Motivation
editLet be a metric space. Then it can be viewed as a category: the objects are the points of and to each pair of objects , there is a unique morphism; namely, the distance . 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 is
Under the natural map , each variable goes to some constant . Then goes to zero under ; i.e., . Since is a maximal ideal (as ), this implies
- .
Hence, for any ideal of ,
- .
where we wrote for the common zero set of elements of .
The converse also holds in the following sense: every closed subset of a manifold is the zero set of a smooth function.[1] To see this, let be an open cover of . Then, at expense of changing the indexing set,
Finally, here is an example of an implication to Fourier analysis. For a discrete subset , we write , 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
editOver the complex numbers
editLet be an algebraic variety over the base field , the field of complex numbers. If is smooth, also has a structure of complex-analytic manifold. In fact, the partial converse is also true:
Over a finite field
editThe important invariant that can be defined here is a Zeta function.
Representations of real forms
editWe have . That is, has two real forms and , called split and compact forms, respectively. Now, consider a complex finite-dimensional representation of . It restricts to the real representations: , and .
Ramification
editFor example, consider a finite separable morphism between smooth connected curves over an algebraically closed field. Because of separability, there is an exact sequence of coherent sheaves on X:
where is a torsion sheaf and the rest invertible sheaves. Now, the exact sequence says that, for , the length of as an -module is the same as the rank of the free -module ; which in turns is given as:
the sheaf is isomorphic to the structure sheaf of an effective divisor , called the ramification divisor, given as
Then f is unramified if and only if .
Elliptic operator associated to a Casimir element
editLet
Flat model
editGiven a non-archimedean local field , let be the valuation ring of it. Then the ring of restricted power series with coefficients in plays a role of a flat model of a Tate algebra.
Unramified ring
editA local ring of mixed characteristic is called unramified if
Clarification of the definition
editIf a ring is a direct sum of additive subgroups of R, then the structure of a graded ring is the direct sum decomposition plus the multiplications induced by the multiplication on R.
When a graded ring is a direct sum of not-necessarily additive subgroups, then are natural maps. For example, if
Split
editAn injective ring homomorphism is said to split if it is a section of a surjective ring homomorphism ; in other words, .
Similarly, a subjective ring homomorphism is said to split if it admits a section.
Proof
editFirst, 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
editThe Segre class of the normal cone C to
The meaning of a locally free module
editThere is a module that is free at each maximal ideal but not is not locally free. As an example,
Lie's formula
editLie's formula states: for a Lie group G with Lie algebra , a smooth function f on G and an element X of of small norm:
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 ,
The formula follows from introducing coordinates using the exponential map and then invoking the usual Taylor formula.
Relation to universal algebra
editIn universal algebra, there is the notion of -algebra, which is quite similar to an operad algebra. The difference is that ...
Artinian ring
editLet
- (formal smooth) For each finite generated
Height example
edit- 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 be a two-dimensional integral domain such that
Definition à la Joyal
editRoughly this approach uses formal power series.
A motivating simple example
editThe notion of "operad" gives a precise meaning to the following statement:
- Given a topological space , the loop space = the space of loops in is an algebra of some kind.
That is, an “operad” defined below fits into “some kind”.
Let denote the open interval, choose a base point * on and, for definitiveness, view as the set of all continuous maps such that (the topology on we ignore). In algebraic topology, the composition of two loops are defined as
An automorphism group acting on isomorphisms
editGiven some category and objects in it, let denote the set of all isomorphisms from X to Y. Then acts on it, say, from the right. Moreover, this action is free and transitive: is a torsor for .
To see how this works in a concrete situation, take to be a cyclic group of order n and .
Direct sum of ideals
editA direct sum of ideals is that of modules.
From a Lie group to a formal group law
editThe functor 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.[2] Let G be a Lie group; then it is real-analytic manifold (not differentiable one) with the group operations
Choose
See also: Lie operad.
Exponential map and matrix exponential
editAlthough 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 . By Ado's theorem, there is a faithful representation into the general linear group over real numbers and, through this, can be viewed as a Lie subalgebra of ; in particular, consists of matrices. Now, for each matrix , the exponential is an invertible n-by-n matrix and generates a subgroup of . Now, have the same Lie algebra and thus are locally isomorphic near the respective identity elements; that is there is a homeomorphism from a neighborhood U of to a neighborhood U' of such that and if and only if .
Determinant of Adg
editSomething like .
Examples
editFiberwise criterion for flatness
editHartshorne's theorem on regular sequences
edit"étale morphism" in coordinates
editIn general, a finite set of ring homomorphisms is called a flat covering of A (more precise of if is a faithfully flat ring homomorphism.
The category of descent data (= stack)
editTo each faithfully flat ring homomorphism one associates the category where
- An object is
Derivative of Ad
editOne can show that
for all
namely, if are vector fields on G that are x, y at the identity, then is the value of the commutators at the identity.)
