User kiseki - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T21:15:29Z http://mathoverflow.net/feeds/user/5274 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/130335/pro-ell-etale-fundamental-group-of-a-semi-abelian-variety pro-$\ell$ etale fundamental group of a semi-abelian variety kiseki 2013-05-11T13:36:58Z 2013-05-11T13:55:49Z <p>Let $A$ be a semi-abelian variety over $K$, $\ell$ a prime number which is not equal to char($K$).</p> <p>Does the abelianization of geometrically pro-$\ell$ etale fundamental group $(\pi_{1}(A\otimes\overline K)^{\ell})^{ab}$ isomorphic to the $\ell$-adic Tate module of $A$?</p> http://mathoverflow.net/questions/129286/morphism-between-two-elliptic-curves-over-a-local-field morphism between two elliptic curves over a local field kiseki 2013-05-01T07:04:27Z 2013-05-01T07:39:15Z <p>Let $X,Y$ be two elliptic curves over $K:=K(R)$ which have good models over $R$, Char($K$)=$0$, where $R$ complete DVR with algebraically residue field $k$.</p> <p>If $L$ is a finite extension of $K$, such that $X \otimes L \cong Y \otimes L$.</p> <p>My question is:</p> <p>Is there $X \cong Y$ ?</p> http://mathoverflow.net/questions/129061/morphism-from-adic-spaces-to-schemes morphism from adic spaces to schemes kiseki 2013-04-29T04:01:35Z 2013-04-29T04:01:35Z <p>Let $X:=Spa A$ be an affinoid adic space, and $\underline X$ the ringed space of $X$. Let $Y:=Spec B$ be an affine scheme, $f: \underline X \longrightarrow Y$ a morphism of ringed spaces.</p> <p>How to define a morphism of rings $g: B \longrightarrow A$ such that $f$ is induced by $g$?</p> http://mathoverflow.net/questions/126697/trivial-deformation-of-a-smooth-affine-scheme-over-complete-dvr trivial deformation of a smooth affine scheme over complete DVR kiseki 2013-04-06T07:13:18Z 2013-04-07T08:13:03Z <p>Let $X$ be a affine smooth scheme finite type over $A/pA$, where $A$ a complete DVR and $chk=p>0$.</p> <p>I know that since $H^{2}=0=H^{1}$, we have a unique lifting to $A/p^{2}$. In algebraic schemes case, we can obtain a trivial deformation by fiber product. But in this case, for example, $A$ is a p-adic number ring, What is the trivial deformation over $A/p^{2}$ ?</p> <p>PS: I added some smooth conditions to $X$.</p> http://mathoverflow.net/questions/125988/questions-of-localization-of-topos questions of localization of topos kiseki 2013-03-30T09:46:52Z 2013-03-30T11:03:30Z <p>Let $T$ be a topos, and $F \in T$, $T/F$ a localization of $T$. So we have a natural morphism $i: T/F \longrightarrow T$.</p> <p>My questions are:</p> <p>1.What are the definitions of $i_{\ast}$ and $i^{\ast}$ without using site ?</p> <p>2.Dose there exist a site $S$ such that $Sh(S) \cong T$ and there is an $U \in S$ which $U^{\sim} \cong F$ ?</p> http://mathoverflow.net/questions/125822/morphism-of-injective-objects morphism of injective objects kiseki 2013-03-28T12:57:50Z 2013-03-28T20:48:35Z <p>Let $A,B$ be two bounded below complexes in module category, and $A \longrightarrow I$ (resp. $B \longrightarrow J$) a injective resolution. If $f: A \longrightarrow B$ is a morphism of complexes.</p> <p>My question is: how to construct a morphism from $I$ to $J$ which induced by $f$?</p> http://mathoverflow.net/questions/125799/question-of-topos-and-site question of topos and site kiseki 2013-03-28T07:12:42Z 2013-03-28T10:37:57Z <p>Let $T, P$ be two topoi, and $f:T \longrightarrow P$.