# Tagged Questions

The concept of scheme mimics the concept of manifold obtained by glueing pieces isomorphic to open balls, but with different "basic" glueing pieces. Properly, a scheme is a locally ringed space locally isomorphic (in the category of locally ringed spaces) to an affine scheme, spectrum of a ...

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### Affine morphism question

For a scheme $X$ and ring $A$, we have $\text{Hom}_{\text{Sch}}(X, \text{Spec}A) \cong \text{Hom}_{\text{Ring}}(A, \Gamma(X,\mathcal{O}_X))$. For $X$ a scheme over $Y$, how can this be generalized ...
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### local equation of a divisor, localization and local ring at a point

Let $X$ be a regular, proper surface (integral, separated, of fin. type) over a perfect field $k$ and let $Z\subset X$ be an integral irreducible subscheme. Moreover $\eta$ is the generic point of $X$ ...
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### Isotrivial family: different definitions

Let $f:X\to B$ a flat morphism of varieties over an algebraically closed field $k$. If $f$ is flat and with connected fibres we say that $f:X\to B$ is a family over $B$. In literature you can find ...
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### Geometrical meaning of $\operatorname{Spec}\widehat{\mathcal O}_{X,x}$

Let $X$ be a complete smooth irreducible variety over a field $K$ and consider a closed point $x \in X$. Moreover let $\widehat{\mathcal O}_{X,x}$ be the $\mathfrak m_x$-adic completion of the local ...
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### Pullback of an invertible sheaf through an isomorphism

Consider an isomorphism of schemes $(f,f^{\#})(X,\mathcal{O}_X)\to(Y,\mathcal{O}_Y)$. Moreover let $\mathcal F$ be an invetible sheaf on $Y$ and let $f^{*}\mathcal{F}$ be its pullback. Is it true ...
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### “Lifting” fibres of morphism of arithmetic schemes to get rid of “nongeometric” ramification

This is a soft question and really a request for pointers towards a certain rigorous formulation of geometric intuition I've had for some "arithmetic schemes". I'm looking for ideas and key references ...
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### Does affine open set equal to distinguished open subset in an affine scheme?

For $A$ a commutative ring, does it always hold that all affine open subschemes of $\text{Spec }A$ lie over a distinguished open subset of $\text{Spec} A$?
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### Is surjectivity preserved in open neighborhoods?

Let $X,S$ be schemes of finite type over a field and let $f:X\times S\to S$ be the projection. Suppose we have a morphism of coherent sheaves $\phi:\mathscr E\to \mathscr F$ on $X\times S$. Is it ...
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### Differential forms on a scheme: unclear equation

Disclaimer: In this question I assume that the reader is familiar with the construction of the module of differentials $\Omega^1_{B|A}$ where $B$ is an $A$-algebra. (If you need more details about ...
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### Is the function field of a variety a function field?

Let $X$ be an integral Noetherian scheme of dimension $n$ over a field $k$ (arbitrary field). The function field of $X$ is defined as $K(X):=\mathcal O_{X,\eta}$ where $\eta$ is the generic point of ...
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### Function field on a regular scheme of dimension $1$

Let $(X,\mathcal O_X)$ be a locally noetherian scheme of dimension $1$ and suppose that $X$ is regular, that is: $\mathcal O_{X,x}$ is a regular local ring. We have no other hypothesis on $X$. What ...
One motivation for affine morphisms I have seen is that: $\operatorname{Spec} A \to \operatorname{Spec} \Bbb Z$ should be an affine morphism for any ring $A$. The set of affine morphisms should be ...
I have often heard it said that it is important to think of properties of a scheme $X$ as really a special case of a property of morphisms applied to the morphism $X\to \operatorname{Spec} \Bbb Z$. ...