User ivan - MathOverflow

Answer by Ivan for Wild Ramification

2013-05-02T12:54:55Z

There is the following work of Kerz and Saito:

In particular, one can view their Coro. II as a higher-dimensional analogue of the existence theorem, which encodes p-extensions (see Artin-Schreier-Witt theory), and hence wild ramification for varieties.

Voevodsky's proof in any characteristic (for motivic and Chow)

2013-02-19T13:27:01Z

Regarding the published version of "Motivic cohomology groups are isomorphic to higher Chow groups in any characterstic" (IMRN) available at:

http://imrn.oxfordjournals.org/content/2002/7/351.full.pdf

I would appreciate any elucidation on the following two points:

(1) a short assertion is made on the second page whose proof is not entirely clear to me

and

(2) how this assertion factors into the proof of Theorem 1.

For (1) the short assertion is (from page 2, second paragraph from top):

For any smooth scheme $X$ (of finite type) over a field $k$, $x$ a point of $X$ and $k_0$ the subfield of constants of $k$, there exists a smooth variety $X_0$ over $k_0$ and a point $x_0$ on $X_0$ such that the local rings $O_{X, x}$ and $O_{X_0, x_0}$ are isomorphic.

As noted, the dimension of $X_0$ may be larger than that of $X$. Here is an example I worked out that I think illustrates the principle of the proof:

Example. Let $k = \mathbb{F}_p(t), X = \mathbb{A}^1_k$ with parameter $T$, and let $x$ correspond to $(T-0)$. Then $k_0 = \mathbb{F}_p$ and put $X_0 = \mathbb{A}^2_{k_0}$ with parameters $u, v$. Then via $u\mapsto T, v\mapsto t$, and $x_0$ corresponding to $(u-0)$, we have the desired isomorphism of local rings. Note that the residue fields of the local rings are $\mathbb{F}_p(t)$ and $\mathbb{F}_p(v)$, respectively.

In general (assuming $k$ is absolutely finitely generated, i.e. is finitely generated over its prime field): since the question is local, we assume $X$ affine: $X = Spec(A)$ with $A = k[T_1, \ldots, T_n]/I$. Then take a transcendence basis $t_1, \ldots, t_r$ of $k$ over $k_0$. Then it seems that $X_0$ is built from $T_1, \ldots, T_n, t_1, \ldots, t_r$, but I haven't worked this out in general.

Problem: what if $k$ does not have finite transcendence degree over $k_0$? In the absolutely finitely generated case, if $F$ is the prime field of $k$, then $k_0$ is a finite extension of $F$ and $tr.deg_F(k) = tr.deg_{k_0}(k)$.

As for (2), the part that is not clear to me is when one passes from $X$ to $X_0$ in the course of the proof of the above cited paper (for example, in Prop 4).

NB: the above cited assertion is not included in the preprint http://www.math.uiuc.edu/K-theory/378/allagree.pdf

Cf. the sentence directly after Corollary 2 on page 1 of this preprint

Answer by Ivan for normal crossing divisor v.s. strict normal crossing divisor

2012-07-05T23:38:33Z

Here is the definition of an sncd (and ncd) from SGA (SGA 5, 3.1.5, pg 24):

Let $X$ be a regular scheme and $D$ an effective divisor on $X$.

DEF: $D$ is an sncd on $X$ if there is a finite family of sections $(f_i)_{i\in I}$, $f_i\in O_X(X)$ such that the following two conditions hold:

i) $D = \sum_{i\in I}$ $div(f_i)$

ii) for each $x\in Supp(D)$, the local restrictions $(f_i)_x$ that satisfy $(f_i)_x \in m_x$ (i.e. those that land in the maximal ideal of $O_{X,x}$) have the property that they form a part of a regular system of parameters for the local ring $O_{X,x}$.

DEF: an effective divisor $D$ is an ncd if (i) and (ii) hold 'etale locally.

