**26**

votes

**1**answer

679 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 ...

**2**

votes

**0**answers

149 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 ...

**3**

votes

**0**answers

161 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 ...

**0**

votes

**0**answers

77 views

### Action of a lattice on abelian varieties

Let $\pi\colon Y\to\mathbb{P}_\mathbb{C}^1$ be a ramified cover of degree two of $\mathbb{P}_{\mathbb{C}}^1$ such that $Y$ is smooth. I fix a point $x$ on $\mathbb{P}^1$, over which the cover is étale,...

**1**

vote

**0**answers

74 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 $X$?

**7**

votes

**0**answers

137 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 $\...

**4**

votes

**1**answer

560 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)+...

**4**

votes

**2**answers

330 views

### An apparent equivalence of the category of affine schemes over $S$ and the category of quasi-coherent $\mathcal{O}_S$-algebras

I had asked something very similar before on math.se (deleted now) but unfortunately it hadn't received a lot of attention. I decided to re-ask here.
Let $S$ be a fixed scheme. Is the following true?
...

**5**

votes

**1**answer

256 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):
"...

**7**

votes

**1**answer

211 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 ...

**4**

votes

**0**answers

178 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
...

**20**

votes

**5**answers

3k views

### Reduced scheme and closed points

In The Geometry of Schemes by Eisenbud and Harris, Exercise I-32 asks one to show that a scheme $X$ is reduced if and only if every local ring $\mathcal{O}_{X,p}$ is reduced for closed points $p \in X$...

**0**

votes

**0**answers

71 views

### Do closed points have “locally maximal” codimensions?

Let $S$ be a Noetherian excellent irreducible scheme of finite Krull dimension; since the question is local it may also be assumed to be affine. Let $s$ be a closed point of $S$. My question is: does ...

**7**

votes

**0**answers

178 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 ...

**8**

votes

**0**answers

289 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 ...

**3**

votes

**0**answers

146 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 ...

**5**

votes

**1**answer

189 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 ...

**155**

votes

**14**answers

11k views

### What elementary problems can you solve with schemes?

I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary self-...

**1**

vote

**0**answers

218 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 $...

**8**

votes

**1**answer

233 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 \...

**2**

votes

**0**answers

105 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 $\...

**2**

votes

**1**answer

165 views

### Some questions about ruled surfaces defined over a number field

definitions:
A non-singular complex projective surface $S$ is a ruled surface if it is birationally equivalent to $C\times_{\text{Spec} \mathbb C}\mathbb P^1_{\mathbb C}$ where $C$ is a non-singular ...

**0**

votes

**0**answers

100 views

### Action on algebraic variety and adjoint bundles

Let $X$ be a complex algebraic variety and let $G$ be a complex algebraic group; I mean that $X$ is a reduced, separated scheme of finite type on $\operatorname{Spec}\mathbb{C}$, and the underlying ...

**2**

votes

**0**answers

90 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 ...

**3**

votes

**2**answers

393 views

### When is the determinant of the push-forward of an ample line bundle ample

Let $f:X\to S$ be a "nice" morphism of "nice" schemes. Let $L$ be an ample line bundle on $X$.
When is $\det f_\ast L$ also ample?
A "nice" morphism could be anything from "finite type separated" to ...

**1**

vote

**0**answers

122 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 ...

**11**

votes

**1**answer

363 views

### Obstructed automorphisms of schemes

Let $X$ be a smooth projective scheme over a field $\mathbf{k}$ of characteristic zero such that $\mathrm{H}^0(X, \mathrm{T}X)$ vanishes, and let $f$ be an automorphism of $X$. I would like to have an ...

**3**

votes

**0**answers

87 views

### extending local systems on a neighbourhood

Let $Y$ an affine finite type scheme over an algebraically closed field $k$.
Let $S$ be a closed subscheme of $Y$ and $Y'$ the henselization of $Y$ along $S$.
If we have a $\mathbb{Z}_{\ell}$ local ...

**1**

vote

**1**answer

161 views

### l-adic local system. on hensel schemes

Let $k$ be a field, $\ell$ a prime different from the characteristic.
If I take $S$ a closed subscheme of $Y$, which is a $k$-scheme of finite type, is it true that any $\mathbb{Z}_{\ell}$-local ...

**6**

votes

**0**answers

145 views

### Comparison of sheaves of modular forms

Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$:
$e^*\Omega^1_{E/X}$ and $\...

**3**

votes

**0**answers

89 views

### henselizations along closed subscheme

Where can I find some references about henselizations ablong a closed subscheme?
For example if I take a map $Y\times\mathbb{A}^{1}\rightarrow Y$ and $Z$ a closed subscheme.
Let $Y_{Z}^{h}$ the ...

**4**

votes

**1**answer

159 views

### infinitesimal lifting criterion for non-noetherian schemes

We have the "standard criterion" which says that a morphism $f:X\rightarrow Y$ is smooth if:
1/ $Y$ is locally noetherian.
2/ $f$ is locally of finite type and satisfies lifting criterion for ...

