### 数学代写|交换代数代写commutative algebra代考|MATH2322

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## 数学代写|交换代数代写commutative algebra代考|Extension of Coefficients

Let $\varphi: R \longrightarrow R^{\prime}$ be a ring homomorphism and $N^{\prime}$ an $R^{\prime}$-module. Then $N^{\prime}$ can be viewed as an $R$-module via $\varphi ;$ just look at $N^{\prime}$ as an additive group and define the scalar multiplication by $r x^{\prime}=\varphi(r) x^{\prime}$ for elements $r \in R$ and $x^{\prime} \in N^{\prime}$, where $\varphi(r) x^{\prime}$ is the product on $N^{\prime}$ as an $R^{\prime}$-module. We say that the $R$-module structure on $N^{\prime}$ is obtained by restriction of coefficients with respect to $\varphi$. In particular, $R^{\prime}$ itself can be viewed as an $R$-module via $\varphi$, and we see that the tensor product $M \otimes_{R} R^{\prime}$, for any $R$-module $M$, makes sense as an $R$-module. It is easily seen that $M \otimes_{R} R^{\prime}$ can even be viewed as an $R^{\prime}$-module. Indeed, let $a^{\prime} \in R^{\prime}$ and consider the $R$-bilinear map

$$M \times R^{\prime} \longrightarrow M \otimes_{R} R^{\prime}, \quad\left(x, b^{\prime}\right) \longmapsto x \otimes a^{\prime} b^{\prime},$$
as well as the induced $R$-linear map
$$M \otimes_{R} R^{\prime} \longrightarrow M \otimes_{R} R^{\prime}, \quad x \otimes b^{\prime} \longmapsto x \otimes a^{\prime} b^{\prime} .$$
Taking the latter as multiplication by $a^{\prime} \in R^{\prime}$ on $M \otimes_{R} R^{\prime}$, it is straightforward to see that the additive group $M \otimes_{R} R^{\prime}$ becomes an $R^{\prime}$-module this way. We say that $M \otimes_{R} R^{\prime}$, viewed as an $R^{\prime}$-module, is obtained from $M$ by extension of coefficients with respect to $\varphi$. More generally, given any $R^{\prime}$-module $N^{\prime}$, we can view the tensor product $M \otimes_{R} N^{\prime}$ as an $R^{\prime}$-module, just by using the map
$$R^{\prime} \times\left(M \otimes_{R} N^{\prime}\right) \longrightarrow M \otimes_{R} N^{\prime}, \quad\left(a^{\prime}, x \otimes y^{\prime}\right) \longmapsto x \otimes a^{\prime} y^{\prime},$$
as scalar multiplication.

## 数学代写|交换代数代写commutative algebra代考|Faithfully Flat Descent of Module Properties

For any ring homomorphism $R \longrightarrow R^{\prime}$, the process $\cdot \otimes_{R} R^{\prime}$ of tensoring with $R^{\prime}$ over $R$ can be viewed as an assignment that attaches to an $R$-module $M$ the $R^{\prime}$-module $M \otimes_{R} R^{\prime}$ and to a morphism of $R$-modules $\varphi: M^{\prime} \longrightarrow M$ the corresponding morphism of $R^{\prime}$-modules $\varphi \otimes \mathrm{id}: M^{\prime} \otimes_{R} R^{\prime} \longrightarrow M \otimes_{R} R^{\prime}$. We talk about a so-called functor from the category of $R$-modules to the category of $R^{\prime}$-modules. As will be seen in Section $4.6$, the general problem of descent is, in some sense, to find an inverse to this process. As a preparation, categories and their functors will be discussed more extensively in Section 4.5. At this place we start descent theory by looking at several module properties that behave well when switching back and forth between $R$ – and $R^{\prime}$-modules by means of extension of coefficients via tensor products.

Proposition 1. Let $M$ be an $R$-module and $R \longrightarrow R^{\prime}$ a ring homomorphism.
(i) If $M$ is of finite type over $R$, then $M \otimes_{R} R^{\prime}$ is of finite type over $R^{\prime}$.
(ii) If $M$ is of finite presentation over $R$, then $M \otimes_{R} R^{\prime}$ is of finite presentation over $R^{\prime}$.
(iii) If $M$ is a flat $R$-module, then $M \otimes_{R} R^{\prime}$ is a flat $R^{\prime}$-module.
(iv) If $M$ is a faithfully flat $R$-module, then $M \otimes_{R} R^{\prime}$ is a faithfully flat $R^{\prime}-$ module.
(v) If $R \longrightarrow R^{\prime}$ is faithfully flat in the sense that $R^{\prime}$ is a faithfully flat $R$-module via $R \longrightarrow R^{\prime}$, then the reversed implications of (i)-(iv) hold as well.
Proof. If
$$R^{n} \longrightarrow M \longrightarrow 0$$
or
$$R^{m} \longrightarrow R^{n} \longrightarrow M \longrightarrow 0$$
are exact sequences of $R$-modules, then the sequences obtained by tensoring with $R^{\prime}$ over $R$ are exact by $4.2 / 1$. Using the fact that the isomorphisms $R^{m} \otimes_{R} R^{\prime} \simeq\left(R^{\prime}\right)^{m}$ and $R^{n} \otimes_{R} R^{\prime} \simeq\left(R^{\prime}\right)^{n}$ furnished by $4.1 / 9$ are, in fact, isomorphisms of $R^{\prime}$-modules, assertions (i) and (ii) are clear.

