## Ag. Algebraic Geometry – Are smooth, coherent sheaves of rank 1 always identical to line bundles?

Accept $$mathcal {L}$$ over $$mathbb {CP} ^ 2$$ is given by the following short exact sequence,

$$0 rightarrow mathcal {L} rightarrow mathcal {O} (4) oplus mathcal {O} (2) oplus mathcal {O} (8) oplus mathcal {O} (1) oplus mathcal {O} (1) rightarrow mathcal {O} (14) oplus mathcal {O} (8) oplus mathcal {O} (5) oplus mathcal {O} (2) rightarrow 0.$$

That's easy to check $$mathcal {E} xt ^ i ( mathcal {L}, mathcal {O}) = 0$$ to the $$i 1$$, So $$mathcal {L}$$ is a rank one coherent coherent sheaf. So I expect $$mathcal {L}$$ should be a line bundle, but $$Ch_2 ( mathcal {L}) ne frac {1} {2} c_1 ( mathcal {L})$$in contrast to what happens with bundles of cables!

I do not know what's going on here …

## stacks – Requirements for understanding the algebraic geometry of "algebraic tannins"

I'm trying to learn something about the algebraic geometry of Gerbes.

I am familiar with the construction of Gerbes in differential geometry. Although there is some similarity between differentiable tanning and tanning as mentioned above, they are not quite the same.

What are the prerequisites for getting to know Gerbs in differential geometry? I know the term stack. Which other Grothendieck topologies and stacks of categories with these Grothendieck topologies should be familiar to me to understand and use the above-mentioned concept of tanning.

## Algebraic Geometry – Torsional modulus has finite length (by Liu)

I have a question about an argument in the proof of Theorem 7.1.38 from Liu's "Algebraic Geometry and Arithmetic Curves" (page 263):

We have two noether rings $$A, B = O_ {X} (X)$$ with break fields $$K (Y) = Frac (A), K (X) = Frac (B)$$,

Since $$(K (X): K (Y)) = n$$ Finally $$K (X)$$ is a finite-dimensional vector space over $$k (Y)$$ generated by $$b_1, …, b_n in B$$,

That's why $$K (X) = langle b_1, …, b_n rangle_ {K (Y)}$$,

Let's say $$M: = sum_ {1 / lei le n} Ab_i$$

By the construction $$B / M$$ is a torsion $$A$$-Module.

My QUESTION is why that implies that $$B / M$$ Has finite length as $$A$$-Module?

## Ag. Algebraic Geometry – Bertini's Theorem for Singular Varieties

I have extended the Bertini Theorem (Hartshorne II.8.18) for singular projective varieties to show this, this or that.
(I think they use Bertini for a singular variety, but Hartshorne shows that only for a not singular variety.)

To let $$k$$ be an arbitrary algebraically closed field and $$X subseteq mathbb {P} ^ N$$ a closed normal subvariety of dimension $$d$$, (possibly singular)
(To use liner systems, I think we need $$X$$ normal.)
Then we will show it $$mathfrak {d} = {H subseteq mathbb {P}: text {a hyperplane with} X not subseteq H text {and} X cap H text {is regular} }$$ is open in $$| H |$$, a complete linear system of a hyperplane.

Almost all parts are identical to those of Harthorne.
For a closed point $$x in X$$, To let $$B_x = {H | X subseteq H text {or} (X not subseteq H text {and} x in X cap H text {and} X cap H text {is unique to} x) }$$,
By the following lemma, if $$x$$ is a singular point, $$B_x = {x in H }$$,

Let A be a singular noetheric local domain of dimension $$n$$. $$f in A$$,
Then $$A / f$$ is also unique.

(The proof is very simple, assuming that $$dim A / f = n -1$$ in this case )

So by the proof of Bertini in Hartshorne, $$dim B_x = N – d – 1$$ if $$x$$ is a not singular point and is $$= N – 1$$ if $$x$$ is a singular point.

Finally, look at the subset $$B = {(x, H) in X times | H | : H in B_x }$$,
This is closed $$X times | H |$$, (I can not show that …)
So, consider it as a closed subschema.
Then the first projection $$h: B to X$$ is surjective.
So after the general proposition about morphisms and fibers for every closed point $$b in B$$ and his picture $$x = h (b)$$. $$dim mathscr {O} _ {h ^ {- 1} (x), b} ge dim mathscr {O} _ {B, b} – dim mathscr {O} _ {X, x }$$,
Now for $$X$$ is a strain over a field, $$dim mathscr {O} _ {X, x} = d$$, and since $$dim mathscr {O} _ {h ^ {- 1} (x), b} le dim h ^ {- 1} (x)$$, it is $$le N -d – 1$$,
That's why $$dim B le N – 1$$,
So under the second projection $$p: B to | H |$$the picture is a correct subset.
Since $$p$$ is a projective morphism, this picture is closed.
Because this picture is accurate $$mathfrak {d}$$we have made the desired set.

That's the way it is $$B$$ closed?

Any help is greatly appreciated !!

