Germ (mathematics)

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In mathematics, the notion of a germ of an object in/on a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
is an equivalence class of that object and others of the same kind that captures their shared local properties. In particular, the objects in question are mostly functions (or maps) and
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s. In specific implementations of this idea, the functions or subsets in question will have some property, such as being analytic or smooth, but in general this is not needed (the functions in question need not even be
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
); it is however necessary that the space on/in which the object is defined is a topological space, in order that the word ''local'' has some meaning.

# Name

The name is derived from ''
cereal germ Cereal germ or Wheat germ: The germ of a cereal is the reproductive part that germinates to grow into a plant; it is the embryo of the seed. Along with bran, germ is often a by-product of the milling that produces refined grain products. Cer ...
'' in a continuation of the
sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * '' The Sheaf'', a student-run newspaper ...
metaphor, as a germ is (locally) the "heart" of a function, as it is for a grain.

# Formal definition

## Basic definition

Given a point ''x'' of a topological space ''X'', and two maps $f, g: X \to Y$ (where ''Y'' is any
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
), then $f$ and $g$ define the same germ at ''x'' if there is a
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
''U'' of ''x'' such that restricted to ''U'', ''f'' and ''g'' are equal; meaning that $f\left(u\right)=g\left(u\right)$ for all ''u'' in ''U''. Similarly, if ''S'' and ''T'' are any two subsets of ''X'', then they define the same germ at ''x'' if there is again a neighbourhood ''U'' of ''x'' such that :$S \cap U = T \cap U.$ It is straightforward to see that ''defining the same germ'' at ''x'' is an equivalence relation (be it on maps or sets), and the equivalence classes are called germs (map-germs, or set-germs accordingly). The equivalence relation is usually written :$f \sim_x g \quad \text \quad S \sim_x T.$ Given a map ''f'' on ''X'', then its germ at ''x'' is usually denoted 'f'' sub>''x''. Similarly, the germ at ''x'' of a set ''S'' is written 'S''sub>''x''. Thus, : A map germ at ''x'' in ''X'' that maps the point ''x'' in ''X'' to the point ''y'' in ''Y'' is denoted as :$f:\left(X,x\right) \to \left(Y,y\right).$ When using this notation, ''f'' is then intended as an entire equivalence class of maps, using the same letter ''f'' for any
representative Representative may refer to: Politics *Representative democracy, type of democracy in which elected officials represent a group of people *House of Representatives, legislative body in various countries or sub-national entities *Legislator, someon ...
map. Notice that two sets are germ-equivalent at ''x'' if and only if their
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
s are germ-equivalent at ''x'': :$S\sim_x T \Longleftrightarrow \mathbf_S \sim_x \mathbf_T.$

## More generally

Maps need not be defined on all of ''X'', and in particular they don't need to have the same domain. However, if ''f'' has domain ''S'' and ''g'' has domain ''T'', both subsets of ''X'', then ''f'' and ''g'' are germ equivalent at ''x'' in ''X'' if first ''S'' and ''T'' are germ equivalent at ''x'', say $S \cap U = T\cap U \neq \emptyset,$ and then moreover $f, _ = g, _$, for some smaller neighbourhood ''V'' with $x\in V \subseteq U$. This is particularly relevant in two settings: # ''f'' is defined on a subvariety ''V'' of ''X'', and # ''f'' has a pole of some sort at ''x'', so is not even defined at ''x'', as for example a rational function, which would be defined ''off'' a subvariety.

## Basic properties

If ''f'' and ''g'' are germ equivalent at ''x'', then they share all local properties, such as continuity, differentiability etc., so it makes sense to talk about a ''differentiable or analytic germ'', etc. Similarly for subsets: if one representative of a germ is an analytic set then so are all representatives, at least on some neighbourhood of ''x''. Algebraic structures on the target ''Y'' are inherited by the set of germs with values in ''Y''. For instance, if the target ''Y'' is a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
, then it makes sense to multiply germs: to define 'f''sub>''x'' 'g''sub>''x'', first take representatives ''f'' and ''g'', defined on neighbourhoods ''U'' and ''V'' respectively, and define 'f''sub>''x'' 'g''sub>''x'' to be the germ at ''x'' of the pointwise product map ''fg'' (which is defined on $U\cap V$). In the same way, if ''Y'' is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
,
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but c ...
, or
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, then so is the set of germs. The set of germs at ''x'' of maps from ''X'' to ''Y'' does not have a useful
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
, except for the
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
one. It therefore makes little or no sense to talk of a convergent sequence of germs. However, if ''X'' and ''Y'' are manifolds, then the spaces of jets $J_x^k\left(X,Y\right)$ (finite order Taylor series at ''x'' of map(-germs)) do have topologies as they can be identified with finite-dimensional vector spaces.

