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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4
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1answer
328 views

Pushouts of schemes along closed immersions

Is there a closed immersion $i:Z\to X$ and a finite morphism $f:Z\to Y$ of schemes such that the pushout of the span $Y\stackrel{f}{\leftarrow} Z\stackrel{i}{\rightarrow} X$ does not exist in the ...
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0answers
190 views

Formal smoothness implies local freeness of the sheaf of relative differentials

What is the least restrictive finiteness assumptions guaranteeing that for a formally smooth morphism of schemes $f:X\rightarrow Y$, the sheaf of relative differentials $\Omega _{X/Y}$ would be ...
2
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1answer
256 views

Does the fundamental group of the normalization of a scheme inject into the fundamental group of the scheme

Let $X$ be an integral noetherian finite type scheme over an algebraically closed field $k$. Let $X'\to X$ be its normalization. Is the induced homomorphism of etale fundamental groups $\pi_1(X')\to\...
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1answer
213 views

characterisation of regular morphisms

Recall that a morphism of schemes $X\rightarrow Y$ is regular if it's flat with geometric fibres that are regular schemes. Fix a field $k$ and consider a morphism $f:X\rightarrow Y$ of noetherian $k$-...
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0answers
289 views

An Explicit Example of Galois Theory for Schemes

I'm currently attempting to understand Galois theory for schemes, largely following the books Galois Theory for Schemes by Henrik Lenstra and Galois Groups and Fundamental Groups by Tamas Szamuely. ...
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1answer
399 views

Fiber Dimension Theorem for infinite-dimensional schemes

For varieties $X,Y$ over an algebraically closed field, and a surjective morphism $f:X\rightarrow Y$, $\dim f^{-1}(y)\geq\dim X-\dim Y$ for all closed $y\in Y$, and $\dim f^{-1}(y)=\dim X-\dim Y$ for ...
3
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1answer
203 views

Differential map of a dominant morphism in char zero

Let $k$ be a field of characteristic zero and $X,Y$ be integral schemes of finite type. Assume we have a dominant morphism $\pi\colon X\to Y$. Then we know that $\pi$ is generically smooth (i.e. on ...
2
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1answer
224 views

Constructible sets II (Grothendieck rings)

Here is my second question on constructible sets, now on Grothendieck rings. Let $K_0(Sch_k)$ be the Grothendieck ring of schemes over $k$. I have read that if $S$ is a constructible set in a ...
2
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0answers
159 views

Constructible sets, I (Morphisms)

I have a number of questions on constructible sets. The first one is on morphisms: suppose $X$ and $Y$ are constructible sets, respectively in projective spaces $\mathbf{P}_1$ and $\mathbf{P}_2$ over ...
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1answer
1k views

A book on elliptic curves using scheme theory?

I'm interested in learning some stuff about elliptic curves. I've been learning scheme theory, and I'm interested in seeing these tools "in action". It seems that the standard introduction to elliptic ...
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0answers
239 views

When is a coherent subsheaf determined by its global sections

I am reading an article in which a proof is based on defining a subsheaf by only giving its global sections. The exact setting is that, one has a surjective finite morphism $f:Y\to X$ between ...
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1answer
194 views

When are free modules on sheaves of sets quasicoherent?

This question was previously asked over at math.SE. Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of sets on $X$. Then we can define $\mathcal{O}_X\langle\mathcal{E}\rangle$, the free module over ...
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41 views

Special unitary group of an affine algebra is integral

Let $R$ be an affine $\mathbb C-$algebra with a linear involution $x\rightarrow \bar x=\iota(x)$, let $S=R/\iota$ and $\psi:R^n\times R^n\rightarrow R$ be an $R/S-$hermitian form. Finaly let $$SU_n(R)=...
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1answer
226 views

How do I find the algebra representing the projective bundle of a direct sum of line bundles over a projective space?

