User:TakuyaMurata/sandbox

Let $(X,\; d)$ be a metric space. Then it can be viewed as a category: the objects are the points of $X$ and to each pair of objects $x,\; y$, there is a unique morphism; namely, the distance $d(x,\; y)$. The composition is the triangular inequality.

In his 1990 notes, MacPherson gives the following definition of a perverse sheaf (which is similar to the definition of a homology functor). The idea is to define a perverse sheaf as a contravariant functor on a suitable category of pairs satisfying the axioms analogous to those of cohomology. By definition, an opposed pair $(R,\; G)$ is

Under the natural map $\backslash pi\; :\; A\; \backslash to\; A/\backslash mathfrak\{m\}\; =\; k$, each variable $x\_i$ goes to some constant $a\_i\; \backslash in\; k$. Then $I\; =\; (x\_1\; -\; a\_1,\; \backslash dots,\; x\_n\; -\; a\_n)$ goes to zero under $\backslash pi$; i.e., $I\; \backslash subset\; \backslash mathfrak\{m\}$. Since $I$ is a maximal ideal (as $A/I\; =\; k$), this implies

:$\backslash mathfrak\{m\}\; =\; (x\_1\; -\; a\_1,\; \backslash dots,\; x\_n\; -\; a\_n)$.

Hence, for any ideal $I$ of $A$,

:$\backslash sqrt\{I\}\; =\; \backslash bigcap\_\{\backslash mathfrak\{m\}\; \backslash supset\; I\}\; \backslash mathfrak\{m\}\; =\; \backslash bigcap\_\{a\; \backslash in\; V(I)\}\; (x\_1\; -\; a\_1,\; \backslash dots,\; x\_n\; -\; a\_n)\; =\; I(V(I))$.

where we wrote $V(I)\; \backslash subset\; k^n$ for the common zero set of elements of $f$.

The converse also holds in the following sense: every closed subset $Z$ of a manifold $M$ is the zero set of a smooth function.Theorem 4.13. in https://people.math.rochester.edu/faculty/iosevich/partitionofunity.pdf To see this, let $U\_i$ be an open cover of $M\; -\; Z$. Then, at expense of changing the indexing set,

Finally, here is an example of an implication to Fourier analysis. For a discrete subset $E\; \backslash subset\; \backslash mathbb\{R\}^n$, we write $\backslash delta\_E\; =\; \backslash sum\_\{a\; \backslash in\; E\}\; \backslash delta\_a$, which is well-defined since, when it applied to test functions, the sum is finite.

Like a derivative, the Fourier transform of a distribution is defined in terms of test functions.

(Incidentally, the above can be used to obtain the Fourier inversion formula.)

Let $X$ be an algebraic variety over the base field $k\; =\; \backslash mathbb\{C\}$, the field of complex numbers. If $X$ is smooth, $X$ also has a structure of complex-analytic manifold. In fact, the partial converse is also true:

The important invariant that can be defined here is a Zeta function.

We have $\backslash mathfrak\{sl\}\_2\; \backslash mathbb\{C\}\; =\; \backslash mathfrak\{sl\}\_2\; \backslash mathbb\{R\}\; \backslash otimes\_\backslash mathbb\{R\}\; \backslash mathbb\{C\}\; =\; \backslash mathfrak\{su\}\_2\; \backslash mathbb\{R\}\; \backslash otimes\_\backslash mathbb\{R\}\; \backslash mathbb\{C\}$. That is, $\backslash mathfrak\{sl\}\_2\; \backslash mathbb\{C\}$ has two real forms $\backslash mathfrak\{sl\}\_2\; \backslash mathbb\{R\}$ and $\backslash mathfrak\{su\}\_2$, called split and compact forms, respectively. Now, consider a complex finite-dimensional representation of $\backslash pi\; :\; \backslash mathfrak\{sl\}\_2\; \backslash mathbb\{C\}\; \backslash to\; \backslash mathfrak\{gl\}(V)$. It restricts to the real representations: $\backslash pi\_s\; :\; \backslash mathfrak\{sl\}\_2\; \backslash mathbb\{R\}\; \backslash to\; \backslash mathfrak\{gl\}(V\_\{\backslash mathbb\{R\}\})$,$\backslash pi\_c\; :\; \backslash mathfrak\{su\}\_2\; \backslash mathbb\{R\}\; \backslash to\; \backslash mathfrak\{gl\}(V\_\{\backslash mathbb\{R\}\})$ and .

For example, consider a finite separable morphism $f:\; X\; \backslash to\; Y$ between smooth connected curves over an algebraically closed field. Because of separability, there is an exact sequence of coherent sheaves on X:

:$0\; \backslash to\; f^*\; \backslash Omega\_Y\; \backslash to\; \backslash Omega\_X\; \backslash to\; \backslash Omega\_\{X/Y\}\; \backslash to\; 0$

where $\backslash Omega\_\{X/Y\}$ is a torsion sheaf and the rest invertible sheaves. Now, the exact sequence says that, for $Q\; =\; f(P)$, the length of $\backslash Omega\_\{X/Y\}$ as an $\backslash mathcal\{O\}\_P$-module is the same as the rank of the free $\backslash mathcal\{O\}\_P$-module $\backslash Omega\_P/f^*\backslash Omega\_Q$; which in turns is given as:

the sheaf $\backslash Omega\_\{X/Y\}$ is isomorphic to the structure sheaf of an effective divisor $R\; \backslash subset\; X$, called the ramification divisor, given as

:$R\; =\; \backslash sum\_\{P\; \backslash in\; X\}\; \backslash operatorname\{length\}\_\{\backslash mathcal\{O\}\_P\}(\backslash Omega\_\{X/Y\})\; P.$

Then f is unramified if and only if $R\; =\; 0$.

