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# 数学代写|同调代数代写Homological Algebra代考|MA3204 Basic Definitions

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## 数学代写|同调代数代写Homological Algebra代考|Basic Definitions

This book deals with groups and algebras. In order to fix the notation, to recall the basic facts and to keep the book as self-contained as possible we shall briefly give the necessary definitions of the most important objects and some of their properties.
Algebras
A field in this book is always commutative, a ring is always associative and has a unit, but may be non-commutative. The centre of a ring $A$ is denoted by $Z(A)$ and is defined to be
$$Z(A):={b \in A \mid a \cdot b=b \cdot a \forall a \in A} .$$
It is clear that $Z(A)$ is a commutative subring of $A$. Ring homomorphisms are always assumed to preserve the unit.

Definition 1.1.1 Let $K$ be a commutative ring and let $A$ be a ring. Then a $K$-algebra is the structure of the ring $A$ together with a ring homomorphism $\epsilon_A: K \longrightarrow Z(A)$.
Remark 1.1.2

• Observe that a $\mathbb{Z}$-algebra is nothing else than a ring.
• Very often we are interested in the case of a field $K$ and then mostly the case where $A$ is of finite dimension over $K$. However, sometimes we need to use a broader concept, and it will be important to be able to pass to general commutative rings $K$.
• For all $\lambda \in K$ and all $a \in A$ we simply write $\lambda \cdot a:=\epsilon_A(\lambda) \cdot a$.
Example 1.1.3 A few constructions are used frequently in the sequel.
1. Let $K$ be a field and let $A=E n d_K\left(K^n\right)$ be the set of square $n$ by $n$ matrices over $K$. Then this is a $K$-algebra of finite dimension $n^2$ as a $K$-vector space, with additive ring structure being the sum of matrices defined by adding each coefficient separately and multiplicative structure the matrix multiplication. The homomorphism of $K$ to the centre of $A$ is given by sending $k \in K$ to the diagonal matrix with diagonals entries all being equal to $k$.
2. If $A_1$ and $A_2$ are both $K$-algebras, then $A_1 \times A_2$ is a $K$-algebra as well. Indeed, if $\lambda_1: K \longrightarrow Z\left(A_1\right)$ is the homomorphism defining the algebra structure of $A_1$, and if $\lambda_2: K \longrightarrow Z\left(A_2\right)$ is the homomorphism defining the algebra structure of $A_2$, then
$\lambda_1 \times \lambda_2: K \ni k \mapsto\left(\lambda_1(k), \lambda_2(k)\right) \in A_1 \times A_2$
defines an algebra structure on $A_1 \times A_2$.
3. If $L$ is a field extension of $K$, then $L$ is a $K$-algebra. If $D$ is a skew field with centre $L$, then $D$ is an $L$-algebra. Moreover, if $A$ is an $L$-algebra, then by restricting the mapping $L \longrightarrow Z(A)$ to a subfield $K$ of $L$, one sees that $A$ is also a $K$-algebra.
As usual, once we have defined the objects, we are interested in structure preserving maps.

## 数学代写|同调代数代写Homological Algebra代考|Modules

We shall be interested mainly in the action of algebras. The corresponding concept is a module, and this will be the central object studied in this book.

Definition 1.1.6 Let $K$ be a commutative ring and let $A$ be a $K$-algebra. Then a left A-module $M$ is an abelian group and a mapping $\mu: A \times M \longrightarrow M$ (we shall write $\mu(a, m)=: a \cdot m)$ such that