</p> <p>Does there exist two site $S_{T}, S_{P}$ and a morphism $g: S_{T} \longrightarrow S_{P}$ such that $f$ is induced by $g$ ? </p> http://mathoverflow.net/questions/123857/how-to-see-the-geometry-and-arithmetic-of-tannakian-fundamental-groups How to see the geometry and arithmetic of tannakian fundamental groups? kiseki 2013-03-07T11:05:54Z 2013-03-09T17:34:26Z <p>The etale fundamental group is an inverse limit of automorphism groups of finite etale coverings. We can see the geometry of etale fundamental group very well from etale coverings just like topologically fundamental group. But the tannakian fundamental groups are defined as the automorphism of a fiber functor of a tensor category, this is definition abstract for me. </p> <p>I want ask that how to see the geometry of tannakian fundamental groups? </p> <p>What is the relation between first etale cohomology and pro-unipotent fundamental group?</p> <p>I would like to extend my question. If we consider a hyperbolic curve $X$ over a local field $K$ with valuation ring $R$, there is an natural Galois action (i.e., outer Galois action) of $G_{K}$ on the etale fundamental group $\pi_{1}(X)$. From this outer Galois action, we can understand some geometry of $X$ and their reduction. For example, there is a good reduction criterion in terms of pro-$l$ fundamental groups (Oda, Tamagawa).</p> <p>My question is: Dose there exist some similar Galois actions or criterions for Tannakian fundamental groups? or dose there exist some anabelian type theorems for Tannakian fundamental groups?</p> http://mathoverflow.net/questions/122162/formal-power-series-over-a-henselian-ring Formal power series over a henselian ring kiseki 2013-02-18T12:48:25Z 2013-02-19T08:53:51Z <p>Let $R$ be a henselian ring. Is $R[[x]]$ also a henselian ring?</p> http://mathoverflow.net/questions/121986/relations-between-log-schemes-and-toric-varieties relations between log schemes and toric varieties kiseki 2013-02-16T13:37:55Z 2013-02-17T12:46:54Z <p>The theory of regular log scheme, roughly specking is that scheme theory with singularities like those of toric varieties.</p> <p>Dose anyone could explain this idea in detail or give a reference about the relations between log schemes and toric varieties? </p> http://mathoverflow.net/questions/119736/deformation-of-stable-curve deformation of stable curve kiseki 2013-01-24T09:15:14Z 2013-01-30T23:58:06Z <p>Let $R$ be a DVR, and $k$ residue field of $R$. We suppose that $X_{0}$ is a stable curve over Spec$k$.</p> <p>Dose there exist a stable model $X$ over $R$ such that the special fiber isomorphic to $X_{0}$ ?</p> <p>If we assume $R=C[[t]]$, where C is complex number field, how to find a deformation which make the generic fiber is smooth? </p> http://mathoverflow.net/questions/120016/galois-action-on-special-fiber-of-a-stable-model Galois action on special fiber of a stable model kiseki 2013-01-27T11:56:52Z 2013-01-28T05:29:01Z <p>Let $X_{K}$ be a curve over a complete DVR $R$, $R/m:=k$ an algebraically closed field. We suppose the minimal field extension $L$ of $K$ such that $X_{L}$ has stable model $X_{R_{L}}$, and the special fiber is $X_{k}$. We obtain an action of $Gal(L/K)$ on $X_{k}$.</p> <p>My question is:</p> <p>Does $Gal(L/K) \longrightarrow Aut(X_{k})$ is an injection ?</p> http://mathoverflow.net/questions/117180/tannakian-fundamental-group-for-finitely-linear-representation-of-group Tannakian fundamental group for finitely linear representation of group kiseki 2012-12-25T02:49:00Z 2012-12-26T00:35:37Z <p>Let $G$ be an arbitrary group and $k$ a field. Denote by $Rep_{k}(G)$ the category of finite dimensional representations of $G$ over $k$. The usual tensor product and dual operations for representations equip $Rep_{k}(G)$ with the structure of a $k$-linear rigid abelian tensor category. The forgetful functor $Rep_{k}(G) \longrightarrow Vec_{k}$ as fiber functor. So we get a neutral Tannakian category.