User - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:36:56Z http://mathoverflow.net/feeds/user/12832 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103101/frobenius-eigenvalues-of-abelian-variety Frobenius eigenvalues of abelian variety unknown (google) 2012-07-25T12:49:57Z 2012-07-25T12:58:18Z <p>Let $A$ be an abelian variety over a finite field <code>$\mathbb{F}_q$</code> and $x_i$ the Frobenius eigenvalues on $H^1$. Does $x_i \mapsto q/x_i$ permute the $x_i$, and why? It should follow from Poincare duality.</p> http://mathoverflow.net/questions/76979/cocharacters-of-glv cocharacters of GL(V) unknown (google) 2011-10-02T13:28:09Z 2011-10-02T13:28:09Z <p>Can someone give me a hint how to determine the cocharacters of $GL(V)$?</p> http://mathoverflow.net/questions/74570/formally-smooth-smooth formally smooth => smooth unknown (google) 2011-09-05T10:25:02Z 2011-09-05T22:04:45Z <p>A morphism of set-valued functors $\eta: F \to G$ on $\mathcal{C}$ is called smooth if for all epimorphisms $B \to A$, the natural morphism $F(B) \to F(A) \times_{G(A)} G(B)$ is surjective.</p> <p>Obviusly "smooth => formally" smooth for $\mathcal{C} = \mathrm{Sch}$.</p> <p>Now my question: Does the converse hold?</p> <p>My thoughts: 1. Assume the morphism is of finite presentation.</p> <ol> <li><p>Assume it is in the local standard form: an étale morphism followed by an affine projection</p></li> <li><p>It is clear that an affine projection is smooth in the above sense, so we have reduced the problem to étale morphisms.</p></li> </ol> http://mathoverflow.net/questions/71893/cohomological-dimension-for-coarser-finer-topologies cohomological dimension for coarser/finer topologies unknown (google) 2011-08-02T15:37:59Z 2011-08-02T16:14:11Z <p>Given a sheaf $\mathcal{F}$ with respect to some Grothendieck topology, is the cohomological dimension for this sheaf less than or equal to the cohomological dimension of a finer topology?</p> <p>Example: $cd_{Zar} \leq cd_{Nis} \leq cd_{ét}$.</p> http://mathoverflow.net/questions/70331/book-on-linear-algebraic-groups-in-scheme-language Book on linear algebraic groups in scheme language unknown (google) 2011-07-14T15:02:42Z 2011-07-14T15:12:28Z <p>Is there a book on linear algebraic groups using the scheme language (i.e. not Springer or Borel, but like Waterhouse, but more in-depth)?</p> <p>The book should discuss topics like Borel subgroups etc.</p> <p>(Related question: <a href="http://mathoverflow.net/questions/17662/books-on-reductive-groups-using-scheme-theory" rel="nofollow">http://mathoverflow.net/questions/17662/books-on-reductive-groups-using-scheme-theory</a>)</p> http://mathoverflow.net/questions/67688/why-does-one-consider-the-dual-of-the-steenrod-algebra Why does one consider the dual of the Steenrod algebra? unknown (google) 2011-06-13T18:00:02Z 2011-06-14T05:36:37Z <p>Why does one consider the dual of the Steenrod algebra?</p> http://mathoverflow.net/questions/66099/h-2-of-a-simply-connected-lie-group-vanishes $H_2$ of a simply connected Lie group vanishes unknown (google) 2011-05-26T19:54:19Z 2011-06-11T03:06:00Z <p>How do I show that the $H_2$ of a simply connected Lie group vanishes? (I don't want to use that $\pi_2(Lie group) = 0$, since this is what I want to prove. And I don't want to use the classification of compact simply-connected Lie groups.)</p> http://mathoverflow.net/questions/65792/applications-of-postnikov-approximation applications of Postnikov approximation unknown (google) 2011-05-23T19:56:40Z 2011-05-23T19:56:40Z <p>What are applications of Postnikov approximation? As I understand it, it is dual to a cell decomposition and can, e.g. be used for computing some homotopy groups of spheres.