Short

Short expository notes to capture insights.
 

The Grand Dictionary of Galois Correspondences

In the Language of Adjoint Functors
math
category-theory

A comprehensive dictionary of Galois correspondences, in the language of adjoint functors.

2026/10/16
en
short
 

Adjoints, Exactness, and (Co)limits in Abelian Categories

math
category-theory
The rest of this post is devoted to
2026/10/15
en
short
 

\(\Delta \implies \otimes\)

Comultiplication makes tensor product great again
math
algebra
In general, unlike group representations or Lie algebra representations, the tensor product of two representations of a generic associative algebra is not naturally a…
2026/04/20
en
short
 

An Intrinsic, Boilerplate-Free Definition of Affine Varieties

math
algebra
[1, definition 2.1.6] gives an intrinsic, boilerplate-free definition of affine varieties as follows (note our “variety” is not necessarily irreducible):
2026/04/01
en
short
 

Lie 定理

math
algebra
lie
关于 Lie 定理的简明证明．主要参考 [1, Sec. 1.2]．
2026/03/31
en
short
 

置换速算技巧

math
algebra
左乘 \(\sigma \mapsto g \sigma\)：
2026/03/26
zh
short
 

三句话证明迹的循环不变性

math
algebra
如题，今天是 \(\pi\) day，我们三句话为迹的循环不变性 \[ \operatorname{Tr}(ABC) = \operatorname{Tr}(BCA) = \operatorname{Tr}(CAB) \] 提供一种圆润的理解．接受这一理解的前置条件是掌握自然同构 \(\operatorname{Hom}(V,W) \cong…
2026/03/14
zh
short
 

Spec，可约与连通性

math
algebra
commalg
本文中环均为交换幺环．
2026/02/27
zh
short
 

Comparing Free Modules via Homomorphisms

math
algebra
commalg
We discuss some well-known results that compare finitely-generated free modules via homomorphisms. Let \(R\) be a nonzero commutative ring with \(1\).
2026/02/25
en
short

透镜成像乱炖

math
physics

基于 Fermat 原理和 Snell 定律的双面薄透镜成像公式推导．

2025/12/18
zh
short
 

Factorials and Integer Partitions

math
combinatorics
Factorials are commonly seen in combinatorics, and today we talk about some variants of them related to integer partitions. Recall that if one wish to count the number of…
2025/06/15
en
short
 

推广的 Cayley-Hamilton 定理及其应用

math
algebra
线性代数中的 Cayley-Hamilton 定理指出，域 \(K\) 上线性变换 \(\varphi\) 的特征多项式 \(f(\lambda) = \det(\lambda I_n - \Phi)\) 是它的一个零化多项式，这里 \(\Phi \in \operatorname{Mat}_{n \times n}(K)\) 是 \(\varp…
2025/06/04
zh
short
 

有限群表示的 Maschke 定理是分裂模正合列的提升

math
algebra
所谓 Maschke 定理，是指有限群表示的半单（可完全分解）性．可以考虑如下正合列风格的理解．设 \(G\) 是有限群，\(K\) 是特征不为 \(|G|\) 的域，\(K[G]\) 是 \(G\) 的群代数；\(V\) 是 \(K[G]\)-模，\(U\) 为 \(V\) 的子模．我们有 \(K[G]\)-模正合列 \[ 0 \to U…
2025/05/25
zh
short
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