formulasearchengine - User contributions [en]

Markov's principle

2013-11-06T17:07:33Z

155.247.166.29: Added image

In [[functional analysis]], a discipline within [[mathematics]], the '''Szász–Mirakyan operators''' (also spelled "Mirakjan" and "Mirakian") are generalizations of [[Bernstein polynomials]] to infinite intervals, introduced by [[Otto Szász]] in 1950 and [[G. M. Mirakjan]] in 1941. They are defined by
:<math>\left[\mathcal{S}_n(f)\right](x)</math>=<math>e^{-nx}\sum_{k=0}^\infty{\frac{(nx)^k}{k!}f\left(\frac{k}{n}\right)}</math>
where <math>x\in[0,\infty)\subset\mathbb{R}</math> and <math>n\in\mathbb{N}</math>.<ref name="Szász">{{cite journal| last=Szász | first=Otto | year=1950 | title=Generalizations of S. Bernstein's polynomials to the infinite interval | journal=Journal of Research of the National Bureau of Standards | volume=45 | issue=3 | pages=239–245 | url=http://nvl.nist.gov/pub/nistpubs/jres/045/3/V45.N03.A09.pdf}}</ref><ref name="Walczak">{{cite journal| last=Walczak | first=Zbigniew | year=2003 | title=On modified Szasz–Mirakyan operators | journal=Novi Sad Journal of Mathematics | volume=33 | issue=1 | pages=93–107 | url=http://www.emis.de/journals/NSJOM/33_1/rad-08.pdf}}</ref>

==Basic results==
In 1964, Cheney and Sharma showed that if <math>f</math> is convex and non-linear, the sequence <math>(\mathcal{S}_n(f))_{n\in\mathbb{N}}</math> decreases with <math>n</math> (<math>\mathcal{S}_n(f)\geq f</math>).<ref name="Cheney">{{cite journal|last=Cheney|first=Edward W.|coauthors=A. Sharma|year=1964|title=Bernstein power series|journal=Canadian Journal of Mathematics|volume=16|issue=[http://books.google.com/books?id=RSNqggY5Q5cC&dq 2]|pages=241–252}}</ref> They also showed that if <math>f</math> is a polynomial of degree <math>\leq m</math>, then so is <math>\mathcal{S}_n(f)</math> for all <math>n</math>.

A converse of the first property was shown by Horová in 1968 (Altomare & Campiti 1994:350).

==Theorem on convergence==
In Szász's original paper, he proved the following:
:: If <math>f</math> is [[Continuous function|continuous]] on <math>(0,\infty)</math>, having a finite limit at infinity, then <math>\mathcal{S}_n(f)</math> [[Uniform convergence|converges uniformly]] to <math>f</math> as <math>n\rightarrow\infty</math>.<ref name="Szász"/>
This is analogous to [[Bernstein polynomial#Approximating continuous functions|a theorem stating that Bernstein polynomials approximate continuous functions on [0,1]]].

==Generalizations==
A [[Leonid Kantorovich|Kantorovich]]-type generalization is sometimes discussed in the literature. These generalizations are also called the [[Szász–Mirakjan–Kantorovich operator]]s.

In 1976, C. P. May showed that the [[Baskakov operators]] can reduce to the Szász–Mirakyan operators.<ref name="May">{{cite journal|last=May|first=C. P.|year=1976|title=Saturation and inverse theorems for combinations of a class of exponential-type operators|journal=Canadian Journal of Mathematics|volume=28|issue=6|pages=1224–1250|url=http://books.google.com/books?hl=en&lr=&id=irg7sNuSTT8C&oi=fnd&pg=PA1224&ots=cdhSVISxAs&sig=OW7T5zOvK9PSaEsmT9_40BG0Pdc}}</ref>

