User:TakuyaMurata/sandbox

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 $\pi :A\to A/{\mathfrak {m}}=k$, each variable $x_{i}$ goes to some constant $a_{i}\in k$. Then $I=(x_{1}-a_{1},\dots ,x_{n}-a_{n})$ goes to zero under $\pi$; i.e., $I\subset {\mathfrak {m}}$. Since $I$ is a maximal ideal (as $A/I=k$), this implies

${\mathfrak {m}}=(x_{1}-a_{1},\dots ,x_{n}-a_{n})$.

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

${\sqrt {I}}=\bigcap _{{\mathfrak {m}}\supset I}{\mathfrak {m}}=\bigcap _{a\in V(I)}(x_{1}-a_{1},\dots ,x_{n}-a_{n})=I(V(I))$.

where we wrote $V(I)\subset k^{n}$ 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. 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 $E\subset \mathbb {R} ^{n}$, we write $\delta _{E}=\sum _{a\in E}\delta _{a}$, 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.)

Varieties over specific fields

Over the complex numbers

Let $X$ be an algebraic variety over the base field $k=\mathbb {C}$, 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:

Over a finite field

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

Representations of real forms

We have ${\mathfrak {sl}}_{2}\mathbb {C} ={\mathfrak {sl}}_{2}\mathbb {R} \otimes _{\mathbb {R} }\mathbb {C} ={\mathfrak {su}}_{2}\mathbb {R} \otimes _{\mathbb {R} }\mathbb {C}$. That is, ${\mathfrak {sl}}_{2}\mathbb {C}$ has two real forms ${\mathfrak {sl}}_{2}\mathbb {R}$ and ${\mathfrak {su}}_{2}$, called split and compact forms, respectively. Now, consider a complex finite-dimensional representation of $\pi :{\mathfrak {sl}}_{2}\mathbb {C} \to {\mathfrak {gl}}(V)$. It restricts to the real representations: $\pi _{s}:{\mathfrak {sl}}_{2}\mathbb {R} \to {\mathfrak {gl}}(V_{\mathbb {R} })$,$\pi _{c}:{\mathfrak {su}}_{2}\mathbb {R} \to {\mathfrak {gl}}(V_{\mathbb {R} })$ and .

Ramification

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

$0\to f^{*}\Omega _{Y}\to \Omega _{X}\to \Omega _{X/Y}\to 0$

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

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

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

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

Elliptic operator associated to a Casimir element

Let

Flat model

Given a non-archimedean local field $K$, let $R={\mathfrak {o}}_{K}$ be the valuation ring of it. Then the ring $R\langle \xi _{1},\dots ,\xi _{n}\rangle$ of restricted power series with coefficients in $R$ plays a role of a flat model of a Tate algebra.

Unramified ring

A local ring of mixed characteristic is called unramified if

Clarification of the definition

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 $R_{i}\times R_{j}\to R_{i+j}$ induced by the multiplication on R.

When a graded ring is a direct sum of not-necessarily additive subgroups, then $R_{i}\times R_{j}\to r_{j}$ are natural maps. For example, if

Split

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

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

Proof

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.)

Segre class and intersection product

The Segre class of the normal cone C to

The meaning of a locally free module

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

Lie's formula

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

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

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 $e^{X}=\operatorname {exp} (X)$,

$f(e^{t_{1}X_{1}}\cdots e^{t_{r}X_{r}})=$

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

Relation to universal algebra

See also: Universal algebra § Generalizations

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

Artinian ring

Let

• (formal smooth) For each finite generated

Height example

• 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

Definition à la Joyal

Roughly this approach uses formal power series.

A motivating simple example

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

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

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

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

An automorphism group acting on isomorphisms

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

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

Direct sum of ideals

A direct sum of ideals is that of modules.

