废液缸

Academic Waste

Drafts WIP and shelved writings.
 

零点定理乱炖

Suspicious Nullstellensatz Soup
math
algebra

On Hilbert’s Nullstellensatz and the ideal–variety correspondence.

2025/08/12
zh
dedicated
 

Artin–Rees–Krull 乱炖

Artin–Rees Lemma and Krull Intersection Theorem
math
algebra
本文所指环均为交换幺环．Krull 交定理刻画了 Noether 环中任意理想幂次组成的降链的极限形态：
2025/08/04
zh
short
 

DVR 乱炖

math
algebra
当局部 Noether 环 \(A\) 的极大理想 \(\mathfrak m\) 是主理想时，\(A\) 的理想结构将变得异常简单．
2025/08/04
zh
short
 

Artin–Noether 乱炖

On the commutative Akizuki–Hopkins–Levitzki Theorem
math
algebra

交换幺环 Artinian 等价于零维 Noetherian．

2025/08/01
zh
short
 

从一般线性群 \(\operatorname{GL}_n(\mathbb F_q)\) 的 Sylow \(p\)-子群谈起

Rambling from the Sylow \(p\)-subgroups of the General Linear Group \(\operatorname{GL}_n(\mathbb F_q)\)
math
algebra
设 \(p\) 为素数，\(q = p^\alpha\)．我们引入一点先进的 \(q\)-analog 记号： \[ \begin{aligned}[] [n]_q &:= \frac{q^n - 1}{q - 1} = q^{n-1} + q^{n-2} + \cdots + q + 1 \\ [n]_q! &:=…
2025/07/12
zh
short
 

诱导表示乱炖

math
algebra
在日常生产生活中，我们经常需要将一个群表示从一个子群扩展到整个群．在所有可能的选择中，诱导表示是最自然的一种．
2025/06/07
zh
short

神奇流形在哪里

Fantastic Manifolds and Where to Find Them
math
geometry
\[ \gamma(t) := (e^{2\pi i t}, e^{2 \pi i \alpha t}) \] is an \(\mathbb R \to \mathbb T^2 \subset \mathbb C^2\) immersion (Note that it’s injective!). The corresponding…
2025/06/02
short
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