==References==
*{{cite book|last=Altomare|first=Francesco|coauthors=Michele Campiti|year=1994|title=Korovkin-Type Approximation Theory and Its Applications|publisher=Walter de Gruyter|isbn=3-11-014178-7}}
*{{cite journal| last=Favard | first=Jean | authorlink=Jean Favard | year=1944 | title=Sur les multiplicateurs d'interpolation | language=French | journal=Journal de Mathematiques Pures et Appliquees | volume=23 | issue=9 | pages=219–247}} (See also: [[Favard operators]])
*{{cite journal|last=Horová|first=Ivana|year=1968|title=Linear positive operators of convex functions|journal=Mathematica (Cluj)|volume=10|issue=33|pages=275–283|zbl=0186.11101}}
*{{cite journal| last=Kac | first=Mark | authorlink=Mark Kac | year=1938 | title=Une remarque sur les polynomes de M. S. Bernstein | language=French | journal=Studia Mathematica | volume=7 | pages=49–51 | zbl=0018.20704 | url=http://matwbn.icm.edu.pl/ksiazki/sm/sm7/sm715.pdf}}
*{{cite journal| last=Kac | first=M. | authorlink=Mark Kac | year=1939 | title=Reconnaissance de priorité relative à ma note 'Une remarque sur les polynomes de M. S. Bernstein' | language=French | journal=Studia Mathematica | volume=8 | pages=170 | jfm=65.0248.03 | url=http://matwbn.icm.edu.pl/ksiazki/sm/sm8/sm8111.pdf}}
*{{cite journal| last=Mirakjan | first=G. M. | year=1941 | trans_title=Approximation of continuous functions with the aid of polynomials of the form <math>e^{-nx}\sum_{k=0}^{m_n}{C_{k,n}x^k}</math> | title=Approximation des fonctions continues au moyen de polynômes de la forme <math>e^{-nx}\sum_{k=0}^{m_n}{C_{k,n}x^k}</math> | language=French | journal=[[Comptes rendus de l'Académie des sciences de l'URSS]] | volume=31 | pages=201–205 | jfm=67.0216.03}}
*{{cite journal|last=Wood|first=B.|year=1969|month=July|title=Generalized Szasz operators for the approximation in the complex domain|journal=SIAM Journal on Applied Mathematics|volume=17|pages=790–801| zbl=0182.08801|issue=4|doi=10.1137/0117071|jstor=2099320}}

===Footnotes===
<references/>

{{DEFAULTSORT:Szasz-Mirakyan Operator}}
[[Category:Approximation theory]]

Equation solving

2013-11-05T19:58:36Z

155.247.31.248: Added image

[[File:Barrel distortion.svg|thumb|left|Curvilinear barrel distortion]]
[[File:Pincushion distortion.svg|thumb|right|Curvilinear pincushion distortion]]'''Curvilinear perspective''' is a [[graphical projection]] used to draw 3D objects on 2D surfaces. It was formally codified in 1968 by the artists and art historians André Barre and Albert Flocon in the book ''La Perspective curviligne'',<ref name=Français>Albert Flocon and André Barre, ''La Perspective curviligne'', Flammarion, Éditeur, Paris, 1968</ref> which was translated into English in 1987 as ''Curvilinear Perspective: From Visual Space to the Constructed Image'' and published by the [[University of California Press]].<ref name=Anglais>Albert Flocon and André Barre, ''CurvilinearPerspective: From Visual Space to the Constructed Image'', (Robert Hansen, translator), [[University of California Press]], Berkely and Los Angeles, California, 1987 ISBN 0-520-05979-4</ref>

== Background ==

In 1959, Flocon had acquired a copy of ''Grafiek en tekeningen'' by [[M. C. Escher]] who strongly impressed him with his use of bent and curved perspective, which influenced the theory Flocon and Barre were developing. They started a long correspondence, in which Escher called Flocon a "kindred spirit".<ref name=Anglais/>

== Horizon and vanishing points ==
{{multiple image|align=center
| footer = A comparison of the same object displayed, on the left using a '''curvilinear''' perspective and on the right, using a [[vanishing point]]
| image1 = Curviligne.svg|width1=230
| image2 = VanishingPerspective.svg|width2=300
}}
[[File:Panotools5618.jpg|thumb|250px|Curvilinearity in photography: Curvilinear (above) and [[rectilinear lens|rectilinear]] (below) image. Notice the [[barrel distortion]] typical for [[fisheye lens]]es in the curvilinear image. While this example has been rectilinear-corrected by software, high quality [[wide-angle lens]]es are built with optical rectilinear correction.]]The system uses curving perspective lines instead of [[Rectilinear grid|straight converging ones]] to approximate the image on the retina of the eye, which is itself spherical, more accurately than the traditional [[linear perspective]], which uses straight lines and gets very strangely distorted at the edges.

It uses either four or five [[vanishing point]]s:

*In five-point ([[Fisheye lens|fisheye]]) perspective: Four vanishing points are placed around in a circle, they are named N, W, S, E, and one vanishing point in the center of the circle.
*Four, or infinite-point perspective is the one that (arguably) most approximates the perspective of the human eye, while at the same time being effective for making impossible spaces, while five point is the curvilinear equivalent of one point perspective, so is four point the equivalent of two point perspective.

This technique can, like two-point perspective, use a vertical line as a horizon line, creating both a worms and birds eye view at the same time. It uses four or more points equally spaced along a horizon line, all vertical lines are made perpendicular to the horizon line, while orthogonals are created using a compass set on a line made at a 90-degree angle through each of the four vanishing points.

