# Questions tagged [algebraic-geometry]

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

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### Regarding the Proof of Hartshorne III.7.11

I got two questions regarding the proof: Why we can just assume $A_\mathfrak{m}$ as $A$ if we pick a sufficiently small neighborhood of $x$? (the 7-th line) Why's the last isomorphism true? Thank ...
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### Motivation of separated scheme

A scheme is called separated if it is separated over $\mathbb{Z}$. Is there a specific reason to define like this, i.e. separated over $\mathbb{Z}$? I know Spec$\mathbb{Z}$ is a final object. But are ...
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### Factoring map of curves over a field of characteristic $p$

I'm reading Silverman's book The Arithmetic of Elliptic Curves and I don't follow proof of a corollary. Corollary 2.12. Every map $\psi: C_1 \rightarrow C_2$ of (smooth) curves over a field of ...
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### Methods for resolutions of sheaves on projective schemes

$\newcommand{\QCoh}{\mathsf{QCoh}} \newcommand{\mod}{\text{-} \mathsf{Mod}} \newcommand{\oh}{\mathcal{O}}$If I want to find a resolution of sheaves on an affine scheme $(X = \mathrm{Spec}(R), \oh_X)$ (...
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### Self product $E \times E$ of an elliptic curve $E$.

Suppose that the self product $E \times E$ of an elliptic curve $E$ contains a compact Riemann surface $C$ of genus $2$. For points $a, b \in E$, we have to $D = a \times E+E \times b$ is ample ...
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### Characterise polynomial roots as intersections of solution sets of its real and imaginary parts

Consider a polynomial equation $p(z)\equiv\sum_k \alpha_k z^k=0$ for $\alpha_k\in\mathbb C$. We can always understand this equation as a system of two polynomial equations, given by real and ...
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### $A$ reduced implies $A\otimes_k K$ reduced, for $k$ perfect.

Let $k$ be a perfect field and $k\subset K$ any field extension. Let $A$ be any reduced $k$-algebra, if it helps we may assume it is finitely generated but the result should be true regardless. How ...
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### $Nil(A)$ is prime implies $Nil(A\otimes_k K)$ is prime, for $k$ separably closed.

Assume $k$ is a separably closed field (i.e. the extension $k\subset\overline{k}$ is purely inseparable). Let $k\subset K$ be any field extension. Let $A$ be any $k$-algebra; if it helps we may assume ...