Let

Given a non-archimedean local field $K$, let $R\; =\; \backslash mathfrak\{o\}\_K$ be the valuation ring of it. Then the ring $R\; \backslash langle\; \backslash xi\_1,\; \backslash dots,\; \backslash xi\_n\; \backslash rangle$ of restricted power series with coefficients in $R$ plays a role of a flat model of a Tate algebra.

A local ring of mixed characteristic is called unramified if

If a ring $R$ is a direct sum of additive subgroups of R, then the structure of a graded ring is the direct sum decomposition plus the multiplications $R\_i\; \backslash times\; R\_j\; \backslash to\; R\_\{i+j\}$ induced by the multiplication on R.

When a graded ring is a direct sum of not-necessarily additive subgroups, then $R\_i\; \backslash times\; R\_j\; \backslash to\; r\_j$ are natural maps. For example, if

An injective ring homomorphism $f\; :\; R\; \backslash to\; S$ is said to split if it is a section of a surjective ring homomorphism $g\; :\; S\; \backslash to\; R$; in other words, $S\; =\; R\; \backslash otimes\; R\text{'}$.

Similarly, a subjective ring homomorphism is said to split if it admits a section.

First, a strictly increasing sequence cannot stabilize, obviously. Conversely, suppose there is an ascending sequence that does not stabilize; then clearly it contains a strictly increasing (necessarily infinite) subsequence. (Notice the proof does not involve the axiom of choice.)

The Segre class of the normal cone C to

There is a module that is free at each maximal ideal but not is not locally free. As an example,

Lie's formula states: for a Lie group G with Lie algebra $\backslash mathfrak\{g\}$, a smooth function f on G and an element X of $\backslash mathfrak\{g\}$ of small norm:

:$f(\backslash exp\; X)\; =\; \backslash sum\_\{k=0\}^n\; \backslash frac\{1\}\{k!\}\; (\backslash widetilde\{X\}^k\; f)(e)\; +\; O(\backslash |X\backslash |^\{n+1\}).$

In other words, it is a version of Taylor's formula for smooth functions on G.

More generally, there is a multi-variable version: with the notation $e^X\; =\; \backslash operatorname\{exp\}(X)$,

:$f(e^\{t\_1\; X\_1\}\; \backslash cdots\; e^\{t\_r\; X\_r\})\; =$

The formula follows from introducing coordinates using the exponential map and then invoking the usual Taylor formula.

{{see also|Universal algebra#Generalizations}}

In universal algebra, there is the notion of $\backslash Omega$-algebra, which is quite similar to an operad algebra. The difference is that ...

Let

• (formal smooth) For each finite generated

• Here is a ring such that the heights of maximal ideals are not constant (i.e., it has a maximal ideal whose height is not the Krull dimension of the ring). Let $A$ be a two-dimensional integral domain such that

Roughly this approach uses formal power series.

The notion of "operad" gives a precise meaning to the following statement:

• Given a topological space $X$, the loop space $\backslash Omega\; X$ = the space of loops in $X$ is an algebra of some kind.

That is, an “operad” defined below fits into “some kind”.

Let $\backslash square^1\; =\; \{(-1,\; 1)\}$ denote the open interval, choose a base point * on $X$ and, for definitiveness, view $\backslash Omega\; X$ as the set of all continuous maps $f:\; \backslash square^1\; \backslash to\; X$ such that $f(-1)\; =\; f(1)\; =\; *$ (the topology on $\backslash Omega\; X$ we ignore). In algebraic topology, the composition of two loops $f,\; g$ are defined as

Given some category and objects $X,\; Y$ in it, let $\backslash operatorname\{Iso\}(X,\; Y)$ denote the set of all isomorphisms $\backslash varphi:\; X\; \backslash overset\{\backslash sim\}\backslash to\; Y$ from X to Y. Then $\backslash operatorname\{Aut\}(X)$ acts on it, say, from the right. Moreover, this action is free and transitive: $\backslash operatorname\{Iso\}(X,\; Y)$ is a torsor for $\backslash operatorname\{Aut\}(X)$.

To see how this works in a concrete situation, take $X$ to be a cyclic group of order n and $Y\; =\; \backslash mathbb\{Z\}/n\backslash mathbb\{Z\}$.

A direct sum of ideals is that of modules.

The functor $\backslash operatorname\{Lie\}$ from the category of (real) Lie groups to that of finite-dimensional real Lie algebras factors through the functor

:{{box|Lie groups}} → {{box|formal group laws}}

where the latter category is the category of formal group laws, as follows.F. Bruhat, [http://www.math.tifr.res.in/~publ/ln/tifr14.pdf Lectures on Lie Groups and Representations of Locally Compact Groups]. Let G be a Lie group; then it is real-analytic manifold (not differentiable one) with the group operations

:$\backslash mu:\; G\; \backslash times\; G\; \backslash to\; G$

:$i:\; G\; \backslash to\; G$

Choose

See also: Lie operad.

Although the exponential map for a general Lie group is not matrix exponential (since the group and the Lie algebra don't even consist of matrices), it can still be viewed locally as such in the following precise sense.