# Relation with sheaves

The idea of germs is behind the definition of sheaves and presheaves. A
presheaf In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
$\mathcal$ of
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
s on a topological space ''X'' assigns an abelian group $\mathcal\left(U\right)$ to each open set ''U'' in ''X''. Typical examples of abelian groups here are: real valued functions on ''U'', differential forms on ''U'', vector fields on ''U'', holomorphic functions on ''U'' (when ''X'' is a complex space), constant functions on ''U'' and differential operators on ''U''. If $V \subseteq U$ then there is a restriction map $\mathrm_:\mathcal\left(U\right)\to \mathcal\left(V\right),$ satisfying certain compatibility conditions. For a fixed ''x'', one says that elements $f\in\mathcal\left(U\right)$ and $g\in \mathcal\left(V\right)$ are equivalent at ''x'' if there is a neighbourhood $W\subseteq U\cap V$ of ''x'' with res''WU''(''f'') = res''WV''(''g'') (both elements of $\mathcal\left(W\right)$). The equivalence classes form the stalk $\mathcal_x$ at ''x'' of the presheaf $\mathcal$. This equivalence relation is an abstraction of the germ equivalence described above. Interpreting germs through sheaves also gives a general explanation for the presence of algebraic structures on sets of germs. The reason is that formation of stalks preserves finite limits. This implies that if ''T'' is a Lawvere theory and a sheaf ''F'' is a ''T''-algebra, then any stalk ''F''''x'' is also a ''T''-algebra.

# Examples

If $X$ and $Y$ have additional structure, it is possible to define subsets of the set of all maps from ''X'' to ''Y'' or more generally sub- presheaves of a given
presheaf In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
$\mathcal$ and corresponding germs: ''some notable examples follow''. *If $X, Y$ are both topological spaces, the subset ::$C^0\left(X,Y\right) \subseteq \mbox\left(X,Y\right)$ :of
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s defines germs of continuous functions. *If both $X$ and $Y$ admit a differentiable structure, the
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
::$C^k\left(X,Y\right) \subseteq \mbox\left(X,Y\right)$ :of $k$-times continuously differentiable functions, the
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
::$C^\infty\left(X,Y\right)=\bigcap\nolimits_k C^k\left(X,Y\right)\subseteq \mbox\left(X,Y\right)$ :of
smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth i ...
s and the
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
::$C^\omega\left(X,Y\right)\subseteq \mbox\left(X,Y\right)$ :of
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s can be defined ($\omega$ here is the ordinal for infinity; this is an abuse of notation, by analogy with $C^k$ and $C^$), and then spaces of germs of (finitely) differentiable, smooth, analytic functions can be constructed. *If $X,Y$ have a complex structure (for instance, are
subsets In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset ...
of complex vector spaces), holomorphic functions between them can be defined, and therefore spaces of germs of holomorphic functions can be constructed. *If $X,Y$ have an algebraic structure, then regular (and
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
) functions between them can be defined, and germs of regular functions (and likewise rational) can be defined. *The germ of ''f'' : ℝ → ''Y'' at positive infinity (or simply the germ of ''f'') is $\$. These germs are used in
asymptotic analysis In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as bec ...
and Hardy fields.

## Notation

The stalk of a sheaf $\mathcal$ on a topological space $X$ at a point $x$ of $X$ is commonly denoted by $\mathcal_x.$ As a consequence, germs, constituting stalks of sheaves of various kind of functions, borrow this scheme of notation: *$\mathcal_x^0$ is the ''space of germs of continuous functions'' at $x$. *$\mathcal_x^k$ for each natural number $k$ is the ''space of germs of $k$-times-differentiable functions'' at $x$. *$\mathcal_x^\infty$ is the ''space of germs of infinitely differentiable ("smooth") functions'' at $x$. *$\mathcal_x^\omega$ is the ''space of germs of analytic functions'' at $x$. *$\mathcal_x$ is the ''space of germs of holomorphic functions'' (in complex geometry), or ''space of germs of regular functions'' (in algebraic geometry) at $x$. For germs of sets and varieties, the notation is not so well established: some notations found in literature include: *$\mathfrak_x$ is the ''space of germs of analytic varieties'' at $x$. When the point $x$ is fixed and known (e.g. when $X$ is a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...
and $x=0$), it can be dropped in each of the above symbols: also, when $\dim X=n$, a subscript before the symbol can be added. As example *$, , , , ,$ are the spaces of germs shown above when $X$ is a $n$-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but c ...
and $x=0$.