</p> <p>My question is :</p> <p>If we consider the $G$ as constant affine group scheme over $k$, does the Tannakian fundamental group isomorphic to $G$? </p> http://mathoverflow.net/questions/114857/an-example-of-almost-etale-extension An example of almost etale extension kiseki 2012-11-29T08:05:53Z 2012-11-29T13:32:56Z <p>In the paper of Faltings' "p-adic Hodge theory", Faltings showed an example of almost etale extension before he proved the almost purity theorem. The example is following:</p> <p>Let $k$ be a perfect field of characteristic $p$ and $W(k)$ the ring of Witt vectors, $W(k)(x_{1},...,x_{d})$ a localization of $W(k)[x_{1},...,x_{d}]$ at $p$. Let $V$ be the completion of $W(k)(x_{1},...,x_{2})$ with function field $K:=K(V)$, and $V_{n}$ be the extension of $V$ which generated by the $p^{n+1}$-th roots of unite together with $p^{n}$-th roots of $x_{i}$. If $W$ is a normalization of $V$ in another finite extension $L$ of $K$, and $W_{n}$ the composite of $W$ and $V_{n}$, and $p^{a_{n}}W_{n}$ the different of $W_{n}$ over $V_{n}$. If we use the notation $W_{\infty}$ (resp. $V_{\infty}$) to denote the union of $W_{n}$ (resp.$V_{n}$).</p> <p>The Theorem 1.2 of Faltings' paper tell us $a_{n}$ converge to $0$ for $n \rightarrow \infty$, and then $V_{\infty} \longrightarrow W_{\infty}$ is almost etale extension.</p> <p>My question is:</p> <p>Why the almost etale extension of $W_{\infty}/V_{\infty}$ is deduced by theorem 1.2 ? Is this checked by the definition of almost etale ? </p> http://mathoverflow.net/questions/114518/a-question-of-line-bundle-for-finite-etale-covering A question of line bundle for finite etale covering kiseki 2012-11-26T11:45:37Z 2012-11-27T02:18:47Z <p>Hi, everyone, I want to ask a question about line bundle.</p> <p>Let $X$ be a smooth curve over a algebraically closed field $k$, and $f:Y \longrightarrow X$ a Galois finite etale covering with Galois group $G$ and degree $n$. Suppose that $L$ is a line bundle on $X$.</p> <p>Dose there exist a line bundle $M$ on $Y$ such that $M^{\otimes n}=f^{*}L$? </p> <p>Thanks.</p> http://mathoverflow.net/questions/110644/does-any-connected-formal-k-group-scheme-is-formally-smooth Does any connected formal $k$-group scheme is formally smooth? kiseki 2012-10-25T10:35:42Z 2012-10-25T10:35:42Z <p>Let $G$ be a connected p-divisible group over a field $k$, then $G$ is a formally smooth group scheme over $k$, since the category of formal lie groups and the category of connected p-divisible groups over a complete noetherian local ring are equivalence. </p> <p>My question is following:</p> <p>Does any connected formal $k$-group scheme is formally smooth?</p> http://mathoverflow.net/questions/106785/reduction-of-elliptic-curves reduction of elliptic curves kiseki 2012-09-10T07:31:22Z 2012-09-10T15:22:46Z <p>Let $X$ be an elliptic curve over a complete local field.</p> <p>The definition of semi-abelian reduction is: "the Neron model of $X$ is a semi-abelian scheme". On the other hand, the definition of semi-stable reduction is: "the minimal regular model of $X$ is semi-stable."</p> <p>For elliptic curves, are the two definitions equivalent? How to prove it?</p> http://mathoverflow.net/questions/106428/relationship-between-etale-cohomology-group-of-vanishing-cycle-and-irreducible-co relationship between etale cohomology group of vanishing cycle and irreducible components kiseki 2012-09-05T12:54:23Z 2012-09-05T12:54:23Z <p>Hi everyone, I want to ask a question about vanishing cycle.