Let G be a Lie group with Lie algebra $\backslash mathfrak\{g\}$. By Ado's theorem, there is a faithful representation $\backslash mathfrak\{g\}\; \backslash hookrightarrow\; \backslash mathfrak\{gl\}\_n\; \backslash mathbb\{R\}$ into the general linear group over real numbers and, through this, $\backslash mathfrak\{g\}$ can be viewed as a Lie subalgebra of $\backslash mathfrak\{gl\}\_n\; \backslash mathbb\{R\}$; in particular, $\backslash mathfrak\{g\}$ consists of matrices. Now, for each matrix $X\; \backslash in\; \backslash mathfrak\{g\}$, the exponential $e^X$ is an invertible n-by-n matrix and $\backslash \{\; e^X\; |\; X\; \backslash in\; \backslash mathfrak\{g\}\; \backslash \}$ generates a subgroup $G\text{'}$ of $\backslash operatorname\{GL\}\_n(\backslash mathbb\{R\})$. Now, $G,\; G\text{'}$ have the same Lie algebra and thus $G,\; G\text{'}$ are locally isomorphic near the respective identity elements; that is there is a homeomorphism $\backslash eta:\; U\; \backslash to\; U\text{'}$ from a neighborhood U of $e\; \backslash in\; G$ to a neighborhood U{{'}} of $e\; \backslash in\; G\text{'}$ such that $\backslash eta(xy)\; =\; \backslash eta(x)\backslash eta(y)$ and $xy\; \backslash in\; U$ if and only if $\backslash eta(x)\backslash eta(y)\; \backslash in\; U\text{'}$.

Something like

$g\; \backslash cdot\; \backslash omega\_R\; =\; \backslash operatorname\{det\}\backslash operatorname\{Ad\}\_g\; \backslash omega\_L$.

In general, a finite set of ring homomorphisms $\backslash \{\; A\; \backslash to\; B\_i\; \backslash \}$ is called a flat covering of A (more precise of $\backslash operatorname\{Spec\}\; A)$ if $A\; \backslash to\; \backslash prod\_i\; B\_i$

is a faithfully flat ring homomorphism.

To each faithfully flat ring homomorphism $f:\; A\; \backslash to\; B$ one associates the category where

1. An object is

One can show that

:$\backslash mathrm\{ad\}\_x(y)\; =\; [x,y]\backslash ,$

for all $x,y\; \backslash in\; \backslash mathfrak\; g$

namely, if $V,\; W$ are vector fields on G that are x, y at the identity, then $[x,\; y]$ is the value of the commutators $V\; W\; -\; WV$ at the identity.)

If $\backslash gamma:\; (-1,\; 1)\; \backslash to\; G$ is a smooth curve with $\backslash gamma(0)\; =\; e,\; \backslash gamma\text{'}(0)\; =\; y$, then

:$d(Ad\_x)\_e(y)\; =\; \backslash lim\_\{t\; \backslash to\; 0\}\; \{1\; \backslash over\; t\}\; \backslash left(\backslash operatorname\{Ad\}\_x(\backslash gamma(t))\; -\; \backslash operatorname\{Ad\}\_x(\backslash gamma(0))\; \backslash right)$

A formal ring law

The above two are special cases of more general Lascoux's formula: (still r = the rank of E)

:$c(\backslash operatorname\{Sym\}^2\; E)\; =\; \backslash sum\_\{\backslash lambda\; \backslash subset\; \backslash epsilon\}\; \backslash left\backslash vert\; \backslash binom\{\backslash lambda\_i\; +\; r\; -\; i\}\{\backslash epsilon\_j\; +\; r\; -\; j\}\; \backslash right\backslash vert\; 2^\{|\backslash lambda|-\backslash binom\{r\}\{2\}\}\; \backslash Delta\_\{\backslash widetilde\{\backslash lambda\}\}\; c(E).${{harvnb|Fulton|loc=Example 14.5.1.}}

Here we sketch the proofs for the first two and gives a more detailed one for the last approach.

{{main|Verma module}}

The idea is to use abstract algebra to explicitly construct a representation of a Lie algebra. By definition, given a linear functional $\backslash lambda$ of $\backslash mathfrak\{h\}$, the Verma module $M(\backslash lambda)$ is the representation induced from the representation of $\backslash mathfrak\{b\}\; =\; \backslash mathfrak\{h\}\; \backslash oplus\; \backslash mathfrak\{g\}\_+$

• Let Γ be a finite group, k a field and $A\; =\; \backslash prod\_\{\backslash sigma\; \backslash in\; \backslash Gamma\}\; k$.

Two notions play fundamental roles: given an associateive algebra A over a commutative ring R',

• The algebra A is called separable algebra if
• The algebra A is called simple if

• Given a finite group G and a commutative R-algebra A, let $\backslash operatorname\{Map\}(G,\; A)$ denote the set of all functions $G\; \backslash to\; A$. Then it is an R-algebra; in fact, a Hopf algebra and, if A is a finite-dimensional algebra over a field, then $B$ is the Hopf-algebra-dual of the group algebra $A[G]$. Geometrically, $\backslash operatorname\{Spec\}(\backslash operatorname\{Map\}(G,\; A))$ and $\backslash operatorname\{Spec\}(A[G])\; =\; G\; \backslash times\_R\; \backslash operatorname\{Spec\}(A)$ are Cartier duals of each other.

An action of a linear algebraic group G on a finite-dimensional vector space V is linear if the action determines linear transformations; i.e., if $\backslash sigma:\; G\; \backslash times\; V\; \backslash to\; V$ denotes the action, then, for each g in G, $\backslash sigma\_g\; =\; \backslash sigma(g,\; \backslash cdot):\; V\; \backslash to\; V$ is a linear transformation (in fact, invertible). In other words, $G\; \backslash to\; GL(V),\; \backslash ,\; g\; \backslash mapsto\; \backslash sigma\_g$ is a linear representation.

