Algebraic Geometry – A close subset of ideals on affine open subsets

Let X be a schema and for each open affine $$U subset X$$ let $$I (U)$$ to be an ideal in $$mathscr O_X (U)$$ as for any identification $$U = spec ~ R$$ we have $$I (U) _f simeq I (D (f))$$ for all $$f$$ in R under the natural map. I am supposed to show that these data determine a closed subsystem Y of X.

It's easy to see that it determines a closed subset Y of X, and we want to clearly define $$mathscr O_Y | _ {U cap Y}: = spec ( mathscr O_X (U) / I (U))$$ whenever U is an affine open subset of X. But I find it difficult to stick them together. What is the simplest (low-tech) way to build the wreath on Y?

Algebraic Theory of Numbers – How Can We Justify the Use of Example 5.4 (from Cohen, Lenstra) Assuming Their Heuristics

These days, I study Cohen-Lenstra's heuristics to understand René Schoof's paper titled "Number of Classes of Real Cyclotomic Fields of First Driver".

On page 932 of Schoof's paper, there is a sentence "According to Cohen-Lenstra, the probability that M does not happen" $$mathbb {Z} ( zeta_ {d})$$-module modulo a random principal ideal "is equal to $$prod_ {2 leq k} (1-q ^ {- k})$$. "

First of all, I do not understand the exact meaning of the sentence. I hope so that someone can explain it to me.

I checked the example 5.4 by myself and tried to understand the sentence as "The probability that $$mathbb {Z} ( zeta_ {d})$$-module to trivial $$rho$$-composing ($$rho$$ is the primordial ideal of $$mathbb {Z} ( zeta_ {d})$$ correspond to $$M$$) is $$eta _ { infty} ( rho) / eta_ {1} ( rho)$$. "

Even if this interpretation is true, I do not know how to deduce it from the fundamental hypothesis 8.1 of the Cohen-Lenstra document. $$d$$ is not necessarily a bonus, so there might not be an abelian group $$Gamma$$ with $$A _ { Gamma}$$ isomorphic to $$mathbb {Z} ( zeta_ {d})$$.

In the text of Schoof, there is a line "The heuristics of Cohen-Lenstra do not really apply to our situation". I hope someone can ask my question and explain the meaning of Schoof's paper.

Thank you very much!

References for the quotient fppf of algebraic groups on a field

I know that for algebraic groups (= smooth group diagram connected to a field) $$H subseteq G$$, the fppf sheafification of $$G / H$$ is a ploy.
Is there a short proof of that?
(If possible, I want one available online.)

In Milne's note, the author only shows it for an almost projective rendering. $$G$$.
(And he works on a small general situation.
So I wish we could prove more easily the existence of a quotient, considering only the quotient of a group by a subgroup …)

But I want to know the general case.

In Conrad's "Modern proof of Chevalley …", the author shows that each algebraic group is quasi-projective.
But in his proof, it seems to me that he uses the existence of quotients for general algebraic groups.

Is there any easy proof of the existence of the quotient of general algebraic groups?

Thank you very much!

Theory nt.number – Implementation of zeta functions of algebraic varieties in SAGE

I'm pretty new to wise, I was studying the zeta functions of hypersurfaces on finite bodies and I do not know how to calculate them in Sage. Are there packages that do most of the work, or maybe similar work that I could consult to get ideas?

group theory – Algebraic criteria for the moment when a \$ gamma: I to Sigma_g \$ curve on a surface is simply closed.

Given a surface $$Sigma_g$$, I wonder about a criterion of when a class of homotopy $$( gamma) in pi_1 ( Sigma_g)$$ has a representative simply closed.

Said differently, given a surface group $$G$$is there a criterion for the moment when an element of the group has a simply closed representative considered as a curve in $$Sigma$$ so that $$pi_1 ( Sigma) = G$$?

When $$g = 1$$, the answer is that $$pi_1 ( Sigma_1) = mathbf Z oplus mathbf Z$$, and a necessary and sufficient condition is that, when they are written in terms of standard generators, we have this $$(a, b) in = mathbf Z oplus mathbf Z$$ are whole coprimes. Is there anything analogous for higher gender surfaces?

at.algebraic topology – Algebraic curve intersecting a square grid

Subdivide the square unit into square mesh cells
with the side $$w$$. That will give us pretty much $$w ^ 2$$ cells.

Officially
$$g_ {ij} = (wi, wj) + (w) ^ 2,$$
for $$i, j = 0, ldots, 1 / w -1$$.
Now consider an algebraic curve $$c$$, described by $$p (x, y) = 0$$ degree
at most $$Delta$$.

How many cells in the grid $$p$$ to cross at most according to $$w$$ and $$deg$$?

We can generalize this question to the dimension $$d$$ easily.