# Applications

The key word in the applications of germs is locality: ''all local properties of a function at a point can be studied by analyzing its germ''. They are a generalization of
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, and indeed the Taylor series of a germ (of a differentiable function) is defined: you only need local information to compute derivatives. Germs are useful in determining the properties of
dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
near chosen points of their phase space: they are one of the main tools in
singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...
and
catastrophe theory In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry. Bifurcation theory studies and classifies phenomena ...
. When the topological spaces considered are Riemann surfaces or more generally complex-analytic varieties, germs of
holomorphic functions In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex der ...
on them can be viewed as power series, and thus the set of germs can be considered to be the analytic continuation of an
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
. Germs can also be used in the definition of
tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are el ...
s in differential geometry. A tangent vector can be viewed as a point-derivation on the algebra of germs at that point.Tu, L. W. (2007). An introduction to manifolds. New York: Springer. p. 11.

# Algebraic properties

As noted earlier, sets of germs may have algebraic structures such as being rings. In many situations, rings of germs are not arbitrary rings but instead have quite specific properties. Suppose that ''X'' is a space of some sort. It is often the case that, at each ''x'' ∈ ''X'', the ring of germs of functions at ''x'' is a local ring. This is the case, for example, for continuous functions on a topological space; for ''k''-times differentiable, smooth, or analytic functions on a real manifold (when such functions are defined); for holomorphic functions on a complex manifold; and for regular functions on an algebraic variety. The property that rings of germs are local rings is axiomatized by the theory of
locally ringed space In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...
s. The types of local rings that arise, however, depend closely on the theory under consideration. The Weierstrass preparation theorem implies that rings of germs of holomorphic functions are
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noeth ...
s. It can also be shown that these are
regular ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ...
s. On the other hand, let $\mathcal_0^\infty\left(\mathbf\right)$ be the ring of germs at the origin of smooth functions on R. This ring is local but not Noetherian. To see why, observe that the maximal ideal ''m'' of this ring consists of all germs that vanish at the origin, and the power ''m''''k'' consists of those germs whose first ''k'' − 1 derivatives vanish. If this ring were Noetherian, then the Krull intersection theorem would imply that a smooth function whose Taylor series vanished would be the zero function. But this is false, as can be seen by considering :$f\left(x\right) = \begin e^, &x \neq 0, \\ 0, &x = 0. \end$ This ring is also not a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is a ...
. This is because all UFDs satisfy the ascending chain condition on principal ideals, but there is an infinite ascending chain of principal ideals :$\cdots \subsetneq \left(x^ f\left(x\right)\right) \subsetneq \left(x^ f\left(x\right)\right) \subsetneq \left(x^ f\left(x\right)\right) \subsetneq \cdots.$ The inclusions are strict because ''x'' is in the maximal ideal ''m''. The ring $\mathcal_0^0\left(\mathbf\right)$ of germs at the origin of continuous functions on R even has the property that its maximal ideal ''m'' satisfies ''m''2 = ''m''. Any germ ''f'' ∈ ''m'' can be written as :$f = , f, ^ \cdot \big\left(\operatorname\left(f\right), f, ^\big\right),$ where sgn is the sign function. Since , ''f'', vanishes at the origin, this expresses ''f'' as the product of two functions in ''m'', whence the conclusion. This is related to the setup of almost ring theory.

* Analytic variety *
Catastrophe theory In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry. Bifurcation theory studies and classifies phenomena ...
* Gluing axiom * Riemann surface *
Sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * '' The Sheaf'', a student-run newspaper ...
* Stalk

# References

*, chapter I, paragraph 6, subparagraph 10 "''Germs at a point''". *, chapter 2, paragraph 2.1, "''Basic Definitions''". *, chapter 2 "''Local Rings of Holomorphic Functions''", especially paragraph A "''The Elementary Properties of the Local Rings''" and paragraph E "''Germs of Varieties''". * Ian R. Porteous (2001) ''Geometric Differentiation'', page 71, Cambridge University Press . *, paragraph 31, "''Germi di funzioni differenziabili in un punto $P$ di $V_n$ (Germs of differentiable functions at a point $P$ of $V_n$)''" (in Italian).