The concept of scheme is from algebraic geometry: A locally ringed space that is glued together (much like a manifold) from affine schemes, that is spectra of rings with Zariski topology and structure sheaf.
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on the adjointness of the global section functor and the Spec functor
In fact, it is an exercise on Hartshorne, Ex 2.4 to the second chapter of it (P79): let $A$ be a ring and $(X,\mathcal{O}_X)$ be a scheme. Given a morphism $f:X\longrightarrow \operatorname{Spec} ...
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2answers
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Lifting sheaves from the special fibre to the generic fibre
Let $R$ be a complete discrete valuation ring and let $S = \operatorname{Spec}(R)$. Let $\mathcal{X}\to S$ be a complete, regular, flat, connected $S$-scheme of finite type whose fibres are smooth, ...
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Can we embed K(X_eta) canonically in K(X)
Let $f:X\longrightarrow S$ be a morphism of schemes. Assume $X$ and $S$ are integral.
Let $\eta$ be the generic point of $S$ and let $X_\eta\longrightarrow \textrm{Spec} \ K(S)$ be the induced ...
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bijective morphism of affine schemes
The following question occurred to me while doing exercises in Hartshorne. If $A \to B$ is a homomorphism of (commutative, unital) rings and $f : \text{Spec } B \to \text{Spec } A$ is the ...
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1answer
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Sheaf cohomology of completion along a subvariety
Suppose we are given a subvariety $Y \subset X$ where $X$ is a projective non-singular variety. We also have a coherent sheaf $\mathcal{F}$ on $Y$. We look at the completion of $X$ along $Y$ and ...
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1answer
151 views
Reduction map from the generic to the special fibre
I have a few basic questions about Liu's [1, Section 10.1.3] description of the reduction map from the closed points of the generic fibre of a proper scheme over a complete DVR to its special fibre. ...
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Diophantine applications of Spec?
Let $f(\bar x)$ be a multivariable polynomial with integer coefficients.
The zeros of that polynomial are in bijection with the homomorphisms $\mathbb Z[\bar x] \rightarrow \mathbb Z$ that factor ...
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1answer
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“Surjectivity is stable under base change” and field compositums
If $f:X\rightarrow Y$ is a surjective morphism of schemes and $g:X'\rightarrow Y$ is another morphism of schemes, one can show that $p_{2}:X\times_{Y}X'\rightarrow X'$ is also surjective. ...
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Isomorphism of schemes and invertible sheaves
I have a question more about terminology than anything else: If $f:X\rightarrow Y$ is an isomorphism of schemes, then what does it mean to say "the invertible sheaf $\mathscr{L}$ on $X$ corresponds ...
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1answer
201 views
Connected components of a fiber product of schemes
The underlying set of the product $X \times Y$ of two schemes is by no means the set-theoretic product of the underlying sets of $X$ and $Y$.
Although I am happy with the abstract definition of fiber ...
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3answers
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Learning schemes
Could someone suggest me how to learn some basic theory of schemes? I have two books from algebraic geometry, namely "Diophantine Geometry" from Hindry and Silverman and "Algebraic geometry and ...
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1answer
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Can an integral scheme have closed points of both positive and zero characteristic?
Background
Recall that an integral scheme $X$ is a scheme which is both irreducible and reduced; equivalently, its ring of functions is an integral domain on every open subset.
Given any point $p$, ...
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Stalks on Projective Scheme
Let $k$ be an algebraic closed field. Let $x$ be a point in $X=P_k^1$. What is $O_{X,x}$?
For example, if I have $x=(t-a)\in \text{Spec }k[t]$. Looking $x$ inside $P_k^1$, does ...
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1answer
160 views
Have I got the right definition of formal smoothness?
I'm trying to work out a basic example where formal smoothness should fail.
I'm considering $\mathbb{R} \to \mathbb{R}[x,y]/(x^2-y^2)$.
The idea is that not every $\mathbb{R}$-homomorphism ...
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1answer
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1answer
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Fiber product of varieties vs schemes reference
Given two complex varieties over a common base, I can take their fiber product in the category of varieties, or I can take their fiber product in the category of schemes and then take the reduced ...
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1answer
288 views
Why do we need noetherianness (or something like it) for Serre's criterion for affineness?
Serre's criterion for affineness (Hartshorne III.3.7) states that:
Let $X$ be a noetherian scheme. Suppose $H^1(X, \mathcal{F})= 0$ for every quasi-coherent sheaf on $X$. Then $X$ is affine.
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