## Ag. Algebraic Geometry – Questioning an Implication of Thomason's étale-Descent Spectral Sequence

On page 5 of this essay by Dwyer and Mitchell it is said that Thomason's étale spectral sequence stems from his essay Algebraic K theory and étale cohomologythat reads

$$H ^ p _ { acute {e} t} (X, mathbb {Z} _l (-q / 2)) Rightarrow pi _ {- q-p} has {L} KX$$

from where $$KX$$ is the algebraic K-theory spectrum of $$X$$ and $$has {L}$$ refers to the $$ell$$-finished Bousefield localization on the topological K-theory implies the natural isomorphism

$$pi_i widehat {L} (KR) cong pi_i text {card} _ { Gamma_F & # 39;} (X _ + ^ theta, has { mathcal {K}})$$

where here, $$R$$ is the ring of integers located at $$ell$$ from a completely real field $$F$$. $$X$$ is a space that realizes the étale homotopy type of $$R$$. $$theta$$ is the character of $$pi_1 (X)$$ According to the $$ell$$-adic cyclotomic character, $$X ^ theta$$ is the cover of $$X$$ according to the kernel of $$theta$$with the corresponding effect of $$theta$$ the Galois / Basic Group. The subscript plus characterizes the unreduced suspension spectrum as usual. The Galois group of $$ell$$-adic cyclotomic extension is particular $$Gamma_F # ;$$and lives inside $$Gamma # cong mathbb {Z} _l ^ times$$ (i.e., the corresponding Galois group over $$mathbb {Q}$$.)

The $$text {Map}$$ is then an equivocal imaging spectrum with the effect of $$Gamma & # 39;$$ about Adam's operations on the target.

(It is believed that we have a rigid effect, that is, not just to homotopy $$Gamma & # 39;$$ on $$has { mathcal {K}}$$.)

So, How does that work?? There is no real explanation why I think it has to be easy. But I am confused as to what the relationship of the localized algebraic k-theory groups to étale-cohomology has to do with this twisted equivariant mapping spectrum.

Sorry, if this is something very formal; I'm pretty inexperienced in these things. I will delete this question if it turns out that it is somehow very obvious.

## Ag. Algebraic Geometry – Slope Filtration vs. Hodge filtration

To let $$X$$ Be a smooth projective change $$mathbb {Z} _p$$With $$X_0$$ his reduction mod $$p$$,

For every elevator $$X & # 39;$$ over $$mathbb {Z} _p$$ from $$X_0$$Is there a canonical isomorphism?
$$H ^ i_ text {crisp} (X_0 / mathbb {Z} _p) cong H ^ i_ text {dR} (X & # 39; / mathbb {Z} _p)$$

The left side of this isomorphism has slope filtering, while the right side has Hodge filtering, but I do not think that this isomorphism is generally filtered.

Is there an example for $$X_0$$ as above so that no $$X & # 39;$$ can be chosen to filter up the isomorphism?

## Algebraic Geometry – Macaulay2: How is the remainder calculated when a polynomial is divided by a set of polynomials (in a particular order)?

I write Buchberger's criterion into a program in Macaulay2 to see if the set of polynomials I have form a Grobner basis for the ideal that produces them. However, I have not been able to find a method that provides me with the rest, for example, if there is a polynomial $$f$$is shared by a lot of polynomials $$G = {g_1, g_2, …, g_t}$$ in a certain order. Would anyone know if such a method exists and if so, what is its name?

Although that's not all, here is the part of the program in which I try to implement the Buchberger criterion:

n=0;
for i to #polynomials-2 do
(
for j from i+1 to #polynomials-1 do
(
remainder := Spair%polynomials;
if remainder == 0 then n=n+1;
);
);
if n == binomial(#polynomials, 2) then print "The polynomials form a Grobner basiss for the ideal it generates." else print "The polynomials don't form a Grobner basis for the ideal it generates."


## Agal Algebraic Geometry – Relative Picard Scheme, Relative Cotangent Sheaf

To let $$X to A$$ Be a gentle, correct morphism of the schemes and let $$Omega_ {X / A} ^ 1$$ denote the relative cotangent bundle of the entire space $$X$$ as a scheme over $$A$$where we accept that $$A$$ is a one-point space.
Accept $$mathcal {L} subseteq Omega_ {X / A} ^ 1$$ is a reversible undergarve of the relative cotane sheaf.

is $$mathcal {L} in operatorname {Pic} _ {X / A} (A)$$?

In other words, the line is subbundle $$mathcal {L}$$ an element of $$A$$Points of the relative Picard scheme of $$X$$ as a scheme over $$A$$,

Suppose also that $$operatorname {pic} _ {x / a} (a)$$ is isomorphic to the Picard group of fiber $$X times_A k$$,

Would this isomorphism mean that every line bundle of $$operatorname {pic} _ {x / a} (a)$$ is a trivial extension of a corresponding trunk group in $$operatorname {pic} (X times k)$$?

## Agal Algebraic Geometry – Flat Maps and Zariski Tangent Spaces

To let $$f: A to B$$ Let be a finite flat local homomorphism of noetheric local rings.

Are there any nice conditions? $$A$$ and $$B$$ The guarantee that the dimension
of the Zariski – Tangensraum of $$A$$ (at its maximum ideal) is smaller or
equal to the dimension of the Zariski tangent space of $$B$$ (at the maximum ideal)?

For example when $$B$$ is then regular $$A$$ is normal, so the above would apply, but I do not want to assume that something is so strong.

I do not know any example where it fails, but I'm particularly interested in the event that $$A$$ and $$B$$ Both are lci and artin.

## ag.algebraic geometry – reference request: Oldest books on algebraic curves with unresolved problems?

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