== History ==

Earlier, less mathematically precise versions can be seen in the work of the miniaturist [[Jean Fouquet]]. [[Leonardo da Vinci]] in a lost notebook spoke of curved perspective lines.<ref name=Anglais/>

Examples of approximated five-point perspective can also be found in the self-portrait of the [[Mannerism|mannerist]] painter [[Parmigianino]] seen through a [[shaving mirror]]. Another example would be the curved mirror in ''Arnolfini's Wedding'' by the Flemish painter [[Jan van Eyck]].

The book ''Vanishing Point: Perspective for Comics from the Ground Up'' by Jason Cheeseman-Meyer teaches five and four (infinite) point perspective.

==Geometric relationship==
{{multiple image|align=center
| footer = '''''Figure 1''''' shows the wall ''1'' and the observer ''2'' from the upper projection
| image1 = Ściana.png
| width1 = 500
}}
Distances '''''a''''' and '''''c''''' between the viewer and the wall are greater than the '''''b''''' distance , so adopting the principle that when an object is a greater distance from the observer, it becomes smaller, the wall should be reduced and thus becomes distorted at the edges.

{{multiple image|align=center
| footer = '''''Figure 2''''' shows the same situation from the observer's view point.
| image1 = Ściana2.png
| width1 = 500
}}

==Mathematics==

If a point has the 3D [[Cartesian coordinates]] <math>\left(x_1,y_1,z_1\right)</math>:

:<math>Pt3D=
\begin{cases}
x1 \\
y1 \\
z1
\end{cases}</math>

the transformation of this point to a curvilinear reference system of radius <math>R</math> is:

:<math>d=\sqrt{x1^2+y1^2+z1^2}</math>

:<math>Pt2D=
\begin{cases}
x =R*(1+x1/d)\\
y =R*(1+y1/d)
\end{cases}</math>

==Examples==
<gallery>
file:Entrée_de_l_empereur_Charles_IV_à_Saint-Denis.jpg|[[Jean Fouquet]], ''Arrival of Emperor Charles IV at the Basilica St Denis''
File:Parmigianino Selfportrait.jpg|[[Parmigianino]], ''Self-portrait in a Convex Mirror''
</gallery>

== See also ==

* [[Graphical projection]]
* [[Perspective projection distortion]]
* [[linear perspective]]
* [[M. C. Escher]]
* [[Curvilinear coordinates]]

== References ==
{{Reflist}}

==External links==
*[http://www.scottmcdaniel.net/drawing/perspective/5_point.html Drawing Comics - 5-Point Perspective]
*[http://www.worldofescher.com/gallery/A23L.html House of Stairs] by [[M. C. Escher]]

{{visualization}}

{{DEFAULTSORT:Curvilinear Perspective}}
[[Category:Graphical projections]]

Commutative diagram

2013-11-05T19:56:24Z

155.247.31.248: Added image

[[File:SnakeLemma01.png|thumb|350px|A diagram used in the [[snake lemma]], a basic result in homological algebra.]]

'''Homological algebra''' is the branch of [[mathematics]] that studies [[homology (mathematics)|homology]] in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in [[combinatorial topology]] (a precursor to [[algebraic topology]]) and [[abstract algebra]] (theory of [[module (mathematics)|modules]] and [[Syzygy (mathematics)|syzygies]]) at the end of the 19th century, chiefly by [[Henri Poincaré]] and [[David Hilbert]].


The development of homological algebra was closely intertwined with the emergence of [[category theory]]. By and large, homological algebra is the study of homological [[functor]]s and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of '''[[chain complex]]es''', which can be studied both through their homology and [[cohomology]]. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological [[invariant (mathematics)|invariants]] of [[ring (mathematics)|rings]], modules, [[topological space]]s, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by [[spectral sequence]]s.

From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes [[commutative algebra]], [[algebraic geometry]], [[algebraic number theory]], [[representation theory]], [[mathematical physics]], [[operator algebra]]s, [[complex analysis]], and the theory of [[partial differential equation]]s. [[K-theory]] is an independent discipline which draws upon methods of homological algebra, as does the [[noncommutative geometry]] of [[Alain Connes]].