I am trying to learn how to compute the projective bundle $\mathbb{P}(\mathcal{O}(a_1)\oplus \cdots \mathcal{O}(a_k))$ over some projective space using relative proj. How can I find a presentation for ...
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1answer
332 views

Example of projective variety that do not contain algebraic curves of genus strictly greater to $1$

Does exist a smooth, complex, projective variety $X$ of dimension $d\geq2$ such that $X$ does not contain smooth, complex, projective curves of wichever genus $g\geq2$? Answer by Bertie: No, it does ...
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1answer
579 views

Sheaf associated to presheaf Aut

Let $S$ be a scheme and let $C$ be the category of schemes flat and locally of finite presentation over $S$. Endow $C$ with the fppf topology (or perhaps any subcanonical topology). Let $\mathcal P$ ...
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444 views

Generic smoothness in positive characteristic

Suppose $\pi: X\rightarrow Y$ is a dominant morphism of integral $k$-schemes, where $k$ is characteristic $p>0$, and $X$ is smooth. What assumptions do we need for there to exist a dense open $U\...
2
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1answer
213 views

Arithmetic projective duality

Projective duality is a duality that associates to a (smooth) subvariety X of $\mathbb{P}^n$ the dual variety $X^*\subset\mathbb{P}^{n*}$ of tangent hyperplanes. What makes the duality interesting ...
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76 views

Geometry of componentially locally strongly separable algebras

Janelidze's categorical Galois theory yields, for nice adjunctions, a good notion of covering morphism. The category of finitely affine schemes admits such an adjunction into the category of ...
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348 views

Chern Classes: two approaches

The following question is closely related to this one. Let $X$ a non singular projective variety over $\mathbb C$, and let $\mathscr E$ a locally free sheaf of rank $r$ (an algebraic vector bundle), ...
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1answer
253 views

Universal property of the Grassmanian [closed]

Does anybody have a good reference on the Grassmanian and its universal property? I am reading this paper on Quot schemes: https://arxiv.org/abs/math/0504590 Where the Grassmanian is constructed, ...
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3answers
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Can $\mathcal O_X$ be recognized abstract-nonsensically?

This question has been asked by Teimuraz Pirashvili many years ago. I forgot about it after a while and remembered only now by accident. He probably knows the answer by now, but I still don't. In the ...
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2answers
666 views

Can the algebraic geometry of schemes be developed internally in topoi?

Using the internal logic of a topos it's often possible to derive newer theorems about sheaves from earlier ones about simpler objects, assuming that you can prove the earlier ones constructively. In ...
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Geometry of schemes by Eisenbud and Harris

I have some background of algebraic geometry. I am now trying to study schemes from book Geometry of schemes by Eisenbud and Harris. I was checking recommendations for books on algebraic geometry ...
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3answers
483 views

Isomorphic schemes over DVR

Let $S, S'$ be flat schemes over a DVR. Their generic fibers are isomorphic and their special fibers are isomorphic as well. Does that imply $S$ and $S'$ isomorphic? If not, what can go wrong? Thanks ...
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815 views

A course on modern algebraic geometry from “The Stacks Project”

I hope this question is viable for this site. I'm sincerely sorry, if you think it isn't. For a lot of time, "EGA" by Alexander Grothendieck and Jean Dieudonne was "the" reference on the basics of ...
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1answer
399 views

Why is it useful for the (relative) Picard functor to be representable?

I have been studying Chapter 8 of Neron models by Bosch et al. The first part deals with the relative Picard functor. A lot of work is done to make it representable. My question would be why this work ...
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167 views

(Ordered) Configuration space in algebraic geometry

Let $X$ be a topological space and denote by $F_n(X)$ the following subspace: $$F_n(X):=\{(x_1,\cdots ,x_n)\in X^n: x_i\neq x_j \forall i\neq j\}.$$ Note that, we are not considering the quotient of $...
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0answers
162 views

Is there an analytic criterion for quasi-compactness of a scheme?

Let $X$ be a locally finite type scheme over $\mathbb C$. I'm looking for the analogue of the notion "finite type" for $X^{an}$ and an SGA 1 Exp. XII type of criterion which says that The scheme $X$...
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2answers
485 views

Smallness of the category of schemes of finite type

Most sources about motivic homotopy theory mention that the category of (smooth) separated schemes of finite type over a (Noetherian of finite Krull dimension) base $S$ is essentially small, which is ...
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318 views

What Spec-like functors are there?

The real spectrum functor is an analog of Spec for partially ordered commutative rings and real closed fields in place of commutative reals and algebraically closed fields. I was hoping that there ...
2
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1answer
443 views

on universal homeomorphisms between schemes

We are taught since when we are young that schemes are cool because they take into account "nilpotents". This means also that we can distinguish between schemes which have the same underlying ...
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264 views

Which locally ringed spaces are schemifiable?

(most of this question is re-asking Schemification (schematization?) of locally ringed spaces, which did not get answered) Given a locally ringed space $X$, say that a schemification of $X$ is a ...
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0answers
239 views

Why define curves over perfect fields?