</p> http://mathoverflow.net/questions/65658/map-defined-by-element-of-function-field-of-a-variety map defined by element of function field of a variety unknown (google) 2011-05-21T16:46:41Z 2011-05-23T06:03:52Z <p>Given a smooth proper variety $V/K$ of dimension $n$ and an element $f \in K(V)$, does this define a map $V \to \mathbf{P}^n$? Probably only a rational map outside a set of codimension $> 1$?</p> http://mathoverflow.net/questions/65580/albanese-variety Albanese variety unknown (google) 2011-05-20T19:04:20Z 2011-05-20T19:04:20Z <p>What are good references for the Albanese variety and its properties?</p> http://mathoverflow.net/questions/65044/quotient-category-cohx-d-cohx-d-1 Quotient category $Coh(X)_d/Coh(X)_{d-1}$ unknown (google) 2011-05-15T15:16:48Z 2011-05-15T15:16:48Z <p>Let $X$ be a regular scheme, and $Coh(X)_d$ be the category of coherent sheaves of $\leq d$ dimensional support.</p> <p>Why is <code>$Coh(X)_d/Coh(X)_{d-1}$</code> equivalent to <code>$\bigoplus_{x \in X_d} \mathcal{A}(\mathcal{O}_{X,x})$</code>?</p> <p>See: <a href="http://www.math.uni-bielefeld.de/~mseverit/algkalgc.pdf" rel="nofollow">http://www.math.uni-bielefeld.de/~mseverit/algkalgc.pdf</a> page 2, 6th equation.</p> <p>(cross post from <a href="http://math.stackexchange.com/questions/39186/quotient-category-coh-d-coh-d-1" rel="nofollow">http://math.stackexchange.com/questions/39186/quotient-category-coh-d-coh-d-1</a> )</p> http://mathoverflow.net/questions/65042/structure-of-t-ell-a-for-a-mathbff-q-an-abelian-variety structure of $T_\ell A$ for $A/\mathbf{F}_q$ an abelian variety unknown (google) 2011-05-15T14:29:28Z 2011-05-15T15:14:24Z <p>Can someone give me references for the structure of the <code>$G_{\mathbf{F}_q}$</code>-module $T_\ell A$, $A/\mathbf{F}_q$ an abelian variety?</p> http://mathoverflow.net/questions/64761/does-every-finite-flat-group-scheme-become-constant-after-finite-base-change Does every finite flat group scheme become constant after finite base change? unknown (google) 2011-05-12T09:16:41Z 2011-05-12T10:48:03Z <p>Does every finite flat group scheme $G/X$ become constant after finite base change? Which additional properties of the base change morphism can we impose?</p> <p>Edit: Which conditions do we have to impose on $G/X$ so that the answer becomes "yes"?</p> http://mathoverflow.net/questions/60295/prove-statement-in-galois-cohomology-by-etale-cohomology prove statement in Galois cohomology by étale cohomology unknown (google) 2011-04-01T14:31:37Z 2011-04-30T05:22:14Z <p>According to Milne's Arithmetic Duality Theorems, Proposition I.6.4: $0 \to H^1(G_S, A[m]) \to H^1(K, A[m]) \to \oplus_{v \not\in S}H^1(K_v, A)$ for an abelian variety $A$ and a nonempty set of primes $S$ containing all infinite primes and the primes of bad reduction.</p> <p>I want to prove this using étale cohomology. My idea was to extend $A$ to an abelian scheme and use the (excision) long exact sequence, but this lead me nowhere. Can someone give me some hints?</p> http://mathoverflow.net/questions/61217/tensor-product-of-motivic-complexes-mathbfzn tensor product of motivic complexes $\mathbf{Z}(n)$ unknown (google) 2011-04-10T15:30:09Z 2011-04-10T15:30:09Z <p>Is the morphism $\mathbf{Z}(n) \otimes^L \mathbf{Z}(m) \to \mathbf{Z}(n+m)$ from the Beilinson-Lichtenbaum conjectures a quasi-isomorphism?</p> http://mathoverflow.net/questions/60310/galois-cohomology-groups-given-by-etale-cohomology Galois cohomology groups given by étale cohomology unknown (google) 2011-04-01T17:10:03Z 2011-04-02T01:48:51Z <p>What are cases when Galois cohomology groups are given by étale cohomology?</p> <p>Example: $S = Spec(K)$ the spectrum of a field, $F \in Sh(K)$, then $H^p(K, F) = H^p(G_K, F_{\bar{K}})$.