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Jian's user avatar
Jian's user avatar
  • Member for 7 years
  • Last seen more than 1 year ago

some questions for myself.


  1. the number of left maximal ideals of Artinian ring may be infinite question about semi-local ring and Artin ring.

  2. a story of the Koszul complex. Motivation for Koszul complex

  3. where we use the cocycle condition in Gluing sheaves. Details of gluing sheaves on a cover

  4. An intuitive approach to the Jordan Normal form.

  5. bimodule structure in $\text{Ext}^i_R(M, N)$ when $R$ is commutative. How do I understand the module structure on Yoneda Ext?

  6. Serre quotient. Quotient of category of f.g. modules by subcategory

  7. Flat vs faithfully flat.

9.GL(n,R) is not connected, GL(n,C) is connected. What's an easy way to show that $GL(n,\mathbb C)$ is connected?
Why is the determinant continuous?
How to show path-connectedness of $GL(n,\mathbb{C})$
How to prove $GL_n(\mathbb{C})$ has no subgroup with finite index?
When are finite-index subgroups of a Lie group closed? subgroup of a lie group with finite index is closed. Zariski topology,by GL(n,C) is open affine variety of M_n(C)=C^{n^2}. C^{n^2} is irreducible, hence GL(n,C) is irreducible and dense. Hence GL(n,C) is connected.

  1. graded free is stronger than free "Graded free" is stronger than "graded and free"? Global dimension and graded global dimension of a graded ring

  2. derived category of $\text{coh}\mathbb{P}^1$ and derived cat of Kronecker algebra.

  3. If $B$ is the Serre subcategory of the abelian category of $A$, then $\textbf{D}(B)\rightarrow \textbf{D}_B(A)$ maybe not be fully faithful. But this is true for bounded(Miyachi) derived category of quotient category

  4. Why is an injective quasi-coherent sheaf's restriction to an open subset still an injective object?

  5. Finite generated filed extension is finite extension(Zariski's version of Nullstellensatz)Finitely generated algebraic field extensions are finite extensions?

  6. Why is the limit of $n^{1/n}$ is 1 How to show that $\lim_{n \to +\infty} n^{\frac{1}{n}} = 1$?

  7. Radical of a ring VS radical of a module, when Rad(R)M=Rad(M)? Jacobson radical of a ring and radical of finitely generated $R$-module

  8. $\text{height}(I)+\text{dim}(R/I)=\text{dim}(R)$ for all ideal $I$ of $R$ iff $\text{height}(\mathfrak{p})+\text{dim}(R/\mathfrak{p})=\text{dim}(R)$ for all prime ideal $\mathfrak{p}$ of $R$. Does codimension equal height in complete local domains?

  9. Homogeneous localization and regularity

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