==History of homological algebra==
Homological algebra began to be studied in its most basic form in the 1800s as a branch of topology, but it wasn't until the 1940s that it became an independent subject with the study of objects such as the [[ext functor]] and the [[tor functor]], among others.<ref name="Weber">History of Homological Algebra, by Chuck Weibel, pp.797-836 in the book The History of Topology, ed. I.M. James, Elsevier, 1999</ref>

== Chain complexes and homology ==
{{main|Chain complex}}
The '''[[chain complex]]''' is the central notion of homological algebra. It is a sequence <math> (C_\bullet, d_\bullet)</math> of [[abelian group]]s and [[group homomorphism]]s,
with the property that the composition of any two consecutive [[map (mathematics)|map]]s is zero:
: <math> C_\bullet: \cdots \longrightarrow
C_{n+1} \stackrel{d_{n+1}}{\longrightarrow}
C_n \stackrel{d_n}{\longrightarrow}
C_{n-1} \stackrel{d_{n-1}}{\longrightarrow}
\cdots, \quad d_n \circ d_{n+1}=0.</math>

The elements of ''C''''n'' are called ''n''-'''chains''' and the homomorphisms ''d''''n'' are called the '''boundary maps''' or '''differentials'''. The '''chain groups''' ''C''''n'' may be endowed with extra structure; for example, they may be [[vector space]]s or [[module (mathematics)|modules]] over a fixed [[ring (mathematics)|ring]] ''R''. The differentials must preserve the extra structure if it exists; for example, they must be [[linear map]]s or homomorphisms of ''R''-modules. For notational convenience, restrict attention to abelian groups (more correctly, to the [[category (mathematics)|category]] '''Ab''' of abelian groups); a celebrated [[Mitchell's embedding theorem|theorem by Barry Mitchell]] implies the results will generalize to any [[abelian category]]. Every chain complex defines two further sequences of abelian groups, the '''cycles''' ''Z''''n'' = Ker ''d''''n'' and the '''boundaries''' ''B''''n'' = Im ''d''''n''+1, where Ker ''d'' and Im ''d'' denote the [[kernel (algebra)|kernel]] and the [[image (mathematics)|image]] of ''d''. Since the composition of two consecutive boundary maps is zero, these groups are embedded into each other as

: <math> B_n \subseteq Z_n \subseteq C_n. </math>

[[Subgroup]]s of abelian groups are automatically [[normal subgroup|normal]]; therefore we can define the ''n''th '''homology group''' ''H''''n''(''C'') as the [[factor group]] of the ''n''-cycles by the ''n''-boundaries,

: <math> H_n(C) = Z_n/B_n = \operatorname{Ker}\, d_n/ \operatorname{Im}\, d_{n+1}. </math>

A chain complex is called '''acyclic''' or an '''[[exact sequence]]''' if all its homology groups are zero.

Chain complexes arise in abundance in [[abstract algebra|algebra]] and [[algebraic topology]]. For example, if ''X'' is a [[topological space]] then the [[singular chain]]s ''C''''n''(''X'') are formal [[linear combination]]s of [[continuous map]]s from the standard ''n''-[[simplex]] into ''X''; if ''K'' is a [[simplicial complex]] then the [[Chain (algebraic topology)|simplicial chain]]s ''C''''n''(''K'') are formal linear combinations of the ''n''-simplices of ''X''; if ''A'' = ''F''/''R'' is a presentation of an abelian group ''A'' by [[Presentation of a group|generators and relations]], where ''F'' is a [[free abelian group]] spanned by the generators and ''R'' is the subgroup of relations, then letting ''C''1(''A'') = ''R'', ''C''0(''A'') = ''F'', and ''C''''n''(''A'') = 0 for all other ''n'' defines a sequence of abelian groups. In all these cases, there are natural differentials ''d''''n'' making ''C''''n'' into a chain complex, whose homology reflects the structure of the topological space ''X'', the simplicial complex ''K'', or the abelian group ''A''. In the case of topological spaces, we arrive at the notion of [[singular homology]], which plays a fundamental role in investigating the properties of such spaces, for example, [[manifold]]s.

On a philosophical level, homological algebra teaches us that certain chain complexes associated with algebraic or geometric objects (topological spaces, simplicial complexes, ''R''-modules) contain a lot of valuable algebraic information about them, with the homology being only the most readily available part. On a technical level, homological algebra provides the tools for manipulating complexes and extracting this information. Here are two general illustrations.
*Two objects ''X'' and ''Y'' are connected by a map ''f '' between them. Homological algebra studies the relation, induced by the map ''f'', between chain complexes associated with ''X'' and ''Y'' and their homology. This is generalized to the case of several objects and maps connecting them. Phrased in the language of [[category theory]], homological algebra studies the [[functor|functorial properties]] of various constructions of chain complexes and of the homology of these complexes.
* An object ''X'' admits multiple descriptions (for example, as a topological space and as a simplicial complex) or the complex <math>C_\bullet(X)</math> is constructed using some 'presentation' of ''X'', which involves non-canonical choices. It is important to know the effect of change in the description of ''X'' on chain complexes associated with ''X''. Typically, the complex and its homology <math>H_\bullet(C)</math> are functorial with respect to the presentation; and the homology (although not the complex itself) is actually independent of the presentation chosen, thus it is an [[invariant (mathematics)|invariant]] of ''X''.