</p> <p>What if $G = \pi_1(X)$ and $F \in Sh(X)$? Under what conditions do we have $H^p(X, F) = H^p(G, [F])$, where $[F]$ denotes a suitable $\pi_1(X)$-module associated with $F$? (Example for this: $X = Spec(O_K)\setminus S$)</p> http://mathoverflow.net/questions/54806/questions-regarding-modular-forms questions regarding modular forms unknown (google) 2011-02-08T19:15:35Z 2011-04-01T16:49:01Z <ol> <li>Let $f$ be modular of level $p^nN$, $(p,n) = 1$, $p > 2$ with character $\chi\psi\eta$, where $\chi$ has conductor dividing $N$, $\psi$ conductor power of $p$ and order power of $p$, and $\eta$ conductor $p$ and order dividing $p-1$. Since $p$ is odd, one can write $\psi = \xi^{-2}$. (i) Why is the character of $f \otimes \xi$ equal to $\chi\eta$; (ii) why is the reduction mod $p$ of this equal to the reduction of $f$; (iii) why is for $r \gg n$ $f \otimes \xi$ modular with respect to $\Gamma_0(p^r) \cap \Gamma_1(pN)$? Can this be proven using the converse theorem?</li> </ol> <p>2.a Why is the twisted Eisenstein series $G = a_0 + \sum_{n=1}^\infty\sum_{d \mid n}\eta^{-1}(d)d^{i-1}q^n$ of Nebentypus $\eta^{-1}$ with respect to $\Gamma_0(p)$?</p> <p>2.b Why is $fG$ modular with respect to $\Gamma_0(p^r) \cap \Gamma_1(N)$, wenn $f \in S_k(\Gamma_0(p^r) \cap\ \Gamma_1(pN), \chi\eta)$ ist?</p> <p>(The article is here: math.berkeley.edu/~ribet/Articles/motives.pdf )</p> http://mathoverflow.net/questions/74570/formally-smooth-smooth Comment by 2011-09-05T11:08:33Z 2011-09-05T11:08:33Z I think my definition of &quot;smooth&quot; is a priori different from this. http://mathoverflow.net/questions/71893/cohomological-dimension-for-coarser-finer-topologies Comment by 2011-08-03T13:28:33Z 2011-08-03T13:28:33Z &quot;Suppose we have one category with two Grothendieck topologies, one finer than the other. Suppose some presheaf is a sheaf with respect to both of them. If Hn=0 for all n&gt;d in the finer case, must the same be true in the other case?&quot; Yes, that is what i meant. http://mathoverflow.net/questions/66401/singular-homology-cohomology-as-a-derived-functor/66406#66406 Comment by 2011-05-30T06:16:28Z 2011-05-30T06:16:28Z &quot;Or perhaps not! If so, ask away.&quot; Yes, please! I'm interested in! http://mathoverflow.net/questions/66099/h-2-of-a-simply-connected-lie-group-vanishes Comment by 2011-05-30T06:14:57Z 2011-05-30T06:14:57Z Isn't there a purely cohomological, non geometrical proof? http://mathoverflow.net/questions/65042/structure-of-t-ell-a-for-a-mathbff-q-an-abelian-variety Comment by 2011-05-15T16:02:12Z 2011-05-15T16:02:12Z OK, $V_\ell A = \mathbf{Q}_\ell^{2g}$, upon which the Frobenius acts semisimply with eigenvalues Weil numbers of weight $-1$. http://mathoverflow.net/questions/65042/structure-of-t-ell-a-for-a-mathbff-q-an-abelian-variety Comment by 2011-05-15T15:32:47Z 2011-05-15T15:32:47Z But this is not all. Let me think about it. http://mathoverflow.net/questions/65042/structure-of-t-ell-a-for-a-mathbff-q-an-abelian-variety Comment by 2011-05-15T15:28:49Z 2011-05-15T15:28:49Z I see: The Tate conjecture gives a description of $End_G(T_\ell A)$, in particular of its $G$-automorphisms, in terms of $End(A) \otimes \mathbf{Z}_\ell$. http://mathoverflow.net/questions/65042/structure-of-t-ell-a-for-a-mathbff-q-an-abelian-variety Comment by 2011-05-15T15:23:50Z 2011-05-15T15:23:50Z Where can I find the description of its <i>Galois module</i> structure? http://mathoverflow.net/questions/64761/does-every-finite-flat-group-scheme-become-constant-after-finite-base-change Comment by 2011-05-15T13:50:28Z 2011-05-15T13:50:28Z Does the &quot;only if&quot; follow from fppf descent? http://mathoverflow.net/questions/54806/questions-regarding-modular-forms/54807#54807 Comment by 2011-02-08T19:29:59Z 2011-02-08T19:29:59Z The article is here: <a href="http://math.berkeley.edu/~ribet/Articles/motives.pdf" rel="nofollow">math.berkeley.edu/~ribet/Articles/motives.pdf</a> page 5.