Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [affine-schemes]

The spectrum of a commutative ring with unit is the set of prime ideals endowed with the Zariski topology. One can define a sheaf of rings on this space : to each Zariski-open set is assigned a commutative ring, thought of as the ring of "polynomial functions" defined on that set. This topological space endowed with this sheaf is called the spectrum of the ring. Every locally ringed space isomorphic to such a spectrum is called an affine scheme.

The spectrum of a commutative ring with unit is the set of prime ideals endowed with the Zariski topology. One can define a sheaf of rings on this space : to each Zariski-open set is assigned a commutative ring, thought of as the ring of "polynomial functions" defined on that set. This topological space endowed with this sheaf is called the spectrum of the ring. Every locally ringed space isomorphic to such a spectrum is called an affine scheme.

646 questions
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What's the "real" reason a finite map has finite fibers?

This is a soft question. I have encountered two very different proofs of what seems like "basically the same theorem," and I want to understand how they relate and "what the real explanation is." Please forgive my imprecision here; but I am…
Ben Blum-Smith
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For a morphism of affine schemes, the inverse of an open affine subscheme is affine

This seems ridiculously simple, but it's eluding me. Suppose $f:X\rightarrow Y$ is a morphism of affine schemes. Let $V$ be an open affine subscheme of $Y$. Why is $f^{-1}(V)$ affine? I noted that $V$ is quasi-compact and wrote it as a finite union…
Potato
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Is there a notion of "schemeification" analogous to that of sheafification of a presheaf?

So this may seem like an odd question, but hear me out. In the Stacks Project, tag 01I4, we find that not only does the category of affine schemes live inside the category of locally ringed spaces, but that limits of affine schemes can be computed…
Joe
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The distinguished open sets are affine subschemes

If $Y=$ Spec$A$ is an affine scheme and $D(f)\subseteq Y$ (with $f\in A$) is a distinguished open, I want to show that $(D(f),\mathscr O_{Y|D(f)})$ is an affine scheme. Below there is my attempt of proof, but I've found a little problem, please…
Dubious
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Can Fermat's Two Squares Theorem be phrased in terms of Schemes?

Fermat's two squares theorem says that a prime number $p = a^2 + b^2$ is the sum of two squares if and only if $p = 4k+1$. How might I phrase this in terms of Schemes? I know that $\mathrm{Spec}(\mathbb{Z}) = \{ primes \}$. And maybe we are saying…
cactus314
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Is the axiom of choice necessary to prove that closed points in the Zariski topology are maximal ideals?

I would like to solve the beginner's standard exercise which claims that a point of $\mathrm{Spec} \ R$ is closed iff it is a maximal ideal. The reverse implication is easy. The direct one seems much subtler. If $P = V(I)$ then, since there exist a…
Alex M.
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What schemes correspond to varieties in the sense of Weil?

Out of (perhaps morbid) curiosity I am trying to learn the basics of Weil's foundations of algebraic geometry. I tried to ask a question earlier but it turned out I had misunderstood some more basic points, so I want to confirm them before re-asking…
Zhen Lin
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Isomorphism of rings induces isomorphism of affine schemes

I am new to schemes and I would be very grateful if someone would check my argument below. Important Note. I am following Ravi Vakil's notes: https://math.stanford.edu/~vakil/216blog/FOAGfeb0717public.pdf I am currently on page 136 (exercise…
user350031
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Closed immersion being an affine-local property on the target.

Assume that $f : X \rightarrow Y$ is a morphism of schemes. Then prove that $f$ is a closed immersion if-f there is an affine cover of $Y$ say $\{ U_i \}$, such that the induced scheme morphisms $f^{-1}(U_i) \rightarrow U_i$, is a closed immersion $…
user321268
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Chinese remainder theorem as sheaf condition?

The chinese remainder theorem in its usual version says that for a finite set of pairwise comaximal ideals $R/\bigcap _jI_j\cong \prod _j R/I_j$. In the binary case, the following general statement holds without conditions on the ideals $R/(I\cap…
Arrow
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Spectrum of infinite product of rings

$\def\Z{{\mathbb{Z}}\,} \def\Spec{{\rm Spec}\,}$ Suppose $R$ a ring and consider $\Spec(\prod_{i \in \mathbb{Z}} R)$. Now for the finite case, I know that holds $\Spec(R \times R) = \Spec(R) \coprod \Spec(R)$. My intutition says that this does not…
Rico
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About the ramification locus of a morphism with zero dimensional fibers

This question arises from my somewhat frustrating attempts to understand what etale means (in the world of algebraic varieties for now) and marry the more advanced algebraic geometry references and the ones from differential geometry. The particular…
Theo Douvropoulos
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Tensor product of injective ring homomorphisms

What is an example of two injective homomorphisms $R \to A$, $R \to B$ of commutative rings such that $R \to A \otimes_R B$ is not injective? Of course neither $R \to A$ nor $R \to B$ can be flat in this example (but this is not enough for an…
Martin Brandenburg
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Questions on scheme morphisms

I have some questions on scheme morphisms. I ask pardon for posting them in one thread as they are most likely not worth to be distributed into several threads. Let $X=Spec R$ be a noetherian scheme. For a maximal ideal $m$ of $R$ one has a…
geometrystudent
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'$R$-rational points,' where $R$ is an arbitrary ring

On page 49 of Liu's Algebraic Geometry and Arithmetic Curves, we find Example 3.32. In it, he shows that if $k$ is a field and $X=k[T_1,\dots,T_n]/I$ is an affine scheme over $k$, the sections $X(k)$ (the $k$-rational points of $X$) are in bijection…
Potato
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