# Questions tagged [schemes]

The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1). It was finalized by Alexandre Grothendieck, during the 1950s and the 1960s.

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### Epi-mono factorisations in schemes via scheme-theoretic image

Suppose that $f : X \rightarrow Y$ is a morphism of schemes. Let $Z \hookrightarrow Y$ the scheme-thereotic image of $f$. Under what conditions is the morphism $X \rightarrow Z$ an epimorphism? If ...
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### Are there nonaffine schemes over which every exact sequence of vector bundles is split?

Is there an example of a non-affine scheme $X$ such that every short exact sequence of vector bundles over $X$ splits? If there are such examples then what if we ask it to be true of all (not ...
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### The weight of a weighted filtration is given (for large $m$) by a polynomial

Let $I$ be an homogeneous ideal of $k[x_0, \dots, x_n]$. Suppose to give integral weights $\lambda_0, \dots, \lambda_n$ to $x_0, \dots, x_n$. We assign a weight to every homogeneous polynomial of ...
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### Motivation for Henselian rings in algebraic geometry

In Andrew Kobin's script on Algebraic Geometry I found on page 355 a comment I would like better understand. It states Another way to view formal smoothness is as an abstraction of Hensel's Lemma. ...
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### Colimits of the infinitesimal neighborhoods of symmetric product in the category of schemes

This problem is highly related to this one and in fact it is the same question applied to a very specific situation. Given a smooth projective curve $C$, let $\text{Sym}^i(C)$ be the $i$-th symmetric ...
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### Foundational question: to nonunitial commutative rings correspond to schemes?

Affine schemes correspond to unitial commutative rings, of course. Further, let us draw upon the Gelfand correspondence, where commutative $C^*$-algebras are dual to compact hausdorff spaces. If we ...
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### Equidimensional Morphism

I am reading the paper "Relative Cycles and Chow Sheaves" due to Suslin and Voevodsky. Here we have the following definition: Definition 2.1.2. A morphism of schemes $p:X\rightarrow S$ is ...
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### How can we generalize the finite type property so that global sections still have the same property?

Motivation: When I was young(er), I was once shocked to learn that, for $X$ a scheme of finite type, $\Gamma(X,\mathscr{O}_X)$ can fail to be of finite type. Now that I am no longer so young and ...
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### Is the blowup of a toric variety corresponding to a subdivision normal?

Toroidal Embeddings 1 by KKMS say subdividing the fan of a toric variety yields the fan of a normalized blowup. How do I avoid normalization? Do I need to choose my subdivision carefully, or is it ...
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### Transversally intersecting divisors $C$ and $D$ in a Hartshorne's AG lemma

Question about proof of lemma V.1.3 in Robin Hartshorne's Algebraic Geometry on page 358. Let $X$ be surface. That's for us a nonsingular projective surface over an algebraically closed field $k$ and ...
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### Rank of the top Chow group

Let $X$ be a regular integal scheme of finite type over $\mathbb Z$ and assume that $X$ has dimension $d$. In general it is not known if the Chow groups $CH^q(X)$ ($q$ is the codimension) are finitely ...
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### Valuation diagram of scheme

Let $f: X\rightarrow Y$ be a morphism of schemes, then if $f$ is quasi-compact, then there exist a valuation ring $A$ and its fraction field $K$ satisfying the following commutative diagram. The proof ...
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### Is an algebraic space having a monomorphism to an affine scheme a scheme?

Definition An algebraic space is a functor $X$ from the opposite of the category of commutative rings to the category of sets satisfying the following conditions: The functor $X$ is a (large) etale ...
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### Finiteness of the integral closure of an integral domain in its field of fractions

I've just started reading John Milne's book ''Etale Cohomology". Prop. 1.1 of Sec 1 in Ch1 reads as follows: {\em If $X$ is a normal scheme and $X'\to X$ is its normalization in a certain ...
Let $C$ be a category, and $\Gamma$ a graph in $C$. Under good conditions it makes sense to talk about the limit $\lim \Gamma$ of $\Gamma$ in $C$. Suppose $\Gamma$ is the union of two subgraphs $\... 0answers 147 views ### Gluing two affine schemes along a different intersection Given a quasi-projective$X$variety that is the union of two affines$\text{Spec(A)}$,$\text{Spec(B)}$with intersection$\text{Spec}(C)$. Let$f\in C$. Then$\text{Spec}(C_f)$is an open in both of ... 2answers 2k views ### Building algebraic geometry without prime ideals$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\ev{ev}$Teaching algebraic geometry, in particular schemes, I am struggling to provide intuitive proofs. In particular, I find it counter-intuitive ... 0answers 92 views ### etale locally infinitesimal lifting property For a morphism$X\rightarrow Y$of qcqs schemes, one has the usual notion of formal smoothness which says that for a pair$(R,I)$with$I^2=0$, if there is a point$y\in Y(R)$such that$y_{\vert R/I}$... 0answers 100 views ### Existence of integral extension of DVR satisfying some conditions Let$X^{\prime}$and$X$be integral noether schemes over$\mathbb{C}$, and$p:X^{\prime}\rightarrow X$be a surjective morphism. Let$R$be any discrete valuation ring over$\mathbb{C}$with its ... 1answer 429 views ### Fpqc-locally constant if and only if étale-locally constant? Also in SE. Let$\mathcal{F}$be sheave over$S_\mathrm{fpqc}$. We say$\mathcal{F}$is a fpqc-locally constant sheaf (of finitely generated abelian groups) if there exists a fpqc covering$(S_i\to S)...
I have a general question about the motivation behind to definition the smooth morphisms as we know it from algebraic geometry. The most common definition of a smooth morphism $: X \to Y$ between two ...