Tagged Questions

Questions about commutative rings, their ideals, and their modules.

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Exercise II-11 from Eisenbud-Harris, subscheme of dimension $0$, degree $3$, supported at origin isomorphic to what?

Suppose that $K$ is algebraically closed, and let $Z = \text{Spec}\,K[x_1, \ldots, x_n]/I \subset \mathbb{A}_K^n$ be any subscheme of dimension $0$ and degree $3$, supported at the origin. How do I ...
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A direct summand of a sequence, Rotman, Homological Algebra, ex. 10.15 [duplicate]

If $0 \rightarrow A' \xrightarrow{\delta} A \rightarrow A'' \rightarrow 0$ is a split short exact sequence in an abelian category $\mathcal{A}$ (if you like, let $\mathcal{A}$ be the category of ...
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When is $\mathbb{Z} [x]/f(x)$ a Dedekind domain?

Given a monic separable irreducible polynomial $f$ with integer coefficients, when $\mathbb{Z} [x]/f(x)$ is a Dedekind domain? And when it happens to be a Dedekind domain, how to know its class ...
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Show that there is no coefficient field containing $k(X+Y^p)$.
Let $k$ be a field of characteristic $p$, let $R = k(Y)[[X]]$ be the power series ring with coefficients in $k(Y)$. Now $R$ is a local ring whose unique maximal ideal $\mathfrak M$ consists of power ...
Given a variety $X=\{F:=x_0^3+t(x_1^3+x_2^3+x_3^3+x_4^3+1)=0\}\subset \mathbb{C}^6$, where $(x_0,...,x_4,t)$ are coordinates of $\mathbb{C}^6$. How to resolve the singularity by only blow-up smooth ...