resolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840152.png" /> (that is, an exact sequence of sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840153.png" /> beginning with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840154.png" />) there is a natural  "comparison"  homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840155.png" />, which is described in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840156.png" /> by using spectral sequences. An important case is when the sheaves of the resolution are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840158.png" />-acyclic, that is, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840159.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840160.png" />; in this case the above homomorphism is an isomorphism. The basic examples of acyclic sheaves are flabby sheaves (for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840161.png" /> the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840162.png" /> are epimorphic) and soft sheaves (any section over a closed set extends to a section over the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840163.png" />). The canonical resolution consists of flabby sheaves. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840164.png" /> is a paracompact space, then every flabby sheaf is also soft.

The universality property enables one to compare cohomologies arising in concrete situations with sheaf cohomology (and consequently also with each other), to discern for them the natural bounds within which their application is effective, and also to apply sheaf-theoretic methods to the solution of concrete problems. For example, Aleksandrov–Čech cohomology (cf. [[Aleksandrov–Čech homology and cohomology|Aleksandrov–Čech homology and cohomology]]) can be defined using cochains obtained from the cochains of a specially selected system of open coverings by taking the direct limit. These cochains turn out to be sections of the sheaves of germs of cochains (defined analogously to the sheaves of germs of functions) constituting a resolution of the group (or even the sheaf) of coefficients, and this sheaf turns out to be soft if the space is paracompact. Thus, for paracompact spaces Aleksandrov–Čech cohomology coincides with a sheaf cohomology. An analogous conclusion holds for Zariski spaces (in particular, for algebraic varieties). Alexander–Spanier cochains also turn out to be sections of the sheaves of a resolution and, moreover, the resolution consists of soft sheaves if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840165.png" /> is paracompact, so in this case, in particular, Alexander–Spanier and Aleksandrov–Čech cohomology are naturally isomorphic. In the case of singular cohomology, identification of cochains which coincide on the  "small"  singular simplices, i.e. subordinated to (arbitrary) open coverings, leads to the so-called localized cochains (giving the same cohomology), which are sections of the sheaves determined by the pre-sheaves of the usual singular cochains. These sheaves turn out to be soft if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840166.png" /> is paracompact (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840167.png" /> is hereditarily paracompact, then they are also flabby), but they form a resolution only under the additional requirement that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840168.png" /> is weakly locally contractible (in every neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840169.png" /> of each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840170.png" /> there is a smaller neighbourhood which is contractible to a point inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840171.png" />). A classic example is de Rham's theorem: The cohomology of the complex of differential forms of a differentiable manifold coincides with the usual cohomology with coefficients in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840172.png" /> of real numbers (the sheaves of germs of differential forms are soft and form a resolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840173.png" />: Sufficiently near to each point each closed differential form is exact).

There are also resolutions corresponding to any open or locally finite closed covering and these enable one to compare the cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840174.png" /> with the cohomology of the coverings (the spectral sequences for coverings). In particular, this gives an isomorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840175.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840176.png" /> for all elements of the covering and their finite intersections (Leray's theorem). Taking the direct limit with respect to open coverings gives an isomorphism between Aleksandrov–Čech cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840177.png" /> and sheaf cohomology, even for non-paracompact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840178.png" />, provided that there are sufficiently many small open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840179.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840180.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840181.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840182.png" /> (Cartan's theorem). This means that the cohomologies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840183.png" />, used in algebraic geometry, with coefficients in coherent sheaves, are also isomorphic to the standard sheaf cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840184.png" />.

General constructions ensuring the existence of a comparison homomorphism enable one to compare also the cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840185.png" /> with the hypercohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840186.png" /> (analogously, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840187.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840188.png" />) in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840189.png" /> is any differential sheaf (that is, a sheaf in which for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840190.png" /> the composition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840191.png" /> is zero) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840192.png" /> acyclic, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840193.png" /> are the derived sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840194.png" /> (these are the quotient sheaves of the kernel by the image in each dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840195.png" />). The corresponding spectral sequences have many applications. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840196.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840197.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840198.png" />. For example, if in place of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840199.png" /> a sheaf of chains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840200.png" /> is taken (the boundary operator lowers the dimension by one, the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840201.png" /> are the chains of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840202.png" />, and the stalk <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840203.png" />), then one obtains the way the homology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840204.png" /> depends on all the possible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840205.png" />. For a manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840206.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840207.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840208.png" />, that is, [[Poincaré duality|Poincaré duality]] holds. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840209.png" /> is an open or closed subset of a locally compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840210.png" />, then the homology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840211.png" /> is determined by the sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840212.png" /> with supports in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840213.png" />, and the homology of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840214.png" /> is determined by the sections of the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840215.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840216.png" />. Conversely (and this is also one of the manifestations of Poincaré duality), if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840217.png" /> is any flabby resolution for the cohomology, then the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840218.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840219.png" /> determines the cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840220.png" /> and the sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840221.png" /> with supports in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840222.png" /> determine the cohomology of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840223.png" />. Since the sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840224.png" /> are flabby for manifolds, the homology sequence of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840225.png" /> coincides up to an inversion of the numbering with the cohomology sequence of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840226.png" />. This means that dualities for manifolds, such as [[Lefschetz duality|Lefschetz duality]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840227.png" />, are particular cases of Poincaré duality. It turns out that the duality relations which are not covered by this scheme are corollaries of Poincaré duality and the acyclicity of the manifold in certain dimensions.