</p> <p>Let $X$ be a stable curve over a complete DVR $A$ with residue field $k=\overline k$, and $X_{k}$ the special fiber. Suppose that $X_{i}$ are irreducible components of $X_{k}$, $S_{i}:=X_{i} \cap (\cup_{j, j \neq i} X_{j})$</p> <p>If we denote $X_{i}-${nodes, $S_{i}$} by $U_{i}$. My question is following.</p> <p>What's the relationship between $H^{1}(X_{k},R\Phi(Q_{\ell})$) and $H^{1}(U_{i},Q_{\ell})$ </p> http://mathoverflow.net/questions/101951/curves-whose-stable-reductions-do-not-contain-rational-curves Curves whose stable reductions do not contain rational curves kiseki 2012-07-11T13:19:33Z 2012-07-11T14:26:06Z <p>Let $X$ be a smooth projective curve over $K:=K(A)$. $A$ is a strict henselian ring, $A/m=k=\bar k$. Suppose $\cal X$ is a stable model of $X$, ${\cal X}_{s}$ is the special fiber. </p> <p>My question is: </p> <p>what are the conditions on $X$ so that $\cal X_{s}$ does not contain any rational curves?</p> http://mathoverflow.net/questions/101933/a-problem-of-graph-theory A problem of graph theory kiseki 2012-07-11T09:47:03Z 2012-07-11T12:06:19Z <p>Hi, everyone, I want to ask a question about graph theory.</p> <p>Let $G$ be a finite graph, and $E$ the set of edges of $G$. For each vertex $a$, we denote that $E_{a}$ by the edges of $G$ which pass to $a$. Suppose for any $a \in G$, $\sharp E_{a} >3$. Let $f: E \longrightarrow$ {$1,2$}.</p> <p>Does there exists a map $f$ such that for each $a$, $\sharp (f^{-1}(1) \cap E_{a})$ is a nonzero even number.</p> <p>Thank you very much! </p> http://mathoverflow.net/questions/100773/references-about-the-grothendiecks-way-of-algebraizing-the-notions-of-calculus-a References about the Grothendieck's way of algebraizing the notions of calculus and differential geometry kiseki 2012-06-27T13:02:50Z 2012-06-27T20:43:05Z <p>Hi everyone,</p> <p>I'm looking for some references about the differential operators on schemes(connection, curvature, etc...). I am reading the EGA IV 16, but EGA does not treats connection, curvature, etc.... </p> <p>Are there any articles/books that deal with the the Grothendieck's way of algebraizing the notions of calculus and differential geometry?</p> <p>Thank you very much! </p> http://mathoverflow.net/questions/21552/roadmap-for-studying-arithmetic-geometry roadmap for studying arithmetic geometry kiseki 2010-04-16T11:01:00Z 2012-06-26T07:30:41Z <p>hi everybody, I have already finished the Hartshorne's algebraic geometry from chapter 1 to chapter 4, so I'd like to find some suggestions about the next step to study arithmetic geometry. I want to know that how to use the scheme theory and their cohomology to solove the arithmeic problem.Would you like to recommend me some of these kind of books and papers?</p> <p>Thank you very much!</p> <p>PS: I also want to learn some materials about moduli theory, if you like, could you recommend me some books or papers ?</p> http://mathoverflow.net/questions/100562/english-version-of-manins-correspondences-motifs-and-monoidal-transformations English version of Manin's "Correspondences, motifs and monoidal transformations " kiseki 2012-06-25T04:10:06Z 2012-06-25T05:54:03Z <p>Hi everyone, I want to find a classical paper "Correspondences, motifs and monoidal transformations " by Manin. But I only found the Russian version. Dose this paper exist a english version? and where I could get it ?</p> <p>Thank you very much!