==Standard tools==

===Exact sequences===
{{Main|Exact sequence}}
In the context of [[group theory]], a sequence
:<math>G_0 \;\xrightarrow{f_1}\; G_1 \;\xrightarrow{f_2}\; G_2 \;\xrightarrow{f_3}\; \cdots \;\xrightarrow{f_n}\; G_n</math>
of [[group (mathematics)|groups]] and [[group homomorphism]]s is called '''exact''' if the [[Image (mathematics)|image]] (or [[Range (mathematics)|range]]) of each homomorphism is equal to the [[Kernel (algebra)|kernel]] of the next:
:<math>\mathrm{im}(f_k) = \mathrm{ker}(f_{k+1}).\!</math>
Note that the sequence of groups and homomorphisms may be either finite or infinite.

A similar definition can be made for certain other [[algebraic structure]]s. For example, one could have an exact sequence of [[vector space]]s and [[linear map]]s, or of [[module (mathematics)|modules]] and [[module homomorphism]]s. More generally, the notion of an exact sequence makes sense in any [[category (mathematics)|category]] with [[kernel (category theory)|kernel]]s and [[cokernel]]s.

====Short exact sequence====

The most common type of exact sequence is the '''short exact sequence'''. This is an exact sequence of the form
:<math>A \;\overset{f}{\hookrightarrow}\; B \;\overset{g}{\twoheadrightarrow}\; C</math>
where &fnof; is a [[monomorphism]] and ''g'' is an [[epimorphism]]. In this case, ''A'' is a [[subobject]] of ''B'', and the corresponding [[quotient]] is [[isomorphism|isomorphic]] to ''C'':
:<math>C \cong B/f(A).</math>
(where ''f(A)'' = im(''f'')).

A short exact sequence of abelian groups may also be written as an exact sequence with five terms:
:<math>0 \;\xrightarrow{}\; A \;\xrightarrow{f}\; B \;\xrightarrow{g}\; C \;\xrightarrow{}\; 0</math>
where 0 represents the [[Initial and terminal objects|zero object]], such as the [[trivial group]] or a zero-dimensional vector space. The placement of the 0's forces &fnof; to be a monomorphism and ''g'' to be an epimorphism (see below).

====Long exact sequence====
A long exact sequence is an exact sequence indexed by the [[natural number]]s.

===The five lemma===
{{Main|Five lemma}}
Consider the following [[commutative diagram]] in any [[abelian category]] (such as the category of [[abelian group]]s or the category of [[vector space]]s over a given [[field (algebra)|field]]) or in the category of [[group (mathematics)|group]]s.

[[image:FiveLemma.png]]

The five lemma states that, if the rows are [[exact sequence|exact]], ''m'' and ''p'' are [[isomorphism]]s, ''l'' is an [[epimorphism]], and ''q'' is a [[monomorphism]], then ''n'' is also an isomorphism.

===The snake lemma===
{{Main|Snake lemma}}
In an [[abelian category]] (such as the category of [[abelian group]]s or the category of [[vector space]]s over a given [[field (algebra)|field]]), consider a [[commutative diagram]]:

[[File:SnakeLemma01.png]]

where the rows are [[exact sequence]]s and 0 is the [[zero object]].
Then there is an exact sequence relating the [[kernel (category theory)|kernels]] and [[cokernel]]s of ''a'', ''b'', and ''c'':

[[File:SnakeLemma02.png]]

Furthermore, if the morphism ''f'' is a [[monomorphism]], then so is the morphism ker ''a'' → ker ''b'', and if ''g''' is an [[epimorphism]], then so is coker ''b'' → coker ''c''.

===Abelian categories===
{{Main|Abelian category}}
In [[mathematics]], an '''abelian category''' is a [[category (category theory)|category]] in which [[morphism]]s and objects can be added and in which [[kernel (category theory)|kernel]]s and [[cokernel]]s exist and have desirable properties. The motivating prototype example of an abelian category is the [[category of abelian groups]], '''Ab'''. The theory originated in a tentative attempt to unify several [[cohomology theory|cohomology theories]] by [[Alexander Grothendieck]]. Abelian categories are very ''stable'' categories, for example they are [[regular category|regular]] and they satisfy the [[snake lemma]]. The class of Abelian categories is closed under several categorical constructions, for example, the category of [[chain complex]]es of an Abelian category, or the category of [[functor]]s from a [[small category]] to an Abelian category are Abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in [[algebraic geometry]], [[cohomology]] and pure [[category theory]]. Abelian categories are named after [[Niels Henrik Abel]].