Just such a situation arises in the case of a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840228.png" />. A resolution for the cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840229.png" /> determines a certain differential sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840230.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840231.png" /> for which the stalks <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840232.png" /> are the direct limits of the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840233.png" /> with respect to neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840234.png" /> of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840235.png" /> (and for closed mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840236.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840237.png" />. The way <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840238.png" /> depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840239.png" /> is described by the Leray spectral sequence of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840240.png" /> (a particular case of this is the spectral sequence of a [[Serre fibration|Serre fibration]]). Acyclic mappings correspond to the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840241.png" /> vanishes, thus ensuring that the cohomologies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840242.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840243.png" /> with corresponding coefficients are isomorphic (Vietoris' theorem and its generalizations). The general constructions referred to above also give the spectral sequence of a mapping and take into account (along with their cohomological structure) the degree of disconnectedness of pre-images of points; this is especially effective for zero-dimensional or finite-to-one mappings (in the case of coverings it becomes the Cartan spectral sequence). There are also special spectral sequences in categories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840244.png" />-spaces (spaces on which a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840245.png" /> acts).

In sheaf cohomology there is a natural way of defining a multiplicative structure. The existence of special flabby resolutions, of which the mappings are determined by a certain semi-simplicial structure, enables one to give explicit formulas for the products of cochains, analogous to the usual ones. At the same time this also makes it possible to define other cohomology operations in sheaf theory.

The apparatus of sheaf theory has many applications wherever abstract homological methods are essential: in topology (homological and cohomological dimension, local homology and duality, the structure of various classes of continuous mappings, including imbeddings onto dense subsets, and, in particular, compactifications, etc.), in the theory of analytic manifolds (homology and cohomology with coefficients in coherent analytic sheaves and their applications, cohomology and analytic differential forms, homology and analytic flows (the analogue of de Rham's theorem), etc.), and also in abstract algebraic geometry (the cohomology of affine, projective and complete algebraic varieties with coefficients in coherent algebraic sheaves, algebraic Serre duality, algebraic (combinatorial) dimension, etc.).

Some basic notions of sheaf theory and spectral sequences appeared in the work of J. Leray (1945 and later) in connection with the study of homological properties of continuous mappings of locally compact spaces, and he also gave the definition of cohomology (with compact support) with coefficients in a sheaf. A fairly complete account of sheaf theory using resolutions was later given by H. Cartan. The proof of the de Rham theorem given by A. Weil (1947) and the work of J.-P. Serre (in the early 1950's) on algebraic varieties greatly influenced the development of sheaf theory. Cohomology with coefficients in a sheaf was first defined by the Aleksandrov–Čech method. A mature view of sheaf theory could be found by the end of the 1950's in the work of A. Grothendieck [[#References|[3]]] and R. Godement [[#References|[2]]], where great generality was achieved and the methods were considerably simplified. E.g., it was shown that the category of sheaves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840246.png" /> has a generator (that is, a sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840247.png" /> admitting non-zero homomorphisms into any non-zero sheaf; for sheaves of Abelian groups, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840248.png" />), and hence that each sheaf can be imbedded in an injective sheaf (Grothendieck's theorem). This is the reason for the formal analogy between cohomology theory with coefficients in sheaves and the theory of derived functors in the module categories: In the category of sheaves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840249.png" /> there are  "enough"  injective objects (although, as a rule, there are few projective objects), and therefore one can freely apply all the corresponding techniques of homological algebra; in particular, one can define the cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840250.png" /> (without any restriction on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840251.png" />) as derived functors of the left exact functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840252.png" /> (or even as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840253.png" />). This also sheds light, for example, on the general nature of such concepts as the cohomological dimension (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840254.png" />) of a space, the algebraic dimension of a variety and the global dimension of a ring. The description given by Grothendieck of the spectral sequence for the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840255.png" /> is essential in algebraic geometry. A much simpler method of constructing injective sheaves was found by Godement. He also showed that to construct a cohomology theory it is entirely sufficient to use his canonical flabby resolution, which, from the point of view of homological algebra, turns out to be simply one of the acyclic resolutions of a sheaf. Godement was the first to apply flabby and soft sheaves (soft sheaves are acyclic only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840256.png" /> paracompact, which explains their use primarily in topology).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.E. Bredon,   "Sheaf theory" , McGraw-Hill  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Godement,   "Topologie algébrique et théorie des faisceaux" , Hermann  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Grothendieck,   "Sur quelques points d'algèbre homologique"  ''Tohoku Math. J.'' , '''9'''  (1957)  pp. 119–221</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Swan,   "The theory of sheaves" , Chicago Univ. Press  (1964)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.G. Sklyarenko,   "Homology and cohomology of general spaces" , Springer  (Forthcoming)  (Translated from Russian)</TD></TR></table>



====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840257.png" /> be a continuous mapping between topological spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840258.png" /> a sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840259.png" />. Then the direct image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840260.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840261.png" /> is the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840262.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840263.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840264.png" /> open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840265.png" />. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840266.png" /> is left-exact on sheaves of modules or Abelian groups. The higher direct images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840267.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840268.png" /> are the right derived functors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840269.png" />. The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840270.png" /> is the sheaf associated to the pre-sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840271.png" />.