A linear action is called separable if each linear map $\backslash sigma\_g:\; V\; \backslash to\; V$ is a separable linear transformation (i.e., in some field extension of the base field, the minimal polynomial has all roots but no repeated root and thus is diagonalizable).

Loosely, a linear representation of a group scheme is a family of linear group representations. Precisely, given a vector bundle $E\; \backslash to\; S$ over a scheme S of finite type over a field k,

For example, if $X\; =\; \backslash operatorname\{Spec\}(A)$, then an action on X, if any, determines the linear action on A, say, as a left regular representation (in this case, [[

An important question is that of linearization; i.e., whether the action comes from a linear action on a vector space.

For associative algebras, standard concepts for rings continue to be valid but they admit formulations in the language of modules, paving a way to use linear algebra.

Given an A-algebra R,

• An element r of R is a unit element if and only if an A-linear map $l\_r:\; R\; \backslash to\; R,\; a\; \backslash mapsto\; ra$ is invertible. (As in linear algebra, the invertability can be formulated using determinant).
• In particular, there is a functor $GL:\; R\; \backslash mapsto\; R^*$ from the category of A-algebra to. If A is a finite-dimension algebra over a field, then this functor is a group scheme over A, called the general linear group.

{{main|Chern class}}

The notion is used to state (a generalization of) the Riemann–Roch theorem

With a suitable definition of a relative dimension of schemes, it is possible to define Chow groups of a scheme over a base other than another scheme (typically a regular scheme) and establish the basic properties.

{{harv|Fulton|1998}} uses the following notion of a relative dimension:

For typical applications, the definition agrees with the one in SGA 6; namely,

Let X be an algebraic scheme, E a vector bundle on it and s a section of E viewed as a linear map $s:\; E^*\; \backslash to\; \backslash mathcal\{O\}\_X$. Let $X\text{'}$ be the blow-up of X along the ideal sheaf $s(E^*)$.

• A Chow group $\backslash operatorname\{CH\}^*(-)$ is a presheaf with transfers: if W is a correspondence from X to Y, then the transfer map is $\backslash varphi\_W:\; \backslash operatorname\{CH\}^*(Y)\; \backslash to\; \backslash operatorname\{CH\}^*(X),\; \backslash ,\; \backslash alpha\; \backslash mapsto\; q\_*(W\; \backslash cdot\; p^*\; \backslash alpha)$ where $q,\; p$ are the projections from $X\; \backslash times\; Y$ to $X,\; Y$.
• Let X be a smooth algebraic scheme and $\backslash mathbb\{Z\}\_\{tr\}(X)$ the contravariant functor on $Cor$ given by $\backslash mathbb\{Z\}\_\{tr\}(X)(U)\; =\; \backslash operatorname\{Cor\}(U,\; X)$. Then it is a (representable) presheaf with transfers.

Example: Let $X\; \backslash subset\; \backslash mathbb\{P\}^2$ be the conic $x^2\; +\; y^2\; =\; z^2$, and $V\; =\; \backslash operatorname\{span\}\; \backslash \{\; x,\; y\; \backslash \}\; \backslash subset\; \backslash Gamma(X,\; \backslash mathcal\{O\}\_X(1))$ the two-dimensional vector subspace. Then V determines

:$\backslash varphi:\; X\; \backslash to\; \backslash mathbb\{P\}^1$

Let X be a scheme over a ring A. Suppose there is a morphism

:$\backslash phi:\; X\; \backslash to\; \backslash mathbf\{P\}^n\_A\; =\; \backslash operatorname\{Proj\}\; A[x\_0,\; \backslash dots,\; x\_n]$.

Then, along this map, the Serre twisting sheaf $\backslash mathcal\{O\}(1)$ pulls-back to a line bundle L on X, which is generated by the global sections $\backslash phi^*(x\_i)$.{{harvnb|Hartshorne|1977|loc=Ch II, Theorem 7.1}} Conversely, any line bundle L which is generated by global sections $s\_0,\; ...,\; s\_n$ defines a morphism

:$\backslash phi:\; X\; \backslash to\; \backslash mathbf\{P\}^n\_A$

which in homogeneous coordinates is given by $\backslash phi(x)\; =\; [s\_0(x):\; \backslash dots\; :\; s\_n(x)].$ This map $\backslash phi$ is such that $L\; \backslash cong\; \backslash phi^*(\backslash mathcal\{O\}(1))$ and $s\_i\; =\; \backslash phi^*(x\_i)$. Furthermore, $\backslash phi$ is a closed immersion if and only if $X\_i$ are affine and $\backslash Gamma(U\_i,\; \backslash mathcal\{O\}\_\{\backslash mathbf\{P\}^n\_A\})\; \backslash to\; \backslash Gamma(X\_i,\; \backslash mathcal\{O\}\_\{X\_i\})$ are surjective.{{harvnb|Hartshorne|1977|loc=Ch II, Proposition 7.2}}

Let $\backslash mathcal\{M\}\_X$ be the sheaf on X associated with $U\; \backslash mapsto$ the total ring of fractions of $\backslash Gamma(U,\; \backslash mathcal\{O\}\_X)$. A global section of $\backslash mathcal\{M\}\_X^*/\backslash mathcal\{O\}\_X^*$ (* means multiplicative group) is called a Cartier divisor on X. The notion actually adds nothing new: there is the canonical bijection

:$D\; \backslash mapsto\; \backslash mathcal\{L\}(D)$

from the set of all Cartier divisors on X to the set of all line bundles on X.-->

Let $G\; =\; \backslash operatorname\{Spec\}(A)$ be a group scheme acting on an affine scheme $X\; =\; \backslash operatorname\{Spec\}(R)$, say, from the right over a field k. Then the action corresponds to a ring homomorphism

:$\backslash sigma^\{\backslash \#\}:\; R\; \backslash to\; R\; \backslash otimes\_k\; A$

satisfying

• Let $\backslash xi$ be a locally free sheaf of a finite rank on a scheme X and $p:\; E(\backslash xi)\; =\; \backslash operatorname\{Spec\}\_X(\backslash mathcal\{S\}ym(\backslash xi^*))\; \backslash to\; X$ its total space (vector bundle associated to it). Then the normal bundle to the zero-section embedding $X\; \backslash hookrightarrow\; E(\backslash xi)$ is
• :$N\_\{X/E(\backslash xi)\}\; =\; E(\backslash xi)$,

:as vector bundles on X.