More concretely, a category is '''abelian''' if
*it has a [[zero object]],
*it has all binary [[Product (category theory)|products]] and binary [[coproduct]]s, and
*it has all [[kernel (category theory)|kernels]] and [[cokernel]]s.
*all [[monomorphism]]s and [[epimorphism]]s are [[normal morphism|normal]].

===The Ext functor===
{{Main|Ext functor}}
Let ''R'' be a [[ring (mathematics)|ring]] and let Mod''R'' be the [[Category (mathematics)|category]] of [[module (mathematics)|modules]] over ''R''. Let ''B'' be in Mod''R'' and set ''T''(''B'') = Hom''R''(''A,B''), for fixed ''A'' in Mod''R''. This is a [[left exact functor]] and thus has right [[derived functor]]s ''RnT''. The Ext functor is defined by

:<math>\operatorname{Ext}_R^n(A,B)=(R^nT)(B).</math>

This can be calculated by taking any [[injective resolution]]

:<math>0 \rightarrow B \rightarrow I^0 \rightarrow I^1 \rightarrow \dots, </math>

and computing

:<math>0 \rightarrow \operatorname{Hom}_R(A,I^0) \rightarrow \operatorname{Hom}_R(A,I^1) \rightarrow \dots.</math>

Then (''RnT'')(''B'') is the [[homology (mathematics)|homology]] of this complex. Note that Hom''R''(''A,B'') is excluded from the complex.

An alternative definition is given using the functor ''G''(''A'')=Hom''R''(''A,B''). For a fixed module ''B'', this is a [[Covariance and contravariance of functors|contravariant]] [[left exact functor]], and thus we also have right [[derived functor]]s ''RnG'', and can define

:<math>\operatorname{Ext}_R^n(A,B)=(R^nG)(A).</math>

This can be calculated by choosing any [[projective resolution]]

:<math>\dots \rightarrow P^1 \rightarrow P^0 \rightarrow A \rightarrow 0, </math>

and proceeding dually by computing

:<math>0\rightarrow\operatorname{Hom}_R(P^0,B)\rightarrow \operatorname{Hom}_R(P^1,B) \rightarrow \dots.</math>

Then (''RnG'')(''A'') is the homology of this complex. Again note that Hom''R''(''A,B'') is excluded.

These two constructions turn out to yield [[isomorphic]] results, and so both may be used to calculate the Ext functor.

===Tor functor===
{{Main|Tor functor}}
Suppose ''R'' is a [[ring (mathematics)|ring]], and denoted by ''R''-'''Mod''' the [[category theory|category]] of [[module (mathematics)|left ''R''-modules]] and by '''Mod'''-''R'' the category of right ''R''-modules (if ''R'' is [[commutative ring|commutative]], the two categories coincide). Pick a fix module ''B'' in ''R''-'''Mod'''. For ''A'' in '''Mod'''-''R'', set ''T''(''A'') = ''A''&otimes;''R''''B''. Then ''T'' is a [[right exact functor]] from '''Mod'''-''R'' to the [[category of abelian groups]] '''Ab''' (in the case when ''R'' is commutative, it is a right exact functor from '''Mod'''-''R'' to '''Mod'''-''R'') and its [[derived functor|left derived functor]]s ''LnT'' are defined. We set

: <math>\mathrm{Tor}_n^R(A,B)=(L_nT)(A)</math>

i.e., we take a [[Projective module#Projective resolutions|projective resolution]]

: <math>\cdots\rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow A\rightarrow 0</math>

then remove the ''A'' term and tensor the projective resolution with ''B'' to get the complex

: <math>\cdots \rightarrow P_2\otimes_R B \rightarrow P_1\otimes_R B \rightarrow P_0\otimes_R B \rightarrow 0</math>

(note that ''A''&otimes;''R''''B'' does not appear and the last arrow is just the zero map) and take the [[homology (mathematics)|homology]] of this complex.

===Spectral sequence===
{{Main|Spectral sequence}}
Fix an [[abelian category]], such as a category of modules over a ring. A '''spectral sequence''' is a choice of a nonnegative integer ''r''0 and a collection of three sequences:
# For all integers ''r'' ≥ ''r''0, an object ''Er'', called a ''sheet'' (as in a sheet of [[paper]]), or sometimes a ''page'' or a ''term'',
# Endomorphisms ''dr'' : ''Er'' → ''Er'' satisfying ''dr'' o ''dr'' = 0, called ''boundary maps'' or ''differentials'',
# Isomorphisms of ''Er+1'' with ''H''(''Er''), the homology of ''Er'' with respect to ''dr''.

