14 votes

Connеcted components of irreducible algebraic varieties

For a (projective) smooth real plane curve $C \subset \mathbb{RP}^2$ the answer is known. Such a curve is a compact smooth one-dimensional manifold without a boundary, so its connected components are ...
Francesco Polizzi's user avatar
7 votes

Frobenius and regular scheme

This is true Zariski-locally by Popescu's desingularisation theorem [Tag 07GC]. Indeed, any regular $\mathbf F_p$-algebra $A$ is geometrically regular [Tag 0381], so by Popescu's theorem it can be ...
R. van Dobben de Bruyn's user avatar
6 votes
Accepted

Criteria for when Gauss-Manin sheaves are vector bundles

If I am reading the assumptions correctly, this should follow from Theorem 6.4 in the paper Luc Illusie, Kazuya Kato and Chikara Nakayama Quasi-unipotent Logarithmic Riemann-Hilbert Correspondences J. ...
Piotr Achinger's user avatar
5 votes
Accepted

Isbell Duality and Dualizing Scheme Objects

Stone Spaces by Peter Johnstone (CUP 1982) is about just this subject. It is an exceptionally well written book. It would be better that you read it than that I make any attempt to summarise it for ...
Paul Taylor's user avatar
  • 7,986
4 votes
Accepted

Some questions on derived pull-back and push-forward functors of proper birational morphism of Noetherian quasi-separated schemes

Let $f \colon X \to Y$ be a morphism between locally ringed spaces. Then (1) Is $Rf_*$ a right-adjoint to $Lf^*$? Yes [L] Proposition (3.2.3), after Spaltenstein, 1988. (2) Is $Lf^* \mathcal O_Y\...
Leo Alonso's user avatar
  • 8,924
4 votes
Accepted

Deligne-Lustzig varieties locally closed schemes

Any set which is open in its topological closure can be endowed with a reduced scheme structure in the way you disguise. The set will also be the intersection of an open set and a closed set, so you ...
Will Sawin's user avatar
  • 135k
4 votes
Accepted

Self-intersection of zero section of line bundle over elliptic base curve

The result in question is a special case of the self-intersection formula (Fulton, intersection theory, p. 103) which states that the intersection of a smooth subvariety $X$ of a smooth variety $Y$ ...
Will Sawin's user avatar
  • 135k
3 votes
Accepted

Existence of a reduced fiber implies generically reduced (Exercise III-74 from Geometry of Schemes)

As noted by Jason Starr, the Generic Principle has been worked out in details by Grothendieck in EGA IV. If your French is not on top these days, the following version looks quite appealing (and can ...
Libli's user avatar
  • 7,210
3 votes
Accepted

Proper birational morphism from a Gorenstein normal scheme to a normal local domain, with trivial higher direct images, implies Cohen-Macaulay?

I started writing this last night, but didn't finish. In addition to Jason Starr's answer, you can use my new vanishing theorems to remove the assumption that $R$ is essentially of finite type over a ...
Takumi Murayama's user avatar
3 votes

Dimension of Zariski closure of a locally closed subscheme

Let $f: X \rightarrow Y$ be a proper dominant morphism between two locally Noetherian integral schemes. By Stacks Lemma 02JX, we have $\dim{X}=\dim{Y}+\delta$, where $\delta$ is the transcendence ...
Aphelli's user avatar
  • 330
3 votes
Accepted

Schemes with open generic point

Let's assume $X = \mathrm{Spec}(R)$ is affine. Since $X$ can be replaced by $X_{\mathrm{red}}$, let's also assume that $X$ is reduced. Then $R$ is an integral domain. The condition that $\{ \eta \}$ ...
3 votes

Find a non-quasi-compact scheme s.t. all finitely generated + globally generated quasi-coherent modules are finitely globally generated

Edit. The argument below applies only to Lindelöf schemes. Proposition. For every Lindelöf non-quasi-compact scheme $X$, there exists a quasi-coherent $\mathcal{O}_X$-module that is globally generated ...
3 votes
Accepted

Find stratification to decompose constructible sheaf to constant parts (example from Wikipedia)

I'm going to use $g$ for the equation of the curve since using $f$ for both the equation of the curve and the map to $\operatorname{Spec}(\mathbb C[s,t])$ leads to ambiguity. The situation just using ...
Will Sawin's user avatar
  • 135k
2 votes

Finite étale cover of factorial ring

Even the baby case can fail in positive characteristics, even with $C$ a field. For example, let $p>2$ be the characteristic of an algebraically closed $C$ and take $C[t][x]/(x^p+x+t^{p-1})$. In ...
Mohan's user avatar
  • 6,117
2 votes

Find a non-quasi-compact scheme s.t. all finitely generated + globally generated quasi-coherent modules are finitely globally generated

Warning. This does not answer the question. See the comments. Let $k$ be a field. For each $n\in\mathbb{N}$, let $X_n$ be a copy of $\mathrm{Spec}(k)$. Put $X:=\coprod_{n\in\mathbb{N}}X_n$. We view $\...
Laurent Moret-Bailly's user avatar
2 votes
Accepted

Dimension of Zariski closure of a locally closed subscheme

I'll interpret your terminology "Dedekind scheme" to mean "regular integral locally Noetherian scheme of dimension one" (or dimension $\leq 1$ if you replace $d + 1$ with $d + \...
351910953's user avatar
  • 261
2 votes
Accepted

Dimension of Zariski closure of a closed point of generic fiber

Probably the easiest way to prove this is via flatness. The closure $\bar{x}$ is integral and dominates $S$, thus is flat over $S$ (see Proposition III.9.7 in Hartshorne). The dimension of the fibres ...
Daniel Loughran's user avatar
2 votes

resolution property and perfect stacks

A quasi compact and quasi-separated scheme has its derived category of sheaves of modules with quasi-coherent cohomology generated by perfect complexes. This is actually a theorem of Bondal and Van de ...
Leo Alonso's user avatar
  • 8,924
2 votes

Base change for fundamental group prime to p in mixed characteristic?

Yes: this kind of question is discussed at length in SGA I, Exposé X. In particular, the specialization homomorphism $\pi_1(X_{\overline K},\overline{\eta})\to\pi_1(X_{\overline k},\overline s)$ is ...
Curious's user avatar
  • 301
2 votes

Proper birational morphism from a Gorenstein normal scheme to a normal local domain, with trivial higher direct images, implies Cohen-Macaulay?

Edit. Thanks to user @Johan for pointing out the indexing mistake. It is now corrected. I read more of Elkik's article. I am just expanding my comments into one answer. Edit. Assume that $R$ is ...
2 votes

How to compute the $G$-theory of the variety $\mathbb{P}^1\times\mathbb{P}^1$?

Edit: Here's another way that gets all the $G$-theory groups. Let $X_0=\mathbb{P}^1\times \mathbb{P}^1$ and $X_1=\mathbb{P}^1\times \infty$. Then $X_0\setminus X_1\cong \mathbb{P}^1\times \mathbb{A}^1$...
Eoin's user avatar
  • 342
1 vote

The weight of a weighted filtration is given (for large $m$) by a polynomial

I'm aware this is an old question, but I'm answering it for the benefit of anyone who comes across this question in the future. There are not one but two proofs of this result in the paper Uniform $K$-...
G Cooper's user avatar

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