</p> http://mathoverflow.net/questions/71875/good-references-in-moduli-stack-and-stable-reduction good references in moduli stack and stable reduction kiseki 2011-08-02T12:58:57Z 2012-06-06T15:50:08Z <p>Hello,everyone. I'm looking for some good references in moduli stack and stable reduction, so I ask here for some advice. </p> <p>I knew the famous paper of Deligne-Mumford, but this paper is hard for me now, I will very happy if someone could give me some suggestions for reading Deligne-Mumford, or tell me some good references about moduli-stack and stable reduction of curves or abelian varieties.</p> <p>THANK YOU VERY MUCH!</p> http://mathoverflow.net/questions/98767/strict-henselian-and-excellent-henselian strict henselian and excellent henselian kiseki 2012-06-04T12:54:51Z 2012-06-06T09:13:03Z <p>Hello, everyone. I want to ask a problem about strict henselian ring.</p> <p>Let $A$ be a strict henselian DVR. </p> <p>Dose there exist subrings $A_{i}$ of $A$, such that $A=lim_{i} A_{i}$ and where $A_{i}$ are excellent strict henselian DVR.</p> http://mathoverflow.net/questions/97461/the-algebraic-closure-of-strict-henselian-dvr the algebraic closure of strict henselian DVR kiseki 2012-05-20T05:48:37Z 2012-05-21T03:57:18Z <p>Let $A$ be a strict henselian DVR, and $\hat A$ is completion of $A$, </p> <p>is $K(A)^{alg} \longrightarrow {K(\hat A)}^{alg}$ a isomorphism? </p> <p>where $K(A)$ and $K(\hat A)$ are quotient fields.</p> http://mathoverflow.net/questions/97357/is-strict-henselian-ring-a-excellent-ring Is strict Henselian ring a excellent ring? kiseki 2012-05-18T23:48:12Z 2012-05-19T08:56:09Z <p>Hi, everyone, I want to ask following problem:</p> <p>Is strict Henselian ring a excellent ring?</p> <p>If not, could you give me a example?</p> http://mathoverflow.net/questions/95661/generic-etale-morphism-from-curve-to-projective-line generic etale morphism from curve to projective line kiseki 2012-05-01T11:02:35Z 2012-05-01T11:02:35Z <p>Let $X$ be a smooth projective curve over $k$, ch$k=p>0$, dose there exist a generic etale morphism from $X$ to projective line ?</p> http://mathoverflow.net/questions/94999/why-find-a-good-field-extension-such-that-the-curve-has-semi-stable-model-is-impo why find a good field extension such that the curve has semi-stable model is important? kiseki 2012-04-24T06:19:17Z 2012-04-24T07:25:40Z <p>Hello everyone,</p> <p>I'd like to ask some question about semi-stable reduction of curves.</p> <p>The Deligne-Mumford theorem tell us "Let $A$ be an Dedekind domain, $K=K(A)$, for any smooth curve $X$ over $K$ and $g(X)>1$, there exist a separable extension $L$ of $K$, such that $X_{L}$ has semi-stable model."</p> <p>But this theorem does not tell us any information about how to find the extension field $L$, but it seems is very important for arithmetic geometers. </p> <p>My question is, in arithmetic geometry, why find the good extension field $L$ (for example:tame extension) such that $X_{L}$ has semi-stable model is so important?</p> http://mathoverflow.net/questions/85119/why-we-need-rigid-geometry why we need rigid geometry? kiseki 2012-01-07T08:34:34Z 2012-04-21T04:44:00Z <p>Hello, everyone.</p> <p>I want to ask some questions about rigid geometry.</p> <p>1.what is the motivation of rigid geometry?</p> <p>2.what is the applications of rigid geometry for solving arithmetic problems, especially for studying the fundamental groups of algebraic curves? what the beautiful theorems which were first proved by rigid geometry method?