• Let $p:\; L\; \backslash to\; X$ be a line bundle and $i:\; X\; \backslash to\; L$ the zero-section embedding. Then{{fact}}

:$i^*\; \backslash mathcal\{O\}\_L(X)\; =\; N\_\{L/X\}\; =\; L.$

Let $F/R$ be a formal O-module. Then a Drinfeld level n structure on it is a homomorphism:http://people.maths.ox.ac.uk/chojecki/gdtScholze1.pdf

:$\backslash phi:\; (\backslash varpi^\{-n\}\; \backslash mathcal\{O\}/\backslash mathcal\{O\})^h\; \backslash to\; F[\backslash varpi^n](R)$

such that the characteristic polynomial $\backslash prod\_\{x\; \backslash in\; \backslash varpi^\{-1\}\; \backslash mathcal\{O\}/\backslash mathcal\{O\}\}\; (T\; -\; \backslash phi(x))$ divides $\backslash varpi\_F(T)$.

The notion of a full level structure was introduced by {{harvnb|Katz–Mazur|1985}}, who were inspired by the works of Drinfeld.

In algebraic topology, an (infinite) infinite projective space plays a role of the classifying space for line bundles.

Let T be a torus acting on a quasi-projective variety X (over an algebraically closed field) and L an ample line bundle on X.

First of all, the linearlizations on L are parameterized by characters of T. Indeed,

Just as $\backslash mathbb\{P\}^n\; =\; (\backslash mathbb\{A\}^n\; -\; 0)/\backslash mathbb\{G\}\_m$,

Let $S\; =\; k[x\_0,\; \backslash dots,\; x\_n]$ be a polynomial ring. If $G\; =\; \backslash mathbb\{G\}\_m$ acts on it by scaling, then $S^G\; =\; S\_0\; =\; k$ and so $\backslash operatorname\{Spec\}(S^G)$ is a point.

To get something more interesting, we consider $X\; =\; \backslash mathbb\{A\}^n\; -\; 0$, which is an open subvariety with affine charts $U\_i\; =\; \backslash operatorname\{Spec\}(S[x\_i^\{-1\}])$. Then $U\_i/\backslash !/\; G\; =\; \backslash operatorname\{Spec\}(S[x\_i^\{-1\}]\_0).$ The GIT quotient $X\; /\backslash !/\; G$ is constructed by gluing the affine GIT quotients

$3.\; \backslash Rightarrow\; 2.$: Let W be the sum of all simple submodules of V (called the socle of V). By 3., it admits a complementary subrepresentation.

A linear algebraic group may be recovered from their finite-dimensional representations.

The basic strategy one can use to determine the decomposition of a semisimple representation $(\backslash pi,\; V)$ into irreducible representations is as follows:Editorial note: This is the approach used for example in {{harv|Fulton–Harris}}, hence not original.

1. Pick some special abelian groups T and then determines the decomposition of V as a T-module.
2. Determine

For example, if G is a symmetric group, one can use cyclic subgroups as T in Step 1. For a Lie group G, taking T to be a maximal torus would be typical.

• The Fourier expansion $f\; \backslash mapsto\; (\backslash widehat\{f\}(n)e^\{2\; \backslash pi\; i\; n\; \backslash bullet\}|n\; \backslash in\; \backslash mathbb\{Z\})$ gives the decomposition
• :$L^2(S^1)\; \backslash simeq\; \backslash widehat\{\backslash bigoplus\_\{n\; \backslash in\; \backslash mathbb\{Z\}\}\}\; \backslash ,\; \backslash mathbb\{C\}\; e^\{2\; \backslash pi\; i\; n\; \backslash bullet\}$
• Let $T\; =\; (\backslash mathbb\{C\}^*)^r$ be a complex torus and V

Let R denote the Grothendieck ring of the category of polynomial functors of bounded degrees. One can define the map

:$\backslash operatorname\{ch\}:\; R\; \backslash to\; \backslash Lambda$

where $\backslash Lambda$ is the ring of symmetric functions by

:$\backslash operatorname\{ch\}(F)(\backslash lambda\_1,\; \backslash dots,\; \backslash lambda\_r)\; =\; \backslash operatorname\{tr\}\; F(\backslash operatorname\{diag\}(\backslash lambda\_1,\; \backslash dots\; \backslash lambda\_r))$

where $\backslash operatorname\{diag\}$ means the diagonal matrix. For example, $\backslash operatorname\{ch\}(\backslash wedge^n)\; =\; e\_n$.

The basic fact is that $\backslash operatorname\{ch\}$ is an isomorphism, having the properties:

For each partition $\backslash lambda$, define the polynomial functor $F\_\{\backslash lambda\}$ by

:$F\_\{\backslash lambda\}(X)\; =\; (M\_\{\backslash lambda\}\; \backslash otimes\; X^\{\backslash otimes\; n\})^\{S\_n\}$.

Then $\backslash operatorname\{ch\}(F\_\{\backslash lambda\})$ is the Schur function corresponding to $\backslash lambda$.