If is a smooth curve with , then
Units in a formal ring law
editThe above two are special cases of more general Lascoux's formula: (still r = the rank of E)
Proofs
editHere we sketch the proofs for the first two and gives a more detailed one for the last approach.
Verma module
editThe idea is to use abstract algebra to explicitly construct a representation of a Lie algebra. By definition, given a linear functional of , the Verma module is the representation induced from the representation of
Borel–Weil–Bott theorem
editInvariant theory
editExamples
edit- Let Γ be a finite group, k a field and .
Structure of an associative algebra
editTwo 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
edit- Given a finite group G and a commutative R-algebra A, let denote the set of all functions . Then it is an R-algebra; in fact, a Hopf algebra and, if A is a finite-dimensional algebra over a field, then is the Hopf-algebra-dual of the group algebra . Geometrically, and are Cartier duals of each other.
Linear representations of a group scheme
editAn 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 denotes the action, then, for each g in G, is a linear transformation (in fact, invertible). In other words, is a linear representation.
A linear action is called separable if each linear map 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 over a scheme S of finite type over a field k,
For example, if , 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
editFor 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 is invertible. (As in linear algebra, the invertability can be formulated using determinant).
- In particular, there is a functor 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
editThe notion is used to state (a generalization of) the Riemann–Roch theorem
Chow group over a regular base
editWith 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
editLet X be an algebraic scheme, E a vector bundle on it and s a section of E viewed as a linear map . Let be the blow-up of X along the ideal sheaf .
Examples
edit- A Chow group is a presheaf with transfers: if W is a correspondence from X to Y, then the transfer map is where are the projections from to .
- Let X be a smooth algebraic scheme and the contravariant functor on given by . Then it is a (representable) presheaf with transfers.
Gysin homomorphisms for divisors and line bundles
editExample: Let be the conic , and the two-dimensional vector subspace. Then V determines
Let X be a scheme over a ring A. Suppose there is a morphism
- .
Then, along this map, the Serre twisting sheaf pulls-back to a line bundle L on X, which is generated by the global sections .[4] Conversely, any line bundle L which is generated by global sections defines a morphism
which in homogeneous coordinates is given by This map is such that and . Furthermore, is a closed immersion if and only if are affine and are surjective.[5] Let be the sheaf on X associated with the total ring of fractions of . A global section of (* means multiplicative group) is called a Cartier divisor on X. The notion actually adds nothing new: there is the canonical bijection
from the set of all Cartier divisors on X to the set of all line bundles on X.-->
Example: a dual action
editLet be a group scheme acting on an affine scheme , say, from the right over a field k. Then the action corresponds to a ring homomorphism
satisfying
Normal bundle example
edit- Let be a locally free sheaf of a finite rank on a scheme X and its total space (vector bundle associated to it). Then the normal bundle to the zero-section embedding is
- ,
- as vector bundles on X.
- Let be a line bundle and the zero-section embedding. Then[citation needed]
Drinfeld level structure
editLet be a formal O-module. Then a Drinfeld level n structure on it is a homomorphism:[6]
such that the characteristic polynomial divides .
Full level structure
editThe notion of a full level structure was introduced by Katz–Mazur 1985 , who were inspired by the works of Drinfeld.
An action in étale topology
editLinear system and projective embedding
editIn algebraic topology, an (infinite) infinite projective space plays a role of the classifying space for line bundles.
A GIT quotient by a torus
editLet 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
editJust as ,
Examples
editLet be a polynomial ring. If acts on it by scaling, then and so is a point.
To get something more interesting, we consider , which is an open subvariety with affine charts . Then The GIT quotient is constructed by gluing the affine GIT quotients
Proof
edit: 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
editA linear algebraic group may be recovered from their finite-dimensional representations.
Worked-out examples
editThe basic strategy one can use to determine the decomposition of a semisimple representation into irreducible representations is as follows:[7]
- Pick some special abelian groups T and then determines the decomposition of V as a T-module.
- 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
edit- The Fourier expansion gives the decomposition
- Let be a complex torus and V
Examples in the compact case
editCharacteristic map
editLet R denote the Grothendieck ring of the category of polynomial functors of bounded degrees. One can define the map
where is the ring of symmetric functions by
where means the diagonal matrix. For example, .
The basic fact is that is an isomorphism, having the properties:
For each partition , define the polynomial functor by
- .
Then is the Schur function corresponding to .
A torsor with a structure groupoid
editOne 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 with
Grothendieck's original definition
editGrothendieck first defines an n -groupoid as data consisting of
- A set for each integer ; "F" refers to "flèche", a French word meaning "arrow",
- For each integer , a pair of functions
- For each integer , a function ,
- The composition
subject to the conditions
For example, let be a set of objects, a set of morphisms
The étale spectrum of a perfect resolving algebra
editOver 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
editBy the correspondence, given a finite-dimensional complex Lie algebra , one knows that there is a simply connected Lie group whose Lie algebra is . This can be constructed explicitly from the Lie algebra representations of .
Projective surfaces
editA 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
edit(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 all have finite length over R, then we let
- .