</p> <p>thank you very much</p> http://mathoverflow.net/questions/129061/morphism-from-adic-spaces-to-schemes Comment by kiseki kiseki 2013-04-29T13:26:39Z 2013-04-29T13:26:39Z @J&#233;r&#244;me Poineau: the global section of $SpaA$ is $\hat A$, so there is $B \longrightarrow \hat A$. http://mathoverflow.net/questions/125988/questions-of-localization-of-topos/125990#125990 Comment by kiseki kiseki 2013-03-30T10:16:54Z 2013-03-30T10:16:54Z Simon, Thank you so much ! http://mathoverflow.net/questions/123857/how-to-see-the-geometry-and-arithmetic-of-tannakian-fundamental-groups Comment by kiseki kiseki 2013-03-07T15:16:45Z 2013-03-07T15:16:45Z @Lars: I want to say pro-unipotent fundamental group. http://mathoverflow.net/questions/119736/deformation-of-stable-curve Comment by kiseki kiseki 2013-01-29T01:16:02Z 2013-01-29T01:16:02Z @pranavk: If we assume $R=C[[t]]$, where $C$ is complex number field, how to find a deformation which make the generic fiber is smooth? http://mathoverflow.net/questions/119736/deformation-of-stable-curve Comment by kiseki kiseki 2013-01-27T12:08:42Z 2013-01-27T12:08:42Z @ayanta: If we assume that $R$ is a complete DVR, is this deformation unique ? http://mathoverflow.net/questions/106785/reduction-of-elliptic-curves/106787#106787 Comment by kiseki kiseki 2012-09-11T09:57:10Z 2012-09-11T09:57:10Z Thank you very much, Pof.Liu. I have found the result in your book. http://mathoverflow.net/questions/101933/a-problem-of-graph-theory Comment by kiseki kiseki 2012-07-11T10:55:11Z 2012-07-11T10:55:11Z mathoverflow.net/questions/98385 showed a conterexamples for my problem. But if the dual graph of a stable curves is the graph of mathoverflow.net/questions/98385, we could construct a abelian covering of the stable curve directly. So maybe we should assume for any vertex $a$ such that ♯Ea&gt;3 http://mathoverflow.net/questions/101933/a-problem-of-graph-theory Comment by kiseki kiseki 2012-07-11T10:20:11Z 2012-07-11T10:20:11Z Thanks for your comment, darij. In fact, this problem arises from arithmetic geometry, I want to construct a special abelian covering of a stable curve which irreducible components are all projective line. http://mathoverflow.net/questions/100773/references-about-the-grothendiecks-way-of-algebraizing-the-notions-of-calculus-a/100782#100782 Comment by kiseki kiseki 2012-06-27T14:57:04Z 2012-06-27T14:57:04Z Thanks so much Lars! http://mathoverflow.net/questions/100562/english-version-of-manins-correspondences-motifs-and-monoidal-transformations Comment by kiseki kiseki 2012-06-25T12:36:02Z 2012-06-25T12:36:02Z Thanks! Thomas. http://mathoverflow.net/questions/98767/strict-henselian-and-excellent-henselian Comment by kiseki kiseki 2012-06-06T09:14:57Z 2012-06-06T09:14:57Z sorry, I modified the problem. http://mathoverflow.net/questions/97357/is-strict-henselian-ring-a-excellent-ring/97382#97382 Comment by kiseki kiseki 2012-05-19T10:37:06Z 2012-05-19T10:37:06Z Thank you for answering! http://mathoverflow.net/questions/95661/generic-etale-morphism-from-curve-to-projective-line Comment by kiseki kiseki 2012-05-01T13:14:41Z 2012-05-01T13:14:41Z Thank you for answering ! http://mathoverflow.net/questions/91067/a-question-about-stable-curve/91088#91088 Comment by kiseki kiseki 2012-03-14T00:44:53Z 2012-03-14T00:44:53Z Thank you for your answer. http://mathoverflow.net/questions/91067/a-question-about-stable-curve/91069#91069 Comment by kiseki kiseki 2012-03-13T14:49:34Z 2012-03-13T14:49:34Z Sorry Dan, could you please explain in detial? If there is a finite morphism from the irreducible component which containing p, say $X_i$, to projective line, this morpism will not be flat since $X_i$ maybe singluar, in this case why this morphism is a isomorphim? On the other hand, the stable curve maybe has a component which isomorphic to projective line.