This definition is unambiguous (unlike Hartshorne, Wikipedia, etc) since, e.g., regularity is not a relative condition (whereas smoothness is a relative condition). Further, you can check that two smooth hyperplanes meeting transversally in an affine neighborhood, e.g. the x-y axis in $A^3_k$ for $k$ a field, form an sncd.

Example / Remark : in SGA 5 they write global sections for an sncd. So here is an example of an ncd that is not an sncd: let $X = \mathbb{P}^1_k$ be the projective line over a field $k$ and consider an affine open $U = \mathbb{A}^1_k = Spec \; k[t]$. Now, the divisor defined by $t$ is not an sncd on $X$ as $t\not\in O_X(X) = k$ but since $U\hookrightarrow X$ is etale, then this divisor is an ncd on $X$.

Perhaps it's a matter of taste whether one defines sncd globally as in SGA 5 or Zariski locally...

(N.B. in EGA IV you can find the definitions of div and Supp)

Definition of Chow groups over Spec Z

2010-03-09T18:13:02Z

Usually (eg, intro. in M. Rost's 'Cycle modules with coefficients'), for a variety, $X$, over a field one can define the Chow group of p-cycles, $CH_p (X)$, as $$CH_p (X) = coker\; \left[\bigoplus_{x\in X_{p+1}} k(x)^\times \rightarrow \bigoplus_{x\in X_p} \mathbb{Z}\; \right]$$.

What about for an arithmetic scheme, eg when $X$ is, say, normal, separated, of finite type and flat over $Spec \; \mathbb{Z} $? Does something go wrong with the above definition?

Peter Arndt had posed part of this question already, but it seems without an answer.

On semi-local schemes

2010-05-31T11:03:08Z

Background I have searched a bit for the definition/constructions on how to "semi-localize" a scheme, but have been unsuccessful in finding a good reference; I apologize in advance if this topic has been covered in detail elsewhere (e.g. in a book or article) and would be happy for a reference!

This question arose from a problem I had been working on in finding an étale morphism into affine space.

Much of the terminology here will be from EGA I.

(Aside: The constructions below take place in the Zariski site but I think some of them go through in the étale/Nisnevich site)

Definitions A local scheme is the spectrum of a local ring and a semi-local scheme the spectrum of a semilocal ring. In the constructions below a ring $O_{X,C}$ is given, so then the candidate semi-local scheme is $Spec \; O_{X,C}$.

NB: for some reason I was having trouble with "\varinjlim" here, so I'm using "lim" below to mean direct limit i.e. colimit.

Question

Given a scheme $X$ we can localize $X$ at a point $x\in X$ by taking $$O_{X,x} := \lim_{U\ni x} O_X(U). $$

Suppose now that we are given a finite set of closed points $x_1,\ldots, x_n \in X$. Let $C :=$ {$x_1,\ldots, x_n $}. How can we 'localize' $X$ around $C$?

There are at least three ways I know how to do this procedure and would be happy to hear about other methods as well as comments (especially geometric ones) regarding the following constructions:

$1.$ Define $$O_{X,C} : = \lim_{U\supset C} O_X(U).$$

This construction is similar to the localization construction above in that we take opens $U$ of $X$ containing $C$ and then take the direct limit; the case $n=1, C = {x_1}$ is then a special case. NB: we can 'see' this direct limit in the sense that for each $x_i$ we find an open $U_i\ni x_i$, then taking the (finite!) union of the $U_i$ we obtain an open $U$ containing $C$. Just as in the local case above, this direct limit is filtered by inclusion.

$2.$ Further assume now that $X$ is locally noetherian and regular. Let $A_i: = O_{X,x_i}$ and then define $$O_{X,C}:= \prod_i A_i .$$

Using the hypothesis that $X$ is regular, we can argue that the maximal ideals here correpsond to the $x_i$: the maximal ideals in $\prod_i A_i$ are generated by elements of the form $(1,1,\ldots, b_{ij},1,\ldots, 1)$ where the $b_{ij}$ generate $x_i$ (here is where we are using the two added hypothesis), i.e. that $(b_{ij})_{1\leq j\leq n_i} = m_i$ where $m_i$ is the max ideal corresponding to $x_i$ and $n_i = dim O_{X,x_i}$.

