25 votes

Can we state the Riemann Hypothesis part of the Weil conjectures directly in terms of the count of points?

(1) We have $$ N_n(X) = \sum_{k = 0}^{2d} (-1)^k \mathrm{tr}\left(\mathrm{Frob}^n \colon H^k(X) \to H^k(X) \right) = \sum_{k = 0}^{2d} (-1)^k \sum_{i=1}^{ h^k(X)} \lambda_{k,i}^n$$ where $\lambda_{k,...
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  • 115k
16 votes

What is the idea behind the proof of the Isogeny theorem and Theorem III.7.9 (Serre) in Silverman's book?

Here is a rough idea of the proof of 1. I'll highlight (with italics) concepts that you may not know now, but will be useful to learn in number theory, which hopefully will be motivated by this ...
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  • 115k
11 votes

Is the ideal product presheaf a sheaf? Do we have any reasons to believe it will be / it won't?

It need not be a sheaf. As an example, consider a space $X$ which is a disjoint union of open subspaces $X_n$, and pick $\mathcal O_X,\mathcal I,\mathcal J$ with the property that some element $c_n$ ...
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  • 23.3k
9 votes

Examples of five-adjoint systems

As noted by Simon Henry, the nLab gives many examples of adjoint chains. You can get an infinitely long adjoint chains from any ambidextrous adjunction. For an example, let $G$ be a finite group ...
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  • 19.4k
8 votes

Is the ideal product presheaf a sheaf? Do we have any reasons to believe it will be / it won't?

So here is a counterexample which is qcqs: Take $X$ the affine line with double origin $a_1$ and $a_2$, then take $I_1$ and $I_2$ the ideal of functions vanishing each at one of the origins ...
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6 votes

Manifolds whose tangent spaces have a special behavior

The answer to the first question is No. The assumption you made is equivalent to stating that for every $q\in M$ that the vector $q\in T_qM$. This is satisfied whenever $M$ is a portion of a cone, ...
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6 votes

Examples of five-adjoint systems

You can build long chains of adjoints by taking functor categories and using Kan extensions. I will give an example. Write $\underline{n}$ for the set $\{1, \ldots, n\}$ considered as a discrete ...
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6 votes

Examples of five-adjoint systems

Edit : there is actualy an nLab page listing exemple of long strings of adjunction. Maybe not what you are after, but there are examples of functors that are both left and right adjoint to each other, ...
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  • 32.2k
5 votes

Functoriality conjectures on the slice filtration

This is also investigated in Shane Kelly's thesis, see Section 4.2.
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  • 3,525
4 votes

Motive of CM elliptic curve and modular forms

Since this question has come alive again, let me point out that the Hecke operators cannot give a splitting of $h^1(E)$ into two pieces over $F$, since the Hecke correspondences on a modular curve are ...
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4 votes

Which varieties are sums of tensor powers of the Lefschetz motive?

One class of examples is already indicated in the comments, and the question itself. I thought it would be good to include this in an official answer. Proposition. Let $X$ be smooth projective ...
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3 votes
Accepted

Invariants of general linear groups under torus action

Call the two matrices ${}_1A$ and ${}_2A$. Your ring can be expressed by taking the $T$-invariants of the free ring in $2n^2$ variables $\mathbb C[{}_kA_{ij}]_{1\leq i,j\leq n, 1\leq k \leq 2}$ and ...
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  • 115k
3 votes

Lie algebroid in algebraic geometry

I suggest having a look at Beĭlinson, A.; Bernstein, J. A proof of Jantzen conjectures. MR: Matches for: MR=1237825 §1.2 . https://people.math.harvard.edu/~gaitsgde/grad_2009/BB%20-%20Jantzen.pdf
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  • 3,733
3 votes

Deformation of (locally) ringed spaces and of their abelian categories of modules

The answer to your second question is "no", I think. Let's assume that sufficiently nice means that it is a smooth algebraic variety over a field of characteristic $0$. Then, as written in ...
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  • 7,513
3 votes
Accepted

On the Artin-Rees Lemma for non-commutative rings

There is some discussion of this in Rowen's "Ring Theory", volume I, Section 3.5, with additional references therein. Exercise 19 on p. 462 in op. cit. states that a polycentral ideal $I$ of ...
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  • 2,624
2 votes

When is a topological fiber bundle Zariski locally trivial?

Since you asked for "comments, suggestions" here are two thoughts off the top of my head: In Torsion Homologique et Sections Rationnelles by Grothendieck (Séminaire Chevalley, 1958), the ...
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  • 8,022
2 votes
Accepted

Concrete descriptions of $S^1$-bundles over smooth manifold $Y$ underying a K3 surface

You can say a fair amount about the topology of the total spaces of the different bundles, although I suspect none of them is a particularly well-known manifold that has a `name'. (Except of course ...
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2 votes
Accepted

Tri-homogenous polynomials of tridegree $(3,3,3)$ to add three points on an elliptic curve

The divisor $D$ is rationally equivalent to an effective divisor, hence $H^0(E^3,D) \neq 0$. To see this, let $p_0,p_1,p_2 \colon E^3 \to E$ be the canonical projections. For $i,j \in \{0,1,2\}$, ...
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2 votes

Gluing along closed subschemes

To complement Laurent Moret-Bailly and Karl Schwede's answers (10 years ago!), the pushout of a closed immersion $Z\to X$ along an affine morphism $Z\to Y$ always exists in the category of algebraic ...
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  • 4,639
2 votes
Accepted

Expressing a vector valued function in terms of its derivatives

$\newcommand{\pa}{\partial}\newcommand{\R}{\mathbb R}$The answer is no. Indeed, suppose the contrary: that for each polynomial $f$ there are functions $a_j,b,c_j$ such that \begin{equation} f(x_1,\...
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1 vote

On the definition of the Cherednik algebra of a variety with a finite group action

Questions 2. and 3. are correct. By Cartan's Lemma $Y$ is smooth. For a point $p\in Y$ one has $$T_{p}Y=(T_{p}X)^{g}\text{.}$$ If $1-\lambda_{Y,g}(p)=0$ then the action of $g$ on $T_{p}X$ is trivial. ...
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  • 261
1 vote

Singularities of arithmetic surface

There seems to be some confusion concerning "regularity" and "smoothness". First of all, if $X$ is a noetherian integral scheme and $x\in X$, then $X$ is regular at $x$ if $$\...
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1 vote

Motive of CM elliptic curve and modular forms

Question 1: The field generated by the Fourier coefficients of an elliptic curve associated to a modular form is $\mathbb Q$. (For example, since the Fourier coefficients can be calculated by counting ...
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