Now let $M$ be a flat $R$-module. To establish (iii), we have to show for every monomorphism of $R^{\prime}$-modules $E^{\prime} \longrightarrow E$ that the tensorized map
$$E^{\prime} \otimes_{R^{\prime}}\left(R^{\prime} \otimes_{R} M\right) \longrightarrow E \otimes_{R^{\prime}}\left(R^{\prime} \otimes_{R} M\right)$$
is injective as well. To do this, look at the commutative diagram
$$E^{\prime} \otimes_{R^{\prime}}\left(R^{\prime} \otimes_{R} M\right) \longrightarrow E \otimes_{R^{\prime}}\left(R^{\prime} \otimes_{R} M\right)$$
where the vertical maps are the canonical isomorphisms from $4.3 / 2$. Since $M$ is a flat $R$-module, the lower horizontal homomorphism is injective and the same holds for the upper horizontal one.

## 数学代写|交换代数代写commutative algebra代考|Categories and Functors

The language of categories and their functors is an essential tool in advanced Algebraic Geometry. Implicitly this concept has already appeared in earlier sections, mostly in the form of “universal properties”. But we need to make more intensive use of it, especially for the descent of modules in Section 4.6. Since pure category theory is not very enlightening by itself, we have chosen to include only basic material al this place.

Definition 1. A category $\mathfrak{C}$ consists of a collection ${ }^{2} \mathrm{Ob}(\mathfrak{C})$ of so-called objects and, for each pair of objects $X, Y \in \mathrm{Ob}(\mathfrak{C})$, of a set $\operatorname{Hom}(X, Y)$ of so-called morphisms (or arrows), together with a law of composition
$$\operatorname{Hom}(Y, Z) \times \operatorname{Hom}(X, Y) \longrightarrow \operatorname{Hom}(X, Z), \quad(g, f) \longmapsto g \circ f$$
for any objects $X, Y, Z \in \mathrm{Ob}(\mathfrak{C})$. The following conditions are required:
(i) The composition of morphisms is associative.
(ii) For all $X \in \mathrm{Ob}(\mathfrak{C})$ there is a morphism $\mathrm{id}{X} \in \operatorname{Hom}(X, X)$ such that $\mathrm{id}{X} \circ f=f$ for all $f \in \operatorname{Hom}(Y, X)$ and $f \circ \mathrm{id}{X}=f$ for all $f \in \operatorname{Hom}(X, Y)$. Note that $\mathrm{id}{X}$ is unique, it is called the identity morphism on $X$.
Sometimes we write $\operatorname{Hom}{\mathfrak{C}}(X, Y)$ instead of $\operatorname{Hom}(X, Y)$, in order to specify the category $\mathfrak{C}$ whose morphisms are to be considered. In most cases, morphisms between objects $X, Y$ are indicated by arrows $X \longrightarrow Y$, thereby appealing to the concept of a (set theoretical) map. However, since only the above conditions (i) and (ii) are required, morphisms can be much more general than just maps. A morphism $f: X \longrightarrow Y$ between two objects of $\mathfrak{C}$ is called an isomorphism if there is a morphism $g: Y \longrightarrow X$ such that $g \circ f=\mathrm{id}{X}$ and $f \circ g=\mathrm{id}_{Y}$. Let us consider some examples.

## 数学代写|交换代数代写commutative algebra代考|Extension of Coefficients

R′×(米⊗Rñ′)⟶米⊗Rñ′,(一个′,X⊗是′)⟼X⊗一个′是′,

## 数学代写|交换代数代写commutative algebra代考|Faithfully Flat Descent of Module Properties

(一) 如果米是有限类型的R， 然后米⊗RR′是有限类型的R′.
(ii) 如果米是有限表示的R， 然后米⊗RR′是有限表示的R′.
(iii) 如果米是一个单位R-模块，然后米⊗RR′是一个单位R′-模块。
(iv) 如果米是一个忠实的平面R-模块，然后米⊗RR′是一个忠实的平面R′−模块。
(v) 如果R⟶R′在这个意义上是忠实平坦的R′是一个忠实的平面R-模块通过R⟶R′，那么 (i)-(iv) 的相反含义也成立。

Rn⟶米⟶0

R米⟶Rn⟶米⟶0

## 数学代写|交换代数代写commutative algebra代考|Categories and Functors

(i) 态射的组合是关联的。
(ii) 对所有人X∈○b(C)存在态射一世dX∈他⁡(X,X)这样一世dX∘F=F对所有人F∈他⁡(是,X)和F∘一世dX=F对所有人F∈他⁡(X,是). 注意一世dX是唯一的，称为恒等态射X.

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