# Questions tagged [algebras]

For questions about algebras, their properties, and their structures.

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### Why were hall cohomological algebras created or what is the motivation behind that algebra?

I can't find a paper that talks about motivation or how they are used in physics, it's that an article by calabi-yau mentions that algebra caught my attention and how I'm learning about topological ...
1 vote
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### Generating the algebra $\mathcal{O}[\mathbf{GL}(n,\mathbb{C})]$ (regular functions)

I recently asked a question about the regular functions in $\mathbf{GL}(n,\mathbb{C})$ but now that I have read the appendix I am again confused; here is my past question: Understanding regular ...
1 vote
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### What are linear coordinate functions?

In the book Symmetry, Representations and Invariants they give the following definition: If $V$ is a finite dimensional $\mathbb{C}$-vector space, a function $f\colon V\to \mathbb{C}$ is a polynomial ...
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### Algebras with surjection and equal length are isomorphic rings

Let $A, B$ be two $\mathcal{O}$-algebras with a surjective ring homomorphism $A \rightarrow B$, such that $\operatorname{length}_{\mathcal{O}}A = \operatorname{length}_{\mathcal{O}}B$. Can we conclude ...
1 vote
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### Tensor product of rings: well-definedness of multiplication

Let $R$ and $S$ be rings, and let $T$ be a ring with ring homomorphisms $\alpha: T \to R$, $\beta: T \to S$. (None of these rings are assumed to be commutative.) Since $R$ and $S$ contain subrings ...
1 vote
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### Show that an algebra $A$ is cyclic under some conditions

For a field $\mathbb{F}$ with $\mathrm{char}\ \mathbb{F} \nmid n$ and $Gal(\mathbb{F}/\mathbb{E})=\left<\sigma\right>$ of order $n$ and $a\in \mathbb{F}^\times$ the cyclic algebra is defined to ...
1 vote
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### Good examples of modules and algebras?

I'm going through Atiyah-Macdonald on my own, and I'm feeling a distinct lack of examples. With rings I can just about get by with quotients of polynomial rings and $\mathbb{Q}$ and so forth. ...
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### When does a Path Algebra give a unique Quiver?

I am studying the text An introduction to Quiver Representations by Derksen. Exercise 1.5.4 asks to prove that if the path algebras $\mathbb{C}Q$ and $\mathbb{C}Q'$ are isomorphic $\mathbb{C}$-...
1 vote
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### Base change to algebraic closure commutes with quotient of polynomial ring by maximal ideal

Let $k$ be a field, $R:=k[x_1, \cdots , x_n]$ and $\mathfrak m$ be a maximal ideal such that $R/\mathfrak m$ is a finite separable field extension of $k$. Consider the algebraic closure $\overline k$ ...
Given a quiver with reduced potential $(Q,W)$. The simple dg module over the complete Jacobian algebra $J(Q,W)$ are in bijection to the set of vertices of Q ? How do the simple modules look like ? And ...