From a Lie group to a formal group law

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

Lie groups

→

formal group laws

where the latter category is the category of formal group laws, as follows. Let G be a Lie group; then it is real-analytic manifold (not differentiable one) with the group operations

$\mu :G\times G\to G$

$i:G\to G$

Choose

See also: Lie operad.

Exponential map and matrix exponential

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

Determinant of Ad_g

Something like $g\cdot \omega _{R}=\operatorname {det} \operatorname {Ad} _{g}\omega _{L}$.

Examples

"étale morphism" in coordinates

In general, a finite set of ring homomorphisms $\{A\to B_{i}\}$ is called a flat covering of A (more precise of $\operatorname {Spec} A)$ if $A\to \prod _{i}B_{i}$ is a faithfully flat ring homomorphism.

The category of descent data (= stack)

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

1. An object is

Derivative of Ad

One can show that

$\mathrm {ad} _{x}(y)=[x,y]\,$

for all $x,y\in {\mathfrak {g}}$

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 $VW-WV$ at the identity.)

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

$d(Ad_{x})_{e}(y)=\lim _{t\to 0}{1 \over t}\left(\operatorname {Ad} _{x}(\gamma (t))-\operatorname {Ad} _{x}(\gamma (0))\right)$

Units in a formal ring law

A formal ring law

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

$c(\operatorname {Sym} ^{2}E)=\sum _{\lambda \subset \epsilon }\left\vert {\binom {\lambda _{i}+r-i}{\epsilon _{j}+r-j}}\right\vert 2^{|\lambda |-{\binom {r}{2}}}\Delta _{\widetilde {\lambda }}c(E).$

Proofs

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

Verma module

Main article: Verma module

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

Examples

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

Structure of an associative algebra

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

Example

• Given a finite group G and a commutative R-algebra A, let $\operatorname {Map} (G,A)$ denote the set of all functions $G\to A$. 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, $\operatorname {Spec} (\operatorname {Map} (G,A))$ and $\operatorname {Spec} (A[G])=G\times _{R}\operatorname {Spec} (A)$ are Cartier duals of each other.

Linear representations of a group scheme

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 $\sigma :G\times V\to V$ denotes the action, then, for each g in G, $\sigma _{g}=\sigma (g,\cdot ):V\to V$ is a linear transformation (in fact, invertible). In other words, $G\to GL(V),\,g\mapsto \sigma _{g}$ is a linear representation.

A linear action is called separable if each linear map $\sigma _{g}:V\to V$ 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 $E\to S$ over a scheme S of finite type over a field k,

For example, if $X=\operatorname {Spec} (A)$, 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.

Ring concepts

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 $l_{r}:R\to R,a\mapsto ra$ is invertible. (As in linear algebra, the invertability can be formulated using determinant).
• In particular, there is a functor $GL:R\mapsto R^{*}$ 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.

Chern classes of a perfect complex

Main article: Chern class

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

Chow group over a regular base

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.

(Fulton 1998) uses the following notion of a relative dimension:

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

Blow-up and projective bundle

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

Examples

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

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

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

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

$\phi :X\to \mathbf {P} _{A}^{n}=\operatorname {Proj} A[x_{0},\dots ,x_{n}]$.

Then, along this map, the Serre twisting sheaf ${\mathcal {O}}(1)$ pulls-back to a line bundle L on X, which is generated by the global sections $\phi ^{*}(x_{i})$. Conversely, any line bundle L which is generated by global sections $s_{0},...,s_{n}$ defines a morphism

$\phi :X\to \mathbf {P} _{A}^{n}$

which in homogeneous coordinates is given by $\phi (x)=[s_{0}(x):\dots :s_{n}(x)].$ This map $\phi$ is such that $L\cong \phi ^{*}({\mathcal {O}}(1))$ and $s_{i}=\phi ^{*}(x_{i})$. Furthermore, $\phi$ is a closed immersion if and only if $X_{i}$ are affine and $\Gamma (U_{i},{\mathcal {O}}_{\mathbf {P} _{A}^{n}})\to \Gamma (X_{i},{\mathcal {O}}_{X_{i}})$ are surjective. Let ${\mathcal {M}}_{X}$ be the sheaf on X associated with $U\mapsto$ the total ring of fractions of $\Gamma (U,{\mathcal {O}}_{X})$. A global section of ${\mathcal {M}}_{X}^{*}/{\mathcal {O}}_{X}^{*}$ (* means multiplicative group) is called a Cartier divisor on X. The notion actually adds nothing new: there is the canonical bijection