One can generalize the notion of a torsor with a structure group to that of a torsor with a structure groupoid ("groupoid" being either a groupoid scheme or a groupoid algebraic-space), as follows. We only consider the algebraic-space case, as the scheme case is a special case of that.

First, given a groupoid object G and a object X in the category of algebraic spaces, a trivial G-torsor is a $G\; \backslash times\; X$ with

Grothendieck first defines an n -groupoid as data consisting of

• A set $F\_i$ for each integer $i\; \backslash ge\; 0$; "F" refers to "flèche", a French word meaning "arrow",
• For each integer $i\; \backslash ge\; 1$, a pair of functions $s\_i,\; t\_i:\; F\_i\; \backslash to\; F\_\{i-1\},$
• For each integer $i\; \backslash ge\; 0$, a function $k\_i:\; F\_i\; \backslash to\; F\_\{i+1\}$,
• The composition

subject to the conditions

1. $s\_\{i\; -1\}\; s\_i\; =\; s\_\{i-1\}\; t\_i,\; \backslash ,\; t\_\{i-1\}$

For example, let $F\_0$ be a set of objects, $F\_1$ a set of morphisms

Over a field k of characteristic zero, Behrend explicitly constructs the étale spectrum in terms of a graded algebra called a perfect resolving algebra. By definition, a perfect resolving algebra is a graded algebra over k such that

The main result of

By the correspondence, given a finite-dimensional complex Lie algebra $\backslash mathfrak\{g\}$, one knows that there is a simply connected Lie group $G^s$ whose Lie algebra is $\backslash mathfrak\{g\}$. This $G^s$ can be constructed explicitly from the Lie algebra representations of $\backslash mathfrak\{g\}$.

{{further|Algebraic surface}}

A projective surface is a projective variety of dimension two. When it is a hypersurface in a projective space, the degree of the defining homogeneous polynomial is the degree of the variety relative to the embedding. Projective surfaces of degree 2, 3 are respectively called a quadratic surface and a cubic surface.

One important operators is the intersections of curves on the surface. In non-degenerate case, the intersection is a finite set and its cardinality is the intersection number.

(This section requires some background in algebraic geometry.)

Let E be a finite-rank free module over a ring R, s: E → R an R-linear map and M a finitely generated module. If $\backslash operatorname\{H\}\_i(K(s,\; M)),\; i\; \backslash ge\; 0$ all have finite length over R, then we let

:$\backslash chi(s,\; M)\; =\; \backslash sum\_\{i=0\}^r\; (-1)^i\; \backslash operatorname\{length\}\_R(\backslash operatorname\{H\}\_i(K(s,\; M))$.

It is the Euler characteristic of the Koszul homology of (s, M).

The notion has a geometric interpretation. Let D₁, ..., D_r be effective Cartier divisors on an algebraic variety X. Also, let W ⊂ X be a closed subvariety of X and $R\; =\; \backslash mathcal\{O\}\_\{W,\; X\}$ the local ring of X at W.

{{math_theorem|name=Lemma|math_statement=Let $(R,\; \backslash mathfrak\{m\})$ be a Noetherian local ring, and x₁, ..., x_r be the minimal generators of the maximal ideal $\backslash mathfrak\{m\}$. Then the Koszul complex K(x₁, ..., x_r) is a subcomplex of the minimal free resolution of the residue field $k\; =\; R/\backslash mathfrak\{m\}$.}}

As it turns out, the Koszul complex $K\_\{\backslash bullet\}(x\_1,\; \backslash dots,\; x\_r;\; M)$ depends only on the ideal generated by $x\_1,\; \backslash dots,\; x\_r$, up to isomorphism. This can be seen as follows.

If $E\; \backslash simeq\; R^r$, then $K\_\{\backslash bullet\}(s;\; M)\; \backslash simeq\; K\_\{\backslash bullet\}(x\_1,\; \backslash dots,\; x\_r;\; M)$ when s corresponds to a row vector $[x\_1\; \backslash cdots\; x\_r]:\; R^r\; \backslash to\; R$. Now, let I be an ideal of R. Choosing a finite sequence $x\_1,\; \backslash dots,\; x\_r$ of elements of R such that $I\; =\; (x\_1,\; \backslash dots,\; x\_r)$ amounts to giving a surjective ring homomorphism:

:

Summarize http://arxiv.org/abs/0804.2242v3 and put the summary to quotient stack

Incorporate http://mathoverflow.net/questions/210068/calculating-the-distinguished-varieties-of-intersection-product/222039#222039 to normal cone

{{term|1=zero-locus}}

{{defn|no=1|1=If f is a regular function, then the (scheme-theoretic) zero-locus is the (scheme-theoretic) fiber $f^\{-1\}(0)$.}}

{{defn|no=2|1=If s is a differential from, then. In particular, the zero-locus of df is the critical set of f.}}

Let R be a local ring. Then, since R has only maximal ideal, to give a morphism $\backslash operatorname\{Spec\}\; R\; \backslash to\; X$ is to give a local homomorphism

:$\backslash mathcal\{O\}\_\{X,\; x\}\; \backslash to\; R$

where x is the image of $\backslash mathfrak\{m\}\_R$.

Let X be an algebraic variety. The Grothendieck group of coherent shaves on X, denoted by G(X), is the group generated by coherent shaves on X with the relations [E] + [G] = [F] whenever there is an exact sequence

:$0\; \backslash to\; E\; \backslash to\; F\; \backslash to\; G\; \backslash to\; 0$.

