Download Homological Algebra by Cartan H., Eilenberg S. PDF

By Cartan H., Eilenberg S.

While this ebook used to be written, equipment of algebraic topology had prompted revolutions on the planet of natural algebra. to elucidate the advances that were made, Cartan and Eilenberg attempted to unify the fields and to build the framework of a completely fledged concept. The invasion of algebra had happened on 3 fronts in the course of the building of cohomology theories for teams, Lie algebras, and associative algebras. This booklet provides a unmarried homology (and additionally cohomology) conception that embodies all 3; lots of effects is hence verified in a normal framework. hence, all of the 3 theories is singled out by way of an appropriate specialization, and its particular houses are studied.The start line is the proposal of a module over a hoop. the first operations are the tensor fabricated from modules and the teams of all homomorphisms of 1 module into one other. From those, "higher order" derived of operations are got, which get pleasure from all of the houses often attributed to homology theories. This leads in a ordinary approach to the research of "functors" and in their "derived functors.This mathematical masterpiece will attract all mathematicians operating in algebraic topology.

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T(A + B) = tA + tB f¨ ur A, B ∈ Mm×n (K) ur λ ∈ K 2. t(λA) = λ(tA) f¨ ur A ∈ Mm×n (K) 3. t(tA) = A f¨ ur A ∈ Mm×n (K), B ∈ Mn× (K) (vgl. 2 1. 4. ). Wir erhalten tA, indem wir die Zeilen von A als Spalten schreiben. 4 61 Die Matrix MCB (f ) einer linearen Abbildung f Seien V , W zwei endlich dimensionale K-Vektorr¨aume und sei HomK (V, W ) := {f : V −→ W | f ist K-linear} Dann ist HomK (V, W ) ein Teilraum von Abb(V, W ), also insbesondere selbst ein K-Vektorraum. Sei dimK V = n und dimK W = m. Wir w¨ahlen eine Basis B = (v1 , .

W¨ f (v1 ), . . , f (vs ) eine Basis von bild(f ) bilden. Dann ist B := {u1 , . . , ur , v1 , . . , vs } eine Basis von V . Beweis. Unabh¨ angigkeit Sei 0 = λ1 u1 + · · · + λr ur + µ1 v1 + · · · µs vs mit λi , µj ∈ K ∀i, j f =⇒ 0 = f (0) = µ1 f (v1 ) + · · · + µs f (vs ), da u1 , . . , ur ∈ kern(f ), =⇒ µ1 = · · · = µs = 0, da f (v1 ), . . f (vs ) linear unabh¨angig sind =⇒ λ1 = · · · = λr = 0, da u1 , . . , ur linear unabh¨angig sind B ist also linear unabh¨angig. Erzeugendensystem Sei v ∈ V , dann gibt es µ1 , .

Beweis. K-Linearit¨ at Seien f, g ∈ HomK (V, W ) zwei K-lineare Abbildungen und seien f¨ ur j = 1, . . n f (vj ) = a1j w1 + a2j w2 + · · · + amj wm g(vj ) = b1j w1 + b2j w2 + · · · + bmj wm die Basisdarstellungen von f (vj ) und g(vj ). Es folgt MCB (f + g) = (aij + bij ) = (aij ) + (bij ) = MCB (f ) + MCB (g) MCB (λf ) = (λaij ) = λ(aij ) = λMCB (f ) ∀λ ∈ K und MCB ist damit K-linear. Analytische Geometrie und Lineare Algebra I, Universit¨ at G¨ ottingen 2005/2006 62 5 Matrizen und lineare Abbildungen Bijektivit¨ at Sei A = (aij ) ∈ Mm×n (K).

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