[[Image:SpectralSequence.png|frame|The E2 sheet of a cohomological spectral sequence]]
A doubly graded spectral sequence has a tremendous amount of data to keep track of, but there is a common visualization technique which makes the structure of the spectral sequence clearer. We have three indices, ''r'', ''p'', and ''q''. For each ''r'', imagine that we have a sheet of graph paper. On this sheet, we will take ''p'' to be the horizontal direction and ''q'' to be the vertical direction. At each lattice point we have the object <math>E_r^{p,q}</math>.

It is very common for ''n'' = ''p'' + ''q'' to be another natural index in the spectral sequence. ''n'' runs diagonally, northwest to southeast, across each sheet. In the homological case, the differentials have bidegree (−''r'', ''r'' − 1), so they decrease ''n'' by one. In the cohomological case, ''n'' is increased by one. When ''r'' is zero, the differential moves objects one space down or up. This is similar to the differential on a chain complex. When ''r'' is one, the differential moves objects one space to the left or right. When ''r'' is two, the differential moves objects just like a [[knight (chess)|knight]]'s move in [[chess]]. For higher ''r'', the differential acts like a generalized knight's move.

===Derived functor===
{{Main|Derived functor}}
Suppose we are given a covariant [[left exact functor]] ''F'' : '''A''' → '''B''' between two [[abelian category|abelian categories]] '''A''' and '''B'''. If 0 → ''A'' → ''B'' → ''C'' → 0 is a short exact sequence in '''A''', then applying ''F'' yields the exact sequence 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if '''A''' is "nice" enough) there is one [[canonical form|canonical]] way of doing so, given by the right derived functors of ''F''. For every ''i''≥1, there is a functor ''RiF'': '''A''' → '''B''', and the above sequence continues like so: 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') → ''R''1''F''(''A'') → ''R''1''F''(''B'') → ''R''1''F''(''C'') → ''R''2''F''(''A'') → ''R''2''F''(''B'') → ... . From this we see that ''F'' is an exact functor if and only if ''R''1''F'' = 0; so in a sense the right derived functors of ''F'' measure "how far" ''F'' is from being exact.

== Functoriality ==
A [[continuous map]] of topological spaces gives rise to a homomorphism between their ''n''th [[homology group]]s for all ''n''. This basic fact of [[algebraic topology]] finds a natural explanation through certain properties of chain complexes. Since it is very common to study
several topological spaces simultaneously, in homological algebra one is led to simultaneous consideration of multiple chain complexes.

A '''morphism''' between two chain complexes, <math> F: C_\bullet\to D_\bullet</math>, is a family of homomorphisms of abelian groups ''F''''n'':''C''''n'' → ''D''''n'' that commute with the differentials, in the sense that ''F''''n'' -1 •  ''d''''n''''C'' = ''d''''n''''D'' • ''F''''n'' for all ''n''. A morphism of chain complexes induces a morphism <math> H_\bullet(F)</math> of their homology groups, consisting of the homomorphisms ''H''''n''(''F''): ''H''''n''(''C'') → ''H''''n''(''D'') for all ''n''. A morphism ''F'' is called a '''[[quasi-isomorphism]]''' if it induces an isomorphism on the ''n''th homology for all ''n''.

Many constructions of chain complexes arising in algebra and geometry, including [[singular homology]], have the following [[functor]]iality property: if two objects ''X'' and ''Y'' are connected by a map ''f'', then the associated chain complexes are connected by a morphism ''F'' = ''C''(''f'') from <math>C_\bullet(X)</math> to <math>C_\bullet(Y),</math> and moreover, the composition ''g'' • ''f'' of maps ''f'': ''X'' → ''Y'' and ''g'': ''Y'' → ''Z'' induces the morphism ''C''(''g'' • ''f'') from <math>C_\bullet(X)</math> to <math>C_\bullet(Z)</math> that coincides with the composition ''C''(''g'') • ''C''(''f''). It follows that the homology groups <math>H_\bullet(C)</math> are functorial as well, so that morphisms between algebraic or topological objects give rise to compatible maps between their homology.

The following definition arises from a typical situation in algebra and topology. A triple consisting of three chain complexes <math>L_\bullet, M_\bullet, N_\bullet</math> and two morphisms between them, <math>f:L_\bullet\to M_\bullet, g: M_\bullet\to N_\bullet,</math>
is called an '''exact triple''', or a '''short exact sequence of complexes''', and written as

: <math> 0 \longrightarrow L_\bullet \stackrel{f}{\longrightarrow}
M_\bullet \stackrel{g}{\longrightarrow}
N_\bullet \longrightarrow 0,</math>

if for any ''n'', the sequence

: <math> 0 \longrightarrow L_n \stackrel{f_n}{\longrightarrow}
M_n \stackrel{g_n}{\longrightarrow}
N_n \longrightarrow 0 </math>

is a [[short exact sequence]] of abelian groups. By definition, this means that ''f''''n'' is an [[injection (mathematics)|injection]], ''g''''n'' is a [[surjection]], and Im ''f''''n'' =  Ker ''g''''n''. One of the most basic theorems of homological algebra, sometimes known as the [[zig-zag lemma]], states that, in this case, there is a '''long exact sequence in homology'''