One may define a curve (e.g. separated scheme of finite type of dim. 1) over an algebraically closed field, as done in Hartshorne's book. A weaker assumption, which is used commonly, is to define a ...
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112 views

functions coming from a perverse sheaf

Let's take a scheme $X$ over a finite field $k$ and $f:X(k)\rightarrow\mathbb{Q}_{\ell}$ What kind of condition do I need on $f$ if I want that it comes from an irreducible perverse sheaf on $X$?
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246 views

$\Bbb A^1$-Localisation of Schemes, and $\Bbb A^1$-Rigid Schemes

Question 1: Are there some publications or preprints that provide $\Bbb A^1$-fibrant replacements of certain classes of (smooth) schemes? Of course, smooth schemes that are $\Bbb A^1$-fibrant are $\...
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1answer
493 views

Algebro-geometric version of {vector fields} $\longleftrightarrow$ {flows} correspondence?

Main Question: What Is the correpondence between flows and vector fields in algebraic geometry? Here is a more precise statement could be an answer If it was true (I have no idea it is): "...
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1answer
265 views

Archimedean fibers “intersecting” curves on arithmetic surfaces

Let's fix a number field $K$ with its ring of integers $O_K$. Moreover consider an arithmetic surface $f:S\to \text{Spec } O_K$. For every archimedean place $\sigma$ in $K$, $K_\sigma$ is the ...
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200 views

A suspected typo, and Deligne's image of the general fiber swallowing the special

In SGA 4.5 (Arcata V.1) Deligne writes: Let $X$ be a complex analytic variety and $f:X\rightarrow D$ map $X$ into the disk. Write $[0,t]$ for closed line segment with extremities 0 and $t$ in ...
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217 views

A Hartogs-type criterion for flatness

Let $U$ be a smooth affine connected variety over $\mathbb C$ and let $V\subset U$ be an open whose complement is of codimension at least two. Now, let $Y$ be a smooth quasi-affine connected variety ...
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932 views

Are all formal schemes *really* Ind-schemes?

I'm trying to understand whether there's a fully faithful functor $LRS \supset FormalSch \to IndSch$ and in what sense. Here's my progress so far: Let $\mathsf{A}$ be the category of adic rings. The ...
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514 views

Equivalent definitions of the Hasse invariant

As probably many others before me, I got stuck in verifying all the nice properties of the Hasse invariant. Let me start by recalling one definition: Let $E\to S$ be an elliptic curve in ...
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1answer
292 views

Spin structures on schemes

This is a very naive question, but I have been wondering about the role of spin geometry and spinor structures in the context of algebraic geometry. I know the definition of spin structures and ...
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1answer
1k views

Useful, non-trivial general theorems about morphisms of schemes

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians. I'm trying to ...
9
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1answer
286 views

Tube of a mod p point on a smooth Z_(p)-scheme

Let $R$ be a smooth, integral, finite-type $\mathbb{Z}_{(p)}$-algebra of relative dimension $n$ and $\overline{f} \colon R \to \mathbb{F}_p$. Then Hensel's lemma tells us that this lifts to a map $R \...
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0answers
148 views

Quotients of quasi affine varieties and extension of scalars

I have some questions about GIT quotients and extensions of scalars of categorical quotients: 1) Let $X$ be a complex algebraic quasi-affine variety, $G$ an algebraic reductive group over $\...
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0answers
106 views

representability of some mapping stack

Let $S$ be an Artin stack of finite type. We assume that it contains a point as an open dense. Is it always true that the mapping stack: $Hom^{0}(\mathbb{P}^{1},S)$ which consists of sections ...
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277 views

About complete residues on curves

Preliminaries: Let $X$ be a projective smooth curve (scheme of finite type, integral and of dimension $1$) over a perfect field $F$. Let $K=K(X)$ be the function field of $X$ and for a closed point $...
4
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1answer
602 views

What sort of ind-scheme is this?

It apparently follows from work of Velu (MathSciNet) that every isogeny between elliptic curves in (long) Weierstrass form over $k$ can be written in the form $$ \left(\frac{u(x)}{v(x)}, \frac{s_1(x)+...
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161 views

Verdier duality on excellent schemes

Let $f:X\rightarrow Y$ be a regular morphism between $k$-schemes which are noetherian and excellent with a funcion of dimension. In the book by Illusie-Laszlo-Orgogozo, there is a theorem (4.4.1 in ...