Given a sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840272.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840273.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840274.png" /> be the associated  "sheaf space"  (so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840275.png" /> sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840276.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840277.png" />). Now consider the fibre product (pull-back) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840278.png" />. The inverse image sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840279.png" /> is the sheaf of sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840280.png" />.

The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840281.png" /> is exact (on sheafs of modules or Abelian groups). The two functors are adjoint to one another: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840282.png" />.

A short sequence of sheaves of Abelian groups or modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840283.png" /> over a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840284.png" /> is exact if and only if the corresponding sequence of stalks <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840285.png" /> is exact for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840286.png" />.

Grothendieck and his school vastly generalized sheaf theory, from sheaves on a space to the more general notion of sheaves on a [[Site|site]] and that of a [[Topos|topos]] (cf. [[#References|[a6]]]–[[#References|[a8]]]).

For the theory of coherent sheaves on algebraic varieties see also the fundamental paper [[#References|[a1]]]. For the theory of sheaves in the étale topology and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084840/s084840287.png" />-adic sheaves see [[#References|[a8]]], [[#References|[a12]]]–[[#References|[a14]]].

For a detailed history of the origins of sheaf theory, see [[#References|[a1]]]. The article above focuses almost entirely on the use of sheaf theory in cohomology theory, but it does have many other applications: for example, in the representation theory of rings and other algebraic systems (see, for example, [[#References|[a3]]], [[#References|[a4]]]), and in logic, in the provision of models for constructive mathematics (see [[#References|[a2]]], and also [[#References|[a9]]], [[#References|[a10]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.W. Gray,   "Fragments of the history of sheaf theory"  M.P. Fourman (ed.)  C.J. Mulvey (ed.)  D.S. Scott (ed.) , ''Applications of sheaves'' , ''Lect. notes in math.'' , '''753''' , Springer  (1979)  pp. 1–79</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.P. Fourman,   D.S. Scott,   "Sheaves and logic"  M.P. Fourman (ed.)  C.J. Mulvey (ed.)  D.S. Scott (ed.) , ''Applications of sheaves'' , ''Lect. notes in math.'' , '''753''' , Springer  (1979)  pp. 302–401</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.J. Mulvey,   "Representations of rings and modules"  M.P. Fourman (ed.)  C.J. Mulvey (ed.)  D.S. Scott (ed.) , ''Applications of sheaves'' , ''Lect. notes in math.'' , '''753''' , Springer  (1979)  pp. 542–585</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F. Borceux,   G. van den Bossche,   "Algebra in a localic topos with applications to ring theory" , ''Lect. notes in math.'' , '''1038''' , Springer  (1983)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.R. Tennison,   "Sheaf theory" , Cambridge Univ. Press  (1975)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Artin,   "Grothendieck topologies" , Harvard Univ. Press  (1962)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P.T. Johnstone,   "Topos theory" , Acad. Press  (1977)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Grothendieck,   et al.,   "Théorie de topos et cohomologie des schémas (SGA 4–5)" , ''Lect. notes in math.'' , '''269, 270, 305, 589''' , Springer  (1972–1977)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M. Makkai,   G.E. Reyes,   "First order categorical logic" , ''Lect. notes in math.'' , '''611''' , Springer  (1977)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  M. Tierney,   "Sheaf theory and the continuum hypothesis"  F.W. Lawvere (ed.) , ''Toposes, algebraic geometry and logic (Dalhousic Univ., Jan. 1971)'' , ''Lect. notes in math.'' , '''274''' , Springer  (1972)  pp. 13–42</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  J.-P. Serre,   "Faisceaux algébriques cohérents"  ''Ann. of Math.'' , '''61'''  (1955)  pp. 197–278</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  P. Deligne,   "Cohomologie étale (SGA 4 1/2)" , ''Lect. notes in math.'' , '''569''' , Springer  (1977)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  J.S. Milne,   "Etale cohomology" , Princeton Univ. Press  (1980)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  E. Freitag,   R. Kiehl,   "Étale cohomology and the Weil conjecture" , Springer  (1988)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  J. Leray,   "Sur la forme des espaces topologiques et sur les points fixes des répresentations"  ''J. Math. Pures Appl.'' , '''24'''  (1945)  pp. 95–167</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  J. Leray,   "L'anneau spectral et l'anneau fibré d'homologie d'un espace localement compact et d'une application continue"  ''J. Math. Pures Appl.'' , '''29'''  (1950)  pp. 1–139</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  A. Weil,   "Foundations of algebraic geometry" , Amer. Math. Soc.  (1946)</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> , ''Sem. H. Cartan'' , '''1-''' , Secr. Math. Inst. H. Poincaré  (1948-)</TD></TR></table>