$D\mapsto {\mathcal {L}}(D)$

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

Example: a dual action

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

$\sigma ^{\#}:R\to R\otimes _{k}A$

satisfying

Normal bundle example

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

  $N_{X/E(\xi )}=E(\xi )$,

as vector bundles on X.

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

$i^{*}{\mathcal {O}}_{L}(X)=N_{L/X}=L.$

Drinfeld level structure

Let $F/R$ be a formal O-module. Then a Drinfeld level n structure on it is a homomorphism:

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

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

Full level structure

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

Linear system and projective embedding

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

A GIT quotient by a torus

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,

Projective variety as a quotient

Just as $\mathbb {P} ^{n}=(\mathbb {A} ^{n}-0)/\mathbb {G} _{m}$,

Examples

Let $S=k[x_{0},\dots ,x_{n}]$ be a polynomial ring. If $G=\mathbb {G} _{m}$ acts on it by scaling, then $S^{G}=S_{0}=k$ and so $\operatorname {Spec} (S^{G})$ is a point.

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

Proof

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

Tannakian duality for liner algebraic groups

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

Worked-out examples

The basic strategy one can use to determine the decomposition of a semisimple representation $(\pi ,V)$ into irreducible representations is as follows:

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.

Examples in the abelian case

• The Fourier expansion $f\mapsto ({\widehat {f}}(n)e^{2\pi in\bullet }|n\in \mathbb {Z} )$ gives the decomposition

  $L^{2}(S^{1})\simeq {\widehat {\bigoplus _{n\in \mathbb {Z} }}}\,\mathbb {C} e^{2\pi in\bullet }$

• Let $T=(\mathbb {C} ^{*})^{r}$ be a complex torus and V

Characteristic map

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

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

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

$\operatorname {ch} (F)(\lambda _{1},\dots ,\lambda _{r})=\operatorname {tr} F(\operatorname {diag} (\lambda _{1},\dots \lambda _{r}))$

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

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

For each partition $\lambda$, define the polynomial functor $F_{\lambda }$ by

$F_{\lambda }(X)=(M_{\lambda }\otimes X^{\otimes n})^{S_{n}}$.

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

A torsor with a structure groupoid

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 $G\times X$ with

Grothendieck's original definition

Grothendieck first defines an n -groupoid as data consisting of

• A set $F_{i}$ for each integer $i\geq 0$; "F" refers to "flèche", a French word meaning "arrow",
• For each integer $i\geq 1$, a pair of functions $s_{i},t_{i}:F_{i}\to F_{i-1},$
• For each integer $i\geq 0$, a function $k_{i}:F_{i}\to F_{i+1}$,
• The composition

subject to the conditions

1. $s_{i-1}s_{i}=s_{i-1}t_{i},\,t_{i-1}$

For example, let $F_{0}$ be a set of objects, $F_{1}$ a set of morphisms

The étale spectrum of a perfect resolving algebra

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

Construction of simply connected Lie groups

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

Projective surfaces

Further information: 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.

The Euler characteristic of a Koszul homology

(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 $\operatorname {H} _{i}(K(s,M)),i\geq 0$ all have finite length over R, then we let

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

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 $R={\mathcal {O}}_{W,X}$ the local ring of X at W.

Free resolution

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

Koszul complex and ideal

As it turns out, the Koszul complex $K_{\bullet }(x_{1},\dots ,x_{r};M)$ depends only on the ideal generated by $x_{1},\dots ,x_{r}$, up to isomorphism. This can be seen as follows.