It is the Euler characteristic of the Koszul homology of (s, M).
The notion has a geometric interpretation. Let D1, ..., Dr be effective Cartier divisors on an algebraic variety X. Also, let W ⊂ X be a closed subvariety of X and the local ring of X at W.
Free resolution
editLemma — Let be a Noetherian local ring, and x1, ..., xr be the minimal generators of the maximal ideal . Then the Koszul complex K(x1, ..., xr) is a subcomplex of the minimal free resolution of the residue field .
Koszul complex and ideal
editAs it turns out, the Koszul complex depends only on the ideal generated by , up to isomorphism. This can be seen as follows.
If , then when s corresponds to a row vector . Now, let I be an ideal of R. Choosing a finite sequence of elements of R such that amounts to giving a surjective ring homomorphism:
TODO
editSummarize 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
editExamples
editLet R be a local ring. Then, since R has only maximal ideal, to give a morphism is to give a local homomorphism
where x is the image of .
The Grothendieck group of coherent sheaves
editLet 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
- .
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[citation needed]
(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
editLet ƒ:X→Y be a morphism of schemes with the accompanying pullback map denoted by . Then, for each x in X, since , it induces
- .
Examples
edit- The conic in P2 is isomorphic to P1 provided the characteristic of k is not 2. (Proof: it admits a rational parametrization.)
- is not flat over . For a proof and an explanation, see for instance [1]
Dominant morphisms
editConversely, if there is an inclusion of the field , then it induces a rational map f from X to Y. (Proof: there is some nonempty open affine subset V of Y such that .) If X is a smooth projective curve, then f" is regular.
Any morphism determines a rational function on X (since 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
editIt is possible to extend the definition to a tensor product of any number of modules. Let R be a commutative ring (for example Z), a family of R-modules indexed by a set I and let (*) be a property on R-multilinear maps. Then, given a family Mi, i ∈ I, of R-modules, the tensor product of the family is an R-module
together with an R-multilinear map satisfying (*)
such that each R-multilinear map satisfying (*)
uniquely factors as
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)
where the ring on the right is the endomorphism ring of M. We say π commutes with another action 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 Ai, i ∈ I, of algebras over R and a family Mi of Ai-modules. If
given some distinct j and k in I, if and 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,
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
editTo give a simple example (a special case of "exact couple" below), we construct a spectral sequence from an injective homomorphism 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:
Taking cohomology we get the long exact sequence:
As a matter of the notation, we let
and d the composition
Notice d has square zero; i.e., it is a differential. Thus, we obtain a new complex to which the above construction applies. We call the result and we iterate.
Remark: The construction here covers the spectral sequence of filtered complexes.[8] If C is a complex with filtration F, then we apply the above construction to
Examples in nature
editSpectral sequence associated to a complex of sheaves
editLet X be a topological space. Then
First properties
editLet where A, B are integral domains that are the quotient of the polynomial ring , k an algebraically closed field.
- A morphism of affine varieties: Each k-algebra homomorphism defines the continuous function by
- .
- (It is true for this particular ring that the pre-image of a maximal ideal is maximal; cf. Jacobson ring) Any function arises in this way is called a morphism of affine varieties. Now, if Y is k, then may be identified with a regular function. By the same logic, if , then can be though of as an n-tuple of regular functions. Since , a morphism between affine varieties in general would have this form.
- Any closed subset of an affine variety has the form ; in particular, it is an affine variety.
- For any f in A, the open set is an affine subvariety of X isomorphic to . Not every open subvariety is of this form.
Let R be a ring and . For any scheme T over , a morphism is called a T-point of . 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 .
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 consists of two points: the closed point s (corresponding to the maximal ideal) and the generic point.
- Let be the polynomial ring over a field k in n variables. Then the points of correspond to the orbits
Hilbert schemes
editThis section needs expansion. You can help by adding to it. |
(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,[9] given a projective scheme X over S and a polynomial P, there exists a projective scheme over S such that, for any S-scheme T,
- to give a T-point of ; i.e., a morphism is the same as to give a closed flat subschemes of with Hilbert polynomial P.
Examples:
- If , then is called the Grassmannian of k-planes in and, if X is a projective scheme over X, is called the Fano scheme of k-planes on X.[10]
- ^ Theorem 4.13. in https://people.math.rochester.edu/faculty/iosevich/partitionofunity.pdf
- ^ F. Bruhat, Lectures on Lie Groups and Representations of Locally Compact Groups.
- ^ Fulton, Example 14.5.1.
- ^ Hartshorne 1977, Ch II, Theorem 7.1
- ^ Hartshorne 1977, Ch II, Proposition 7.2
- ^ http://people.maths.ox.ac.uk/chojecki/gdtScholze1.pdf
- ^ Editorial note: This is the approach used for example in (Fulton–Harris) , hence not original.
- ^ Eisenbud, Appendix 3. 13.3.
- ^ Kollár 1996, Ch I 1.4
- ^ Eisenbud–Harris 2000, VI 2.2