This construction is more ad-hoc (I think) vs. 1. Moreover, the geometry here is slightly more explicit in that this $Spec \; O_{X,C}$ is a finite disjoint union of local schemes, whereas in case 1, the topology is less disjoint when looking at neighborhoods of the $x_i$.

$3.$ With $X$ any scheme (no additional hypothesis as in 2), let $F_i : = O_{X,x_i}/m_i$ where $m_i$ is the maximal ideal corresponding to the closed point $x_i$. Define: $$O_{X,C}: = \prod_i F_i .$$ This construction is the most disjoint of the three in that the spectrum is now we have a finite coproduct of ''points''.

Closing remarks Presently, for me the most useful of the three is 1 and I would appreciate feedback on where the process of semi-localization has been defined. A professor that I admire very much once said (during a lecture) "from now on and for the rest of your life, every time you see something in commutative algebra, try to relate it to geometry, and vice versa" (I'm paraphrasing).

Answer by Ivan for spectral sequences in number theory

2010-05-28T16:30:40Z

I posit the following example, in response to your ambiguous question:

The coniveau spectral sequence seems to play an important role in 'arithmetic geometry'. One instance is in class field theory for schemes:

From W. Raskind's nice survery article "Abelian class field theory of arithmetic schemes" [AMS, 1992, pgs. 100-101]:

Let $X$ be an arithmetic scheme, $n>0$ invertible on $X$. Then there is a coniveau spectral sequence (in the etale site):

$$E^{p,q}_1 = \bigoplus _{ x\in X^{p} } H^{q-p} (k(x), \;\mathbb{Z}/n \; (j-p)) \Rightarrow H^{p+q} (X, \mathbb{Z}/n \;(j)) $$

Without going into more details, this sequence plays an important role in defining a reciprocity map from a class group of $X$ to abelian fundamental groups.

That's all I will say for now in hopes that the above provides for motivation to delve further into studying coniveau, etc.

Finally, one of the best articles I have seen on coniveau is by Colliot-Thélène, Hoobler, and Kahn, "The Bloch-Ogus-Gabber theorem" which can be found at: http://www.math.jussieu.fr/~kahn/preprints/prep.html

It might be nice to have others' remarks/comments on coniveau, but I don't have any precise questions yet.

Answer by Ivan for Etale coverings of certain open subschemes in Spec O_K

2010-05-01T20:45:39Z

@Ariyan and BCnrd: Cf. Exercise 1.9, Section 4.1 of Q. Liu's wonderfully written book, "Algebraic Geometry and Arithmetic Curves". @Ariyan: Exercise: re-write the one-dimensional (i.e. classical) idele class group as a certain complement of $Spec\mathcal{O}_K$...cf. the intro to "Global class field theory" by Kato-Saito.

When does a projective morphism give an etale morphism (into affine space)? (Finite field) (normalization)

2010-02-25T00:07:39Z

Does anyone know of generalizations on what Mumford (Red Book) calls "uniformizing parameters"? For example, given a regular quasi-projective scheme over a finite field $\mathbb{F}$, is there an etale morphism into affine space over $\mathbb{F}$?.

Answer by Ivan for Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$?

2010-02-25T01:12:56Z

Is there a good/intuitive way to generate a stock of examples of non-abelian unramified extensions? The only examples I know of are a bit unintuitive (eg in Janusz's book). Intuition might suggest to start looking at Galois extensions with group equal to semi-direct products. I was originally interested in this question by the necessity of "ab" on the rhs of $H^{1} (X_{Zar}\,, \mathcal{O}_{X} ^ {*}) = \pi_{1} ^{ab} (X), \; \; X = Spec\; \mathcal{O}_K$ (re-interpretation of unramified global cft).

