Questions tagged [ac.commutative-algebra]

By a decomposition of a polynomial $F(x)$ over a field $K$ we mean writing $F(x)$ as $$ F(x)=G(H(x)) \quad(G(x), H(x) \in K[x]), $$ which is nontrivial if $\operatorname{deg} G(x)>1$ and $\...

Let $k$ be an algebraically closed field of characteristic $p > 0 $ and let $A$ be an algebra over $k$, which is a local ring. There is an isomorphism of algebras $\prod_{i=1}^p A \cong A \otimes k[...

Let $R \to A$ be a homomorphism of commutative rings, and let $x\in R$ be an element (or a sequence of elements in $R$, if you prefer) that is both $R$-regular and $A$-regular. Then we have $$ A\...

Let $F$ be an algebraically closed field and suppose that the torus $(F^*)^n$ acts on the Laurent polynomial ring $L$ in $n$ variables $X_1, \dots, X_n$ defined by $X_i \dashrightarrow a_iX_i$ for ...

Let $G$ be an abelian finite group act on $\mathbb C^n$, when the equivariant Hilbert scheme $\mathrm{Hilb}^{R}(\mathbb C^n)^G=\mathrm{Hilb}^{R}([\mathbb C^n/G])$ is connected? Now $R$ is a ...

I have asked this question on mathstacks, but a collegue of mine recommended me to post it here. I am trying to find an optimal system of parameters for a graded ring using Magma. Specifically, I want ...