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## Über dieses Buch

This book is the second volume of an intensive “Russian-style” two-year undergraduate course in abstract algebra, and introduces readers to the basic algebraic structures – fields, rings, modules, algebras, groups, and categories – and explains the main principles of and methods for working with them.

The course covers substantial areas of advanced combinatorics, geometry, linear and multilinear algebra, representation theory, category theory, commutative algebra, Galois theory, and algebraic geometry – topics that are often overlooked in standard undergraduate courses.

This textbook is based on courses the author has conducted at the Independent University of Moscow and at the Faculty of Mathematics in the Higher School of Economics. The main content is complemented by a wealth of exercises for class discussion, some of which include comments and hints, as well as problems for independent study.

## Inhaltsverzeichnis

### Chapter 1. Tensor Products

Abstract
Let K be a commutative ring, and let V 1, V 2, , V n and W be K-modules. A map
$$\displaystyle{ \varphi: V _{1} \times V _{2} \times \,\cdots \, \times V _{n} \rightarrow W }$$
(1.1)
is called multilinear or n - linear if φ is linear in each argument while all the other arguments are fixed, i.e.,
$$\displaystyle{ \varphi (\,\ldots,\,\lambda v' +\mu v'',\,\ldots \,) =\lambda \,\varphi (\,\ldots,\,v',\,\ldots \,) +\mu \,\varphi (\,\ldots,\,v'',\,\ldots \,) }$$
for all λ, μ ∈ K, v′, v″ ∈ V i, $$1\leqslant i\leqslant n$$. For example, the 1-linear maps V → V are the ordinary linear endomorphisms of V, and the 2-linear maps V × V → K are the bilinear forms on V. The multilinear maps (1.1) form a K- module with the usual addition and multiplication by constants defined for maps taking values in a K-module. We denote the K- module of multilinear maps (1.1) by $$\mathop{\mathrm{Hom}}\nolimits (V _{1},V _{2},\ldots,V _{n};W)$$, or by $$\mathop{\mathrm{Hom}}\nolimits _{K}(V _{1},V _{2},\ldots,V _{n};W)$$ when explicit reference to the ground ring is required.
Alexey L. Gorodentsev

### Chapter 2. Tensor Algebras

Abstract
Let V be a vector space over an arbitrary field $$\mathbb{k}$$. We write $$V ^{\otimes n}\stackrel{\mathrm{def}}{=}V \otimes V \otimes \,\cdots \, \otimes V$$ for the tensor product of n copies of V and call it the n th tensor power of V.
Alexey L. Gorodentsev

### Chapter 3. Symmetric Functions

Abstract
The symmetric group S n acts on the polynomial ring $$\mathbb{Z}[x_{1},x_{2},\ldots,x_{n}]$$ by permutations of variables:
$$\displaystyle{ g\,f(x_{1},x_{2},\mathop{\ldots },x_{n}) = f\left (x_{g^{-1}(1)},x_{g^{-1}(2)},\ldots,x_{g^{-1}(n)}\right )\quad \forall \,g \in S_{n}. }$$
(3.1)
A polynomial $$f \in \mathbb{Z}[x_{1},x_{2},\ldots,x_{n}]$$ is called symmetric if gf = f for all g ∈ S n, and alternating if gf = sgn(g) ⋅ f for all g ∈ S n. The symmetric polynomials clearly form a subring of $$\mathbb{Z}[x_{1},x_{2},\ldots,x_{n}]$$, whereas the alternating polynomials form a module over this subring, since the product of symmetric and alternating polynomials is alternating.
Alexey L. Gorodentsev

### Chapter 4. Calculus of Arrays, Tableaux, and Diagrams

Abstract
Fix two finite sets I = { 1, 2, …, n}, J = { 1, 2, …, m} and consider a rectangular table with n columns and m rows numbered by the elements of I and J respectively in such a way that indices I increase horizontally from left to right, and indices j increase vertically from bottom to top.
Alexey L. Gorodentsev

### Chapter 5. Basic Notions of Representation Theory

Abstract
Given a set R and a field $$\mathbb{k}$$, let us write $$R \otimes \mathbb{k}$$ for the vector space with basis R over $$\mathbb{k}$$. It is formed by the formal linear combinations ∑ x r ⋅ r of elements r ∈ R with coefficients $$x_{r} \in \mathbb{k}$$, all but a finite number of which vanish. By definition, the free associative $$\mathbb{k}$$-algebra spanned by the set R is the tensor algebra $$A_{R}\stackrel{\mathrm{def}}{=}\mathsf{T}(R \otimes \mathbb{k})$$ of the vector space $$R \otimes \mathbb{k}$$.
Alexey L. Gorodentsev

### Chapter 6. Representations of Finite Groups in Greater Detail

Abstract
Everywhere in this section, we write by default G for an arbitrary finite group and $$\mathbb{k}$$ for an algebraically closed field such that $$\mathop{\mathrm{char}}\nolimits (\mathbb{k}) \nmid \vert G\vert$$.
Alexey L. Gorodentsev