It is an abelian group since [E] + [F] = [E ⊕ F] is symmetric in E and F. If one repeats the construction with vector bundles (i.e., finite-rank locally free shaves) instead of coherent sheaves, then the result is K(X), which has the structure of a ring with the multiplication given by tensor product. Via tensor product, G(X) is a module over K(X). If X is a smooth variety, then G(X) = K(X).

It is known that{{fact}}

:$K(\backslash mathbf\{P\}^n)\; =\; \backslash bigoplus\_\{i=0\}^n\; \backslash mathbb\{Z\}[\{\backslash mathcal\{O\}\_\{\backslash mathbf\{P\}^n\}(i)\}]$

(for n = 1, this follows from Grothendieck's theorem.)

An algebraic variety is said to have the resolution property if every coherent sheaf on it admits an possibly infinite resolution by vector bundles. A quasi-projective variety has the resolution property.

Let ƒ:X→Y be a morphism of schemes with the accompanying pullback map denoted by $f^\{\backslash \#\}$. Then, for each x in X, since $f^\{\backslash \#\}(\backslash mathfrak\{m\}^n\_\{f(x)\})\; \backslash subset\; \backslash mathfrak\{m\}\_x^n$, it induces

:$\backslash operatorname\{gr\}\; \backslash mathcal\{O\}\_\{Y,\; f(x)\}\; \backslash overset\{\backslash mathrm\{def\}\}=\; \backslash oplus\_\{n=0\}^\{\backslash infty\}\; \{\backslash mathfrak\{m\}^n\}/\{\backslash mathfrak\{m\}^\{n+1\}\}\; \backslash to\; \backslash operatorname\{gr\}\; \backslash mathcal\{O\}\_\{X,\; x\}$.

• The conic $x^2\; +\; y^2\; =\; z^2$ in P² is isomorphic to P¹ provided the characteristic of k is not 2. (Proof: it admits a rational parametrization.)
• $\backslash mathbb\{C\}[x,\; y,\; z]/(y\; -\; xz)$ is not flat over $\backslash mathbb\{C\}[x,\; y]$. For a proof and an explanation, see for instance [http://math.stackexchange.com/questions/110599/why-isnt-mathbbcx-y-z-xz-y-a-flat-mathbbcx-y-module]

Conversely, if there is an inclusion of the field $k(Y)\; \backslash hookrightarrow\; k(X)$, then it induces a rational map f from X to Y. (Proof: there is some nonempty open affine subset V of Y such that $k[V]\; \backslash hookrightarrow\; k(X)$.) If X is a smooth projective curve, then ''f" is regular.

Any morphism $f:\; X\; \backslash to\; \backslash mathbf\{P\}^1$ determines a rational function on X (since $f:\; X\; -\; f^\{-1\}(\backslash infty)\; \backslash to\; \backslash mathbf\{A\}^1$ is regular). Conversely, if f is a rational function on X, then there is the inclusion of the field k(f) ⊂ k(X).

It is possible to extend the definition to a tensor product of any number of modules. Let R be a commutative ring (for example Z), $M\_i$ a family of R-modules indexed by a set I and let (*) be a property on R-multilinear maps. Then, given a family M_i, i ∈ I, of R-modules, the tensor product of the family is an R-module

:$\backslash bigotimes\_\{(i\; \backslash in\; I,\; *)\}\; M\_i$

together with an R-multilinear map satisfying (*)

:$\backslash otimes:\; \backslash prod\_\{i\; \backslash in\; I\}\; M\_i\backslash to\; \backslash bigotimes\_\{(i\; \backslash in\; I,\; *)\}\; M\_i$

such that each R-multilinear map satisfying (*)

:$f:\; \backslash prod\_\{i\; \backslash in\; I\}\; M\_i\; \backslash to\; N$

uniquely factors as

:$\backslash prod\_\{i\; \backslash in\; I\}\; M\_i\; \backslash overset\{\backslash otimes\}\; \backslash to\; \backslash bigotimes\_\{(i\; \backslash in\; I,\; *)\}\; M\_i\; \backslash overset\{\backslash widetilde\{f\}\}\backslash to\; N.$

If (*) is empty, then this is the definition of the tensor product of a family of modules over R. But, using a non-empty (*), it also covers tensoring over non-commutative rings.

First, we make some remark on terminology. Given a ring A and M an A-module, an A-module structure on M is the same thing as a ring homomorphism (ring action)

:$\backslash pi:\; A\; \backslash to\; \backslash operatorname\{End\}(M)$

where the ring on the right is the endomorphism ring of M. We say π commutes with another action $\backslash rho:\; B\; \backslash to\; \backslash operatorname\{End\}(M)$ by a ring B if

:π(a) ρ(b) = ρ(b) π(a)

for all a in A and b in B. For example, if π, ρ are the left and right ring actions on a module, then saying the two action commute is precisely saying that the module is bimodule with π, ρ.

