# Tagged Questions

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. ...

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### Sheafification of singular cochains

Let $S^k$ be the presheaf on a space $X$ that assigns to every open set $U$ the abelian group $S^k(U)$ of singular k- cochains on $U$. This is clearly not a sheaf. Consider the sheafification $F^k$ of ...
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### Open and Closed Set in Zariski Topology

I'm confused about the definition closed and open set in Zariski Topology, it is said that the set $$V(I)=\{P \in \operatorname{Spec}(R)\mid I \subseteq P\}$$ are the closed set in Zariski Topology. ...
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### “Néron-Ogg-Shafarevich criterion” in positive characteristic

The Néron-Ogg-Shafarevich theorem usually seems to be cited to say that an abelian variety over a finite extension $K/{Q}_p$ has good reduction at $\ell \neq p$ if and only if the associated ...
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### Finding the completion of a coordinate ring

Consider $A=\mathbb C[x,y]/(y^2-x(x+1))$, and consider the $\mathfrak m$-adic completion, where $\mathfrak m =(x,y)$. I want to show that this completion is isomorphic to $\mathbb C[[u,v]]/(uv)$, ...
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### Why 7 points on a twisted cubic is enough to fix a quadratic?

From Joe Harris, Algebraic Geometry, Page 10. Show that if seven points $p_{1},\cdots,p_{7}$ on a twisted cubic curve, then the common zero locus of the quadratic polynomials vanishing at the ...
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### Are there non-continuable functions that become continuable when raised to some power?

Let $X$ be a complex algebraic variety (integral, separated scheme of finite type over $\mathbb C$) and $U\hookrightarrow X$ an open subvariety. I will say that $f\in\mathscr O_X(U)$ is continuable if ...
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### Sheaf cohomology in non-commutative setup

Let $X$ be a topological space and $A$ a sheaf of noncommutative associative algebras over a fixed field $k$. My questions are: 1) Does the category of modules over A have enough injective? 2) If we ...