If $E\simeq R^{r}$, then $K_{\bullet }(s;M)\simeq K_{\bullet }(x_{1},\dots ,x_{r};M)$ when s corresponds to a row vector $[x_{1}\cdots x_{r}]:R^{r}\to R$. Now, let I be an ideal of R. Choosing a finite sequence $x_{1},\dots ,x_{r}$ of elements of R such that $I=(x_{1},\dots ,x_{r})$ amounts to giving a surjective ring homomorphism:

TODO

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

Glossary

zero-locus

1.  If f is a regular function, then the (scheme-theoretic) zero-locus is the (scheme-theoretic) fiber $f^{-1}(0)$.

2.  If s is a differential from, then. In particular, the zero-locus of df is the critical set of f.

Examples

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

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

where x is the image of ${\mathfrak {m}}_{R}$.

The Grothendieck group of coherent sheaves

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

$0\to E\to F\to G\to 0$.

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

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

(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.

Differential of a morphism

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

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

Examples

• 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.)
• $\mathbb {C} [x,y,z]/(y-xz)$ is not flat over $\mathbb {C} [x,y]$. For a proof and an explanation, see for instance [1]

Dominant morphisms

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

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

Several modules

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

$\bigotimes _{(i\in I,*)}M_{i}$

together with an R-multilinear map satisfying (*)

$\otimes :\prod _{i\in I}M_{i}\to \bigotimes _{(i\in I,*)}M_{i}$

such that each R-multilinear map satisfying (*)

$f:\prod _{i\in I}M_{i}\to N$

uniquely factors as

$\prod _{i\in I}M_{i}{\overset {\otimes }{\to }}\bigotimes _{(i\in I,*)}M_{i}{\overset {\widetilde {f}}{\to }}N.$

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)

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

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

$f((s\cdot x_{j},x_{i}|i\neq j))=f((s\cdot x_{k},x_{i}|i\neq k)).$

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

Motivational example

To give a simple example (a special case of "exact couple" below), we construct a spectral sequence from an injective homomorphism $f:C\to C$ 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:

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

Taking cohomology we get the long exact sequence:

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

As a matter of the notation, we let

$E_{1}^{p,*}(C)=H^{p}(C/fC)$

and d the composition

$E_{1}^{p,*}(C){\overset {\delta }{\to }}H^{p+1}(C)\to E_{1}(C)^{p+r,*}.$

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. If C is a complex with filtration F, then we apply the above construction to

Examples in nature

Spectral sequence associated to a complex of sheaves

Let X be a topological space. Then

$E_{2}^{p,q}=H^{p}(X;{\mathcal {H}}^{q}(F^{\cdot }))\Rightarrow$

First properties

Let $X=\operatorname {spec} A,Y=\operatorname {spec} B$ where A, B are integral domains that are the quotient of the polynomial ring $k[t_{1},\dots ,t_{n}]$, k an algebraically closed field.

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

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

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

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

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Let R be a ring and $A=R[t_{1},\dots ,t_{n}]/(f_{1},\dots ,f_{r}).$. For any scheme T over $\operatorname {Spec} R$, a morphism $T\to \operatorname {Spec} A$ is called a T-point of $\operatorname {SpecA}$. 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 $A\to B$.

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 $\operatorname {Spec} R$ 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 $\operatorname {Spec} R$ correspond to the orbits

Hilbert schemes

Main article: 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, 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 $T\to H_{X}^{P}$ is the same as to give a closed flat subschemes of $X_{T}$ with Hilbert polynomial P.

Examples:

• If $P(z)={\binom {z+k}{k}}$, then $H_{\mathbf {P} _{S}^{n}}^{P}$ is called the Grassmannian of k-planes in $\mathbf {P} _{S}^{n}$ and, if X is a projective scheme over X, $H_{X}^{P}$ is called the Fano scheme of k-planes on X.