Now, suppose we are given a family A_i, i ∈ I, of algebras over R and a family M_i of A_i-modules. If

given some distinct j and k in I, if $M\_j$ and $M\_k$ have an action of a ring S that is compatible with that of R, then we let (P) be the property: for all s in S,

:$f((s\; \backslash cdot\; x\_j,\; x\_i\; |\; i\; \backslash ne\; j))\; =\; f((s\; \backslash cdot\; x\_k,\; x\_i\; |\; i\; \backslash ne\; k)).$

For example, if I = { 1, 2 }, then this is a fancier way of expressing the early "balanced condition". If I = { 1, 2, 3 } and if

To give a simple example (a special case of "exact couple" below), we construct a spectral sequence from an injective homomorphism $f:\; C\; \backslash to\; C$ between complexes of abelian groups (or modules) that preserves degree. By replacing C by C[f], without loss of generality, we view f as a multiplication on C. Then we have the short exact sequence of complexes:

:$0\; \backslash to\; C\; \backslash overset\{f\}\backslash to\; C\; \backslash to\; C/fC\; \backslash to\; 0.$

Taking cohomology we get the long exact sequence:

:$\backslash dots\; \backslash to\; H^p(C)\; \backslash overset\{f^*\}\backslash to\; H^p(C)\; \backslash to\; H^p(C/fC)\; \backslash overset\{\backslash delta\}\backslash to\; H^\{p+1\}(C)\; \backslash overset\{f^*\}\backslash to\; H^\{p+1\}(C)\; \backslash to\; \backslash dots.$

As a matter of the notation, we let

:$E\_1^\{p,\; *\}(C)\; =\; H^p(C/fC)$

and d the composition

:$E\_1^\{p,\; *\}(C)\; \backslash overset\{\backslash delta\}\backslash to\; H^\{p+1\}(C)\; \backslash to\; E\_1(C)^\{p+r,\; *\}.$

Notice d has square zero; i.e., it is a differential. Thus, we obtain a new complex $E\_1$ to which the above construction applies. We call the result $E\_2$ and we iterate.

Remark: The construction here covers the spectral sequence of filtered complexes.{{harvnb|Eisenbud|loc=Appendix 3. 13.3.}} If C is a complex with filtration F, then we apply the above construction to

Let X be a topological space. Then

:$E^\{p,q\}\_2\; =\; H^p(X;\; \backslash mathcal\{H\}^q(F^\{\backslash cdot\}))\; \backslash Rightarrow$

Let $X\; =\; \backslash operatorname\{spec\}\; A,\; Y\; =\; \backslash operatorname\{spec\}\; B$ where A, B are integral domains that are the quotient of the polynomial ring $k[t\_1,\; \backslash dots,\; t\_n]$, k an algebraically closed field.

• A morphism of affine varieties: Each k-algebra homomorphism $\backslash phi:\; B\; \backslash to\; A$ defines the continuous function $\backslash phi^\{\backslash \#\; \}:X\; \backslash to\; Y$ by

::$\backslash mathfrak\{m\}\; \backslash mapsto\; \backslash phi^\{-1\}(\backslash mathfrak\{m\})$.

:(It is true for this particular ring that the pre-image of a maximal ideal is maximal; cf. Jacobson ring) Any function $X\; \backslash to\; Y$ arises in this way is called a morphism of affine varieties. Now, if Y is k, then $\backslash phi^\{\backslash \#\; \}$ may be identified with a regular function. By the same logic, if $Y\; =\; k^n$, then $\backslash phi$ can be though of as an n-tuple of regular functions. Since $Y\; \backslash subset\; k^n$, a morphism between affine varieties in general would have this form.

• Any closed subset of an affine variety has the form $V(I)\; =\; \backslash \{\; \backslash mathfrak\{m\}\; \backslash in\; X\; \backslash mid\; I\; \backslash subset\; \backslash mathfrak\{m\}\; \backslash \}$; in particular, it is an affine variety.
• For any f in A, the open set $D(f)$ is an affine subvariety of X isomorphic to $\backslash operatorname\{spec\}\; (A[f^\{-1\}])$. Not every open subvariety is of this form.

---------

Let R be a ring and $A\; =\; R[t\_1,\; \backslash dots,\; t\_n]/(f\_1,\; \backslash dots,\; f\_r).$. For any scheme T over $\backslash operatorname\{Spec\}\; R$, a morphism $T\; \backslash to\; \backslash operatorname\{Spec\}\; A$ is called a T-point of $\backslash operatorname\{Spec\; A\}$. If T is affine, say, the spectrum of a ring B over R, then giving such a point is the same thing as giving a R-algebra homomorphism $A\; \backslash to\; B$.

Examples:

• Let R be an integral domain. Then the point corresponding to the zero ideal is dense: it is the generic point.
• Let R be a discrete valuation ring. Then $\backslash operatorname\{Spec\}\; R$ consists of two points: the closed point s (corresponding to the maximal ideal) and the generic point.
• Let $R$ be the polynomial ring over a field k in n variables. Then the points of $\backslash operatorname\{Spec\}R$ correspond to the orbits

{{expand-section}}

{{main|Hilbert scheme}}

(In this section, schemes means locally noetherian schemes.)

Suppose we want to parametrizes all closed subvarieties of a projective scheme. The idea is to construct a scheme so that each "point" (in the functorial sense) of the scheme corresponds to a closed subscheme. (To make the construction to work, one needs to allow for a non-variety.) Such a scheme is called a Hilbert scheme. It is a very deep theorem of Grothendieck that a Hilbert scheme exists at all. Let S be a scheme. One version of the the theorem states that,{{harvnb|Kollár|1996|loc=Ch I 1.4}} given a projective scheme X over S and a polynomial P, there exists a projective scheme $H\_X^P$ over S such that, for any S-scheme T,

:to give a T-point of $H\_X^P$; i.e., a morphism $T\; \backslash to\; H^P\_X$ is the same as to give a closed flat subschemes of $X\_T$ with Hilbert polynomial P.

Examples:

• If $P(z)\; =\; \backslash binom\{z+k\}\{k\}$, then $H\_\{\backslash mathbf\{P\}^n\_S\}^P$ is called the Grassmannian of k-planes in $\backslash mathbf\{P\}^n\_S$ and, if X is a projective scheme over X, $H\_X^P$ is called the Fano scheme of k-planes on X.{{harvnb|Eisenbud–Harris|2000|loc=VI 2.2}}