**5**

votes

**1**answer

261 views

### Mapping scheme from a proper variety

Let $X$ be a proper scheme over a field $k$. Let $T$ be a scheme over $k$. Is it true that morphisms $T \times X \to \mathbb{A}^1$ are in bijection with morphisms $T \to \Gamma (X, \mathcal{O}_X)$ (...

**2**

votes

**0**answers

149 views

### Description of the equalizer of $\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$

This is a crosspost of this MSE question.
I have asked several questions in an attmept to get a general version of the Chinese remainder theorem without conditions on the ideals which will ...

**16**

votes

**0**answers

388 views

### History of the functor of points

Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar.
However, in this note by Lawvere the author writes:
"I myself had learned the ...

**2**

votes

**0**answers

114 views

### Morphisms of locally ringed spaces into affine schemes

In Görtz and Wedhorn's Algebraic Geometry I, there's the following proposition:
Proposition 3.4. Let $(X,\mathcal O_X)$ be a locally ringed space. If $Y$ is an affine scheme then the natural map ...

**33**

votes

**1**answer

3k views

### Why “open immersion” rather than “open embedding”?

When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic ...

**3**

votes

**1**answer

245 views

### The SGA1 version of Riemann's Existence Theorem is about analytic spaces. How does one relate it to topological covering spaces?

In SGA1, Theoreme 5.1 (Riemann's Existence Theorem) states:
Let $X$ be a $\mathbb{C}$-scheme locally of finite type, $X^{\operatorname{an}}$ the associated analytical space. Then the functor which ...

**8**

votes

**0**answers

205 views

### Arithmetic zeta function and local zeta functions

For the arithmetic zeta function of (say) a nonsingular projective variety $X$, one has the following Euler product
\begin{equation}
\zeta_X(s) = \prod_{p\ \mbox{prime}}\zeta_{X\vert\mathbb{F}_p}(s),
...

**8**

votes

**1**answer

253 views

### Base schemes and Bayesian priors

One of Grothendieck's dicta about algebraic geometry is to consider "the relative situation", where one doesn't consider the category of schemes but of schemes over a fixed base scheme.
In Bayesian ...

**8**

votes

**2**answers

343 views

### Are there varieties with non finitely generated Picard group and vanishing irregularity?

Let $X$ be a smooth projective variety over an algebraically closed field $k$.
Can it happen that $q(X) := \dim H^1(X,\mathcal O_X) =0$ and $\textrm{Pic} \,X$ is not finitely generated?
Certainly, ...

**18**

votes

**1**answer

624 views

### Reference for de Rham cohomology in positive characteristic

It is known in characteristic $0$ that (algebraic) de Rham cohomology is a Weil cohomology theory. However, in characteristic $p > 0$ it isn't, if only because it has mod $p$ coefficients, whereas ...

**3**

votes

**1**answer

186 views

### Chevalley devissage

Let $G$ be an algebraic group over a perfect field $k$. Then it is know that it can be written as an extension of an affine algebraic group and a proper algebraic group.
Is there a similar result for ...

**20**

votes

**1**answer

524 views

### Why would one “attempt” to define points of a motive as $\operatorname{Ext}^1(\mathbb{Q}(0),M)$?

I'm a novice when it comes to motives. (I've read multiple introductory texts.)
I'm attempting to read Galois Theory and Diophantine geometry by Minhyong Kim. In it, he says that "One might attempt, ...

**0**

votes

**1**answer

489 views

### Artin approximation of a diagram

Let consider $f:(X,x)\to (Z,z)$ and $g:(Y,y)\to (Z,z)$ morphisms of pointed $k$-schemes of finite type ($k$ is a field). Suppose that there exists a map on the level of formal neighborhoods $\phi:X_{x}...

**7**

votes

**1**answer

230 views

### Higher-dimensional Artin L-functions

I begin by clarifying that the "higher-dimensional" in my question refers to analogues of Artin L-functions over higher dimensional base schemes than $\mathrm{Spec}(\mathbb{Z})$.
Now for the set-up. ...

**14**

votes

**2**answers

822 views

### Homotopy types of schemes

Let $X$ be a scheme over $\mathbb{C}$.
When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?
When does the topological ...

**2**

votes

**0**answers

180 views

### Blow up along a section of a smooth morphism

Let $C$ be a ground locally notherian and quasiprojective scheme. Let $\pi:S\to B$ be a $C$-morphism of finite type $C$-schemes. We call $\pi$ a $C$-smooth morphism if the morphisms $S\to C$ and $B\to ...

**8**

votes

**2**answers

304 views

### Parahoric Group Scheme

I am looking for the definition of a parahoric group scheme in the sense of Bruhat and Tits? I couldn't find a reference for that? at least a "clear" reference!
thanks

**7**

votes

**1**answer

337 views

### About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've asked this question http://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text on math.stackexchange , however I don't think I will receive ...