• $1_A \cdot m=m \forall m \in M$
• $\left(a_1+a_2\right) \cdot m=\left(a_1 \cdot m\right)+\left(a_2 \cdot m\right)$
• $\left(a_1 \cdot a_2\right) \cdot m=a_1 \cdot\left(a_2 \cdot m\right)$
• $a \cdot\left(m_1+m_2\right)=\left(a \cdot m_1\right)+\left(a \cdot m_2\right)$
for all $a, a_1, a_2 \in A$ and $m, m_1, m_2 \in M$.
Definition 1.1.7 Let $K$ be a commutative ring and let $A$ be an $K$-algebra. Then the opposite algebra $\left(A^{o p},+,{ }^{o p}\right.$ ) (or $A^{o p}$ for short) is $A$ as a $K$-module, with multiplication ${ }^{o p}$ defined by $a \cdot{ }^{o p} b:=b \cdot a$ for all $a, b \in A$.
Definition 1.1.8 For an algebra Aa right A-module is a left $A^{o p}$-module.
Example 1.1.9 Let us illustrate this concept by some easy examples.
• For a field $K$, a $K$-module is nothing other than a $K$-vector space.
• For any $K$-algebra $A$, the abelian group ${0}$ with only one element is an $A$-module, which will be denoted by 0 .
• A $\mathbb{Z}$-module is just an abelian group.
• For a field $K$ a $K[X]$-module $M$ is the action of a $K$-linear endomorphism $\psi$ on a $K$-vector space $M$. The endomorphism $\psi$ is given by $\mu(X,-)$.
• Let $\psi: A \longrightarrow B$ be an algebra homomorphism and let $M$ be a $B$-module with structure map $\mu: B \times M \longrightarrow M$. Then $M$ is an $A$-module if we define $A \times M \longrightarrow$ $M$ as the composition $\mu \circ\left(\psi \times i d_M\right)$.
• For any $a \in A$ the set $A \cdot a:={b \cdot a \mid b \in A}$ is a (left) $A$-module by multiplication on the left. If $a=1$, then we call this module the regular A-module.
We would like to compare modules.

## 数学代写|同调代数代写同源代数代考|基本定义

$$Z(A):={b \in A \mid a \cdot b=b \cdot a \forall a \in A} .$$
。很明显，$Z(A)$是$A$的交换子元素。环同态总是假定保留单位。

Let $K$ 做一块田，让 $A=E n d_K\left(K^n\right)$ 是正方形的集合 $n$ by $n$ 矩阵除以 $K$。那么这就是 $K$有限维代数 $n^2$ 作为 $K$-向量空间，加性环结构为各系数分别相加定义的矩阵和，乘性结构为矩阵乘法。的同态 $K$ 到中心 $A$ 是通过发送 $k \in K$ 到对角线矩阵对角线元素都等于 $k$.

If $A_1$ 和 $A_2$ 都是 $K$-代数 $A_1 \times A_2$ 是一个 $K$代数也是。的确，如果 $\lambda_1: K \longrightarrow Z\left(A_1\right)$ 同态定义了的代数结构吗 $A_1$，如果 $\lambda_2: K \longrightarrow Z\left(A_2\right)$ 同态定义了的代数结构吗 $A_2$，则
$\lambda_1 \times \lambda_2: K \ni k \mapsto\left(\lambda_1(k), \lambda_2(k)\right) \in A_1 \times A_2$

If $L$ 是场的延伸吗 $K$，那么 $L$ 是一个 $K$-代数。如果 $D$ 是有中心的斜场吗 $L$，那么 $D$ 是 $L$-代数。此外，如果 $A$ 是 $L$-代数，然后通过限制映射 $L \longrightarrow Z(A)$ 到子字段 $K$ 的 $L$，人们看到 $A$ 也是一个 $K$-代数。和往常一样，一旦定义了对象，我们感兴趣的是保持结构的映射

## 数学代写|同调代数代写同源代数代考|模块

$1_A \cdot m=m \forall m \in M$

$\left(a_1+a_2\right) \cdot m=\left(a_1 \cdot m\right)+\left(a_2 \cdot m\right)$

$\left(a_1 \cdot a_2\right) \cdot m=a_1 \cdot\left(a_2 \cdot m\right)$

$a \cdot\left(m_1+m_2\right)=\left(a \cdot m_1\right)+\left(a \cdot m_2\right)$

$\mathbb{Z}$-module是一个交换群。

Let $\psi: A \longrightarrow B$ 是一个代数同态，让 $M$ 做一个 $B$-模块与结构映射 $\mu: B \times M \longrightarrow M$。然后 $M$ 是 $A$-module，如果我们定义 $A \times M \longrightarrow$ $M$ 作为合成 $\mu \circ\left(\psi \times i d_M\right)$.

$a \in A$ 集合 $A \cdot a:={b \cdot a \mid b \in A}$ 是(左) $A$-module通过左边的乘法运算。如果 $a=1$，那么我们称这个模块为普通的a模块。

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