Comment by Ivan

2013-02-25T15:28:36Z

side remark: (from Liu's Alg. geom. book, 8.2.40 (b)) a regular local ring $R$ is excellent if $Frac(\hat{R})$ is separable over $Frac(R)$ (i.e. if every finitely-generated subextension is separable). Hence, a way to check if a dvr is excellent. As an example, consider the henselization of $k[x,y]$ at the prime (y), for $k$ a field of char $p>0$. This henselization is sometimes denoted by $k(x)\{y\}.$

Comment by Ivan

2013-02-20T12:54:20Z

Thank you for the clear explanation!

Comment by Ivan

2013-02-19T19:34:06Z

Marc: Thanks for your comment. What do you intend to show by saying a field is not of the form of a local ring in your first sentence? Perhaps you meant something about the residue fields? In general, residue fields of local rings can have infinite transcendence degree over the prime field. – Ivan 2 mins ago

Comment by Ivan

2010-06-25T14:26:39Z

Speaking of Mumford and arithmetic approaches, it seems the fabled "Mumford-Lang" type-script has been subsumed by notes of Mumford/Oda (Lang is not mentioned as a coauthor here). In these notes they discuss Galois theory for schemes. Cf. especially Chapter 4 (Part I) of math.upenn.edu/~chai/624_08/math624_08.html For a student with a background in algebraic number theory, Grothendieck's version of Galois theory may provide for added motivation to learn about schemes

Comment by Ivan

2010-05-31T13:23:58Z

@Qing Liu, I think you are correct on (1) especially bc in my application I was lucky enough that I could perform a change of coordinates to ensure that $C$ is contained in an open affine. I don't have any counterexamples yet, but appreciate the comment.

Comment by Ivan

2010-05-30T16:51:25Z

I'm neutral as to whether or not this question is taken down. If people find the comments/answers here useful, then perhaps leaving it up is ok.

Comment by Ivan

2010-05-29T14:05:16Z

Thanks for the spelling corrections; I've incorporated them. I've also placed the authors' names in the order appearing in their paper

Comment by Ivan

2010-05-28T16:32:45Z

also, I second Mariano's question above.

Comment by Ivan

2010-03-09T22:48:16Z

Thank you for the illuminating examples and references Prof. Liu, I am looking forward to the Gabber/Lorenzini article.

Comment by Ivan

2010-03-09T22:46:43Z

Thank you! this is just what I was looking for

Comment by Ivan

2010-03-09T19:26:08Z

while smooth over (say) perfect fields implies regular, regularity does not imply smoothness (eg over inseparable extensions). i'm referencing the definition of smoothness (of a morphism) (EGA IV) for arbitrary schemes (not just varieties). Here is a typical example where regular does not imply smooth: let k be an inperfect field, take a finite inseparable extension L/k, then $\mathbb{P}^1 _L$ is regular (variety over k) but not smooth over k.

Comment by Ivan

2010-03-09T18:41:43Z

thank you, and what does "smooth scheme S" mean? (smooth is always a relative notion), bc if it means S is smooth over a field k, then it coincides with the above definition. I think the first chapter of 'Lectures on Arakelov geometry' is also a good reference, here they treat schemes over Dedekind rings (for example)

Comment by Ivan

2010-03-02T07:13:11Z

Almost, but the crucial point is exactly what "isomorphism" should be. For etale covers $Y\rightarrow X$, the group $Aut(Y/X)$ is the set of scheme isomorphisms of $Y$ fixing $X$. It is then a consequence that IF #Aut(Y/X) = [K(Y):K(X)]$ Then we can talk about permuting geometric fibers. I was looking, eg, for an affine explanation (imprecise), but precisely, the claims in this paper aim at something I was looking for: arxiv.org/abs/0910.4646

Comment by Ivan

2010-02-26T04:43:41Z

Thanks! explicit finite extensions was what I was looking for. In skimming the paper, it looks very well-written and with only elementary methods as you mentioned. jtnb.cedram.org/item?id=JTNB_1997__9_1_51_0

Comment by Ivan

2010-02-25T03:51:11Z

Thank you for the responses. Indeed, the simple example shows that the answer is negative in general; eg, etale morphisms must preserve dimension. I was curious about generalizations of noether normalization.