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### Abelian categories and axiom (AB5)

Let $\mathcal{A}$ be an abelian category. We say that $\mathcal{A}$ satisfies (AB5) if $\mathcal{A}$ is cocomplete and filtered colimits are exact. In Weibel's Introduction to homological algebra, ...
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### Hartshorne Exercise II. 3.19 (c)

Hartshorne Exercise II. 3.19 (c) is as follows. Prove the following theorem of Chevalley by using Exercise II. 3.19 (a) and (b) and noetherian induction on $Y$. How do we prove this? Theorem of ...
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### Ideal class group of a one-dimensional Noetherian domain

Let $A$ be a one-dimensional Noetherian domain. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated $A$-module. It is well-known that B ...
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### On orders of a quadratic number field

Let $K$ be an algebraic number field of degree $n$. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank $n$. Let $\alpha_1, \cdots, \alpha_n$ be a basis of $R$ ...
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### Conductor of a quadratic order

We need some definitions to state the problem. Let $B$ be a commutative ring, $A$ its subring. We denote by $(A : B)$ the set $\{x \in B | xB \subset A\}$. $(A : B)$ is an ideal of $B$. It is ...
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### Canonical basis of an ideal of a quadratic order

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. I am interested in the ideal theory on $R$ because it is closely related to the theory of binary quadratic forms ...
Let $K$ be a quadratic number field. Let $\mathcal{O}_K$ be the ring of algebraic integers in $K$. Let $R$ be an order of $K$, $D$ its discriminant. I am interested in the ideal theory on $R$ because ...