How many cells in the grid $$p$$ to cross at the most in function
of $$w$$, $$d$$ and $$Delta$$?

algebraic topology – Is the volume of relative cycles at least the systole of the variety?

Let $$M$$ to be a variety with limit $$partial M$$. Assume that $$M$$ has a structure for which a notion of volume for strings can be defined. For example, if $$M$$ is triangulated, then the volume of a simplicial string is simply the number of cells with nonzero coefficients in the formal sum (let's work on the field of 2 elements for simplicity). A similar notion can be defined if $$M$$ has a Riemannian metric instead.

Let $$p: C_i (M) rightarrow C_i (M, partial M)$$ to be the canonical card of $$i$$-chains to parent $$i$$-chains, and set for everything $$c in C_i (M, partial M)$$:
$$flight (c): = inf {flight (c) mid p (c) = c }$$

Clearly, $$p$$ can not increase the volume. However:

Is it true that the volume of any non-contractible parent $$i$$-cycle $$c in Im (p)$$ Is at least $$sys_i (M)$$?

Right here $$sys_i (M)$$to know the $$i$$-systole of $$M$$, is by definition the infimal volume of a non-contractible $$i$$ride a bike $$M$$.

The search for channel-level diagrams (not just homology) was the main type of argument I tried to produce to prove such an assertion. But I also suspect that I miss a counter-example not so difficult.

Algebraic Geometry – Degree Limits on Point Coordinates of a Zero-dimensional Variety

Let $$S = {f_1, points, f_s in mathbb {Q} (x_1, points, x_n) }$$ have a zero dimension nullset $$V subset mathbb {C} ^ n$$and suppose that everyone $$f_i$$ at most total degree $$d$$.

Is there a shared root $$( alpha_1, dots, alpha_n) in V$$ such as some $$alpha_i$$ has a degree at most $$dn$$ as an algebraic integer (i.e. $$( mathbb {Q} ( alpha_i): mathbb {Q}) leq dn$$)?

This is vaguely in the sense of this question but concerns the "minimal" coordinate rather than the degree of extension required to contain all the coordinates. I know that there is a limit of $$d ^ n$$ (from the link, or via the inequality of Bezout followed by Shamma Lemma) but I wonder if we can say more after trying some examples and found nothing worse than $$dn$$.

Why is the algebraic semantics of programming languages ​​extinct and not used today?

Algebraic semantics is a type of semantics that uses algebraic expressions to relate the formal descriptions of the initial and final states of an operation defined by the programming language. I can not imagine a more general and versatile semantics, but it is not used in current research. Why is that?

Traditionally, symbolic methods were considered quite expensive, but today's mathematical software, proof assistants, and other automations of mathematical and algebraic manipulations – all of these provide the automation of algebraic semantics. But still – algebraic semantics has lost its favor.

Algebraic Number Theory – Calculation of the Tate Epsilon Factor in the Branched Case

Let $$F$$ to be a non-archimedean local field, $$chi$$ a ramified figure of $$F ^ { ast}$$, $$psi$$ a non-trivial nature of $$F$$, and $$dx$$ a measure of Haar on $$F$$ regarding the Fourier transformation. The local factor of the Tate epsilon $$epsilon ( chi, psi, dx)$$ is defined via the local functional equation
$$epsilon ( chi, psi, dx) int limits_ {F ^ { ast}} f (x) chi (x) d { ast} x = int limits_ {F ^ { ast}} hat {f} (x) chi ^ {- 1} (x) | x | d ^ { ast} x$$
for all the appropriate functions $$f in C_c (F)$$. taking $$f$$ to be a characteristic function of an appropriate open compact subgroup of $$F$$, we can realize $$epsilon ( chi, psi, dx)$$ as "main value integral"

$$epsilon ( chi, psi, dx) = int limits_ {F} chi ^ {- 1} (x) overline { psi (x)} d { ast} x tag { 1}$$
in that the right side becomes constant and equal to the left side when it is evaluated on $$mathfrak p_F ^ {- k}$$ for large enough $$k$$.

Let $$f$$ and $$d$$ to be the drivers of $$chi$$ and $$psi$$. The Tate article in Corvallis II describes $$epsilon ( chi, psi, dx)$$ more specifically as the right side of (1) where the integral is taken only on the ring of $$x in F ^ { ast}$$ with $$operatorname {ord} (x) = -d-f$$.

Right here $$operatorname {ord} (c) = d + f$$.

My question is, what can we say about the summands

$$int limit _ { pi ^ n mathcal O ^ { ast}} chi ^ {- 1} (x) psi (x) dx$$

Are they all equal to zero except for $$n = -d-f$$? In Tate's thesis, he shows that they are zero for $$n> -d-f$$. I wondered if those for $$n <-d-f$$ are individually equal to zero, or if they simply cancel each other in the long run.