: <math> \ldots \longrightarrow H_n(L) \stackrel{H_n(f)}{\longrightarrow} H_n(M) \stackrel{H_n(g)}{\longrightarrow} H_n(N) \stackrel{\delta_n}{\longrightarrow} H_{n-1}(L) \stackrel{H_{n-1}(f)}{\longrightarrow} H_{n-1}(M) \longrightarrow \ldots, </math>

where the homology groups of ''L'', ''M'', and ''N'' cyclically follow each other, and ''δ''''n'' are certain homomorphisms determined by ''f'' and ''g'', called the '''[[connecting homomorphism]]s'''. Topological manifestations of this theorem include the [[Mayer–Vietoris sequence]] and the long exact sequence for [[relative homology]].



== Foundational aspects ==
Cohomology theories have been defined for many different objects such as [[topological space]]s, [[sheaf (mathematics)|sheaves]], [[group (mathematics)|group]]s, [[ring (mathematics)|ring]]s, [[Lie algebra]]s, and [[C*-algebra]]s. The study of modern [[algebraic geometry]] would be almost unthinkable without [[sheaf cohomology]].

Central to homological algebra is the notion of [[exact sequence]]; these can be used to perform actual calculations. A classical tool of homological algebra is that of [[derived functor]]; the most basic examples are functors [[Ext functors|Ext]] and [[Tor functor|Tor]].

With a diverse set of applications in mind, it was natural to try to put the whole subject on a uniform basis. There were several attempts before the subject settled down. An approximate history can be stated as follows:

* [[Henri Cartan|Cartan]]-[[Samuel Eilenberg|Eilenberg]]: In their 1956 book "Homological Algebra", these authors used [[projective resolution|projective]] and [[Injective resolution|injective module resolutions]].
* 'Tohoku': The approach in a celebrated paper by [[Alexander Grothendieck]] which appeared in the Second Series of the [[Tohoku Mathematical Journal]] in 1957, using the [[abelian category]] concept (to include [[sheaf (mathematics)|sheaves]] of abelian groups).
* The [[derived category]] of [[Grothendieck]] and [[Jean-Louis Verdier|Verdier]]. Derived categories date back to Verdier's 1967 thesis. They are examples of [[triangulated category|triangulated categories]] used in a number of modern theories.

These move from computability to generality.

The computational sledgehammer ''par excellence'' is the [[spectral sequence]]; these are essential in the Cartan-Eilenberg and Tohoku approaches where they are needed, for instance, to compute the derived functors of a composition of two functors. Spectral sequences are less essential in the derived category approach, but still play a role whenever concrete computations are necessary.

There have been attempts at 'non-commutative' theories which extend first cohomology as ''[[torsor]]s'' (important in [[Galois cohomology]]).

{{reflist}}

==See also==
* [[Abstract nonsense]], a term for homological algebra and [[category theory]]
* [[Derivator]]
* [[Homotopical algebra]]

== References ==
* [[Henri Cartan]], [[Samuel Eilenberg]], ''Homological algebra''. With an appendix by David A. Buchsbaum. Reprint of the 1956 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. xvi+390 pp. ISBN 0-691-04991-2
* [[Alexander Grothendieck]], ''Sur quelques points d'algèbre homologique''. Tôhoku Math. J. (2) 9, 1957, 119–221
* [[Saunders Mac Lane]], ''Homology''. Reprint of the 1975 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. x+422 pp. ISBN 3-540-58662-8
* [[Peter Hilton]]; Stammbach, U. ''A course in homological algebra''. Second edition. Graduate Texts in Mathematics, 4. Springer-Verlag, New York, 1997. xii+364 pp. ISBN 0-387-94823-6
* Gelfand, Sergei I.; [[Yuri Manin]], ''Methods of homological algebra''. Translated from Russian 1988 edition. Second edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xx+372 pp. ISBN 3-540-43583-2
* Gelfand, Sergei I.; Yuri Manin, ''Homological algebra''. Translated from the 1989 Russian original by the authors. Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences (''Algebra'', V, Encyclopaedia Math. Sci., 38, Springer, Berlin, 1994). Springer-Verlag, Berlin, 1999. iv+222 pp. ISBN 3-540-65378-3
* {{Weibel IHA}}

[[Category:Homological algebra| ]]

Talk:Entire function

2013-10-04T15:24:40Z

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Evenly spaced integer topology

2012-08-23T14:12:55Z

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