### Chapter 7. Representations of Symmetric Groups

Abstract
A Young diagram λ of weight | λ | = n filled by nonrepeating numbers 1, 2,  , n is called a standard filling of shape λ. Given a filling T, we write λ(T) for its shape. Associated with every standard filling T of shape λ = (λ 1, λ 2, …, λ k), ∑ λ i = n, are the row subgroup R T ⊂ S n and the column subgroup C T ⊂ S n permuting the elements 1, 2,  , n only within the rows and within the columns of T respectively. Thus, $$R_{T} \simeq S_{\lambda _{1}} \times S_{\lambda _{2}} \times \,\cdots \, \times S_{\lambda _{k}}$$ and $$C_{T} \simeq S_{\lambda _{1}^{t}} \times S_{\lambda _{2}^{t}} \times \,\cdots \, \times S_{\lambda _{m}^{t}}$$, whereλ t = (λ 1 t, λ 2 t, …, λ m t) is the transposed Young diagram. For example, the standard filling
Alexey L. Gorodentsev

### Chapter 8. -Modules

Abstract
Everywhere in this section we assume by default that $$\mathbb{k}$$ is a field of characteristic zero.
Alexey L. Gorodentsev

### Chapter 9. Categories and Functors

Abstract
A category $$\mathcal{C}$$ is formed by a class of objects $$\mathop{\mathrm{Ob}}\nolimits \mathcal{C}$$ and a class of disjoint sets $$\mathop{\mathrm{Hom}}\nolimits (X,Y ) =\mathop{ \mathrm{Hom}}\nolimits _{\mathcal{C}}(X,Y )$$, one set for each ordered pair of objects $$X,Y \in \mathop{\mathrm{Ob}}\nolimits \mathcal{C}$$. Elements of the set $$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{C}}(X,Y )$$ are called morphisms from X to Y in the category $$\mathcal{C}$$. We will depict them by arrows φ: X → Y and refer to the objects X, Y as the source (or domain) and target (or codomain ) of φ respectively. Morphisms $$\varphi,\psi \in \mathop{\mathrm{Mor}}\nolimits \mathcal{C}$$ are called composable if the source of φ coincides with the target of ψ. For every ordered triple of objects $$X,Y,Z \in \mathop{\mathrm{Ob}}\nolimits \mathcal{C}$$, the composition map
$$\displaystyle{ \mathop{\mathrm{Hom}}\nolimits (Y,Z) \times \mathop{\mathrm{Hom}}\nolimits (X,Y ) \rightarrow \mathop{\mathrm{Hom}}\nolimits (X,Z)\,,\quad (\varphi,\psi )\mapsto \varphi \circ \psi \,, }$$
(9.1)
is defined. It is associative, meaning that (ηφ) ∘ψ = η ∘ (φψ) for all composable pairs η, φ and φ, ψ. Finally, for every $$X \in \mathop{\mathrm{Ob}}\nolimits \mathcal{C}$$, there exists an identity endomorphism
$$\displaystyle{\mathrm{Id}_{X} \in \mathop{\mathrm{End}}\nolimits _{\mathcal{C}}(X)\stackrel{\mathrm{def}}{=}\mathop{\mathrm{Hom}}\nolimits _{\mathcal{C}}(X,X)}$$
such that φ ∘ IdX = φ and IdXψ = ψ for all morphisms φ: X → Y, ψ: Z → X in $$\mathcal{C}$$. It is actually unique for every $$X \in \mathop{\mathrm{Ob}}\nolimits \mathcal{C}$$, because IdX  = IdX ∘ IdX ′ ′ = IdX ′ ′ for every two such endomorphisms $$\mathrm{Id}_{X}^{{\prime}},\mathrm{Id}_{X}^{{\prime\prime}}\in \mathop{\mathrm{Hom}}\nolimits (X,X)$$.
Alexey L. Gorodentsev

### Chapter 10. Extensions of Commutative Rings

Abstract
Everywhere in this section, the term “ring” means by default a commutative ring with unit. All ring homomorphisms are assumed to map the unit to the unit.
Alexey L. Gorodentsev

### Chapter 11. Affine Algebraic Geometry

Abstract
In this chapter we assume by default that $$\mathbb{k}$$ is an algebraically closed field.
Alexey L. Gorodentsev

### Chapter 12. Algebraic Manifolds

Abstract
Everywhere in this chapter we assume by default that the ground field $$\mathbb{k}$$ is algebraically closed.
Alexey L. Gorodentsev

### Chapter 13. Algebraic Field Extensions

Abstract
Recall that a field extension $$\mathbb{k} \subset \mathbb{F}$$ is said to be finite of degree d if $$\mathbb{F}$$ has dimension d <  as a vector space over $$\mathbb{k}$$. We write $$\deg \mathbb{F}/\mathbb{k} = d$$ in this case.
Alexey L. Gorodentsev

### Chapter 14. Examples of Galois Groups

Abstract
Let us identify the Euclidean coordinate plane $$\mathbb{R}^{2}$$ with the field $$\mathbb{C}$$ in the standard way.
Alexey L. Gorodentsev

### Backmatter

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