anon
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Registered User
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10h |
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Orders in number fields The $r=1$ case can be handled as follows. Each order in $R$ of index $p$ is uniquely determined by its image in $R/pR$. Now $R/pR$ is an \'etale $k := \mathbf{F_p}$-algebra of rank $n$ as $p$ is unramified. Now the set of such rings injects into the set of index $p$ $k$-subalgebras of $R/pR \otimes_k \overline{k} \simeq \overline{k}^n$. The latter set is a combinatorial object (it's the set of surjective maps from an $n$-element set to an $(n-1)$-element set), and can be bounded above by the set of index $p$ subrings of $\mathbf{Z}^n$. |
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Apr 21 |
awarded | ● Enlightened |
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Apr 21 |
awarded | ● Nice Answer |
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Apr 20 |
revised |
Is the derived double dual of a quasi-coherent sheaf a sheaf? Complete rewrite with clarification on which derived category we work in |
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Apr 20 |
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pushforward of injective sheaf acyclic for cohomology with supports A variation on Angelo's question: where does the argument in the last paragraph use the Zariski topology? All you need is that $\pi_* F(Y) \to \pi_* F(Y - Z)$ is surjective, which is the same as asking that $F(X) \to F(X - f^{-1} Z)$ be surjective, which is what the last paragraph shows. |
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Apr 20 |
accepted | Is the derived double dual of a quasi-coherent sheaf a sheaf? |
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Apr 19 |
revised |
Is the derived double dual of a quasi-coherent sheaf a sheaf? more clarification |
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Apr 19 |
revised |
Is the derived double dual of a quasi-coherent sheaf a sheaf? Added clarification about missing details |
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Apr 19 |
answered | Is the derived double dual of a quasi-coherent sheaf a sheaf? |
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Apr 11 |
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$k[[x]]$ as a $(k[[x]])^p$ module for ugly fields @Filippo: The $R$-submodule $x^p S \subset S$ is exactly the ideal $(x^p) \subset S$, so I don't understand the question. |
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Apr 10 |
accepted | Why is a proper, affine morphism finite? |
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Apr 10 |
accepted | $k[[x]]$ as a $(k[[x]])^p$ module for ugly fields |
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Apr 10 |
answered | $k[[x]]$ as a $(k[[x]])^p$ module for ugly fields |
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Apr 10 |
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$k[[x]]$ as a $(k[[x]])^p$ module for ugly fields @Karl: By passing to direct summands (which preserves completeness), it suffices to show that $M :== \oplus_{i=1}^\infty R$ is not $x$-adically complete for any ring $R$ with a regular element $x$. Consider the elements $m_n:=\sum_{i=1}^n x^i \cdot e_i$, where $e_i$ is the $i$-th basis vector. Then $\{m_n\}$ is a Cauchy sequence in $M$ with no limit (as any $m := \sum_{i=1}^N a_i \cdot e_i \in M$ admits an open neighbourhood $m + x^{N+2} M$ that does not contain a single $m_n$). |
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Apr 10 |
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$k[[x]]$ as a $(k[[x]])^p$ module for ugly fields It seems to me that the larger ring $S$ is the $x$-adic completion of the extension by scalars along $k^p \to k$ of the power series ring $R$ over $k^p$. Since $k$ is a free $k^p$-module of infinite rank, the question becomes: is the $x$-adic completion of a free $R$-module of infinite rank also free? The answer to this should be no, as an infinite rank free $R$-module is never $x$-adically complete. |
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Apr 6 |
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Algebraic $p$-adic integers mod $p$ The Fontaine-Wintenberger theorem identifies $\overline{\mathbf{Z}_p}/p$ with its characteristic $p$ counterpart, i.e., with $R/t$, where $R$ is the ring of integers in an algebraic closure of $\mathbf{F}_p ((t))$. |
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Mar 28 |
awarded | ● Editor |
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Mar 28 |
comment |
Why is a proper, affine morphism finite? Thanks! I corrected the proof. (This mistake was mine, not Olivier's.) |
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Mar 28 |
revised |
Why is a proper, affine morphism finite? Corrected mistake pointed by xuhan |
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Mar 28 |
answered | Why is a proper, affine morphism finite? |
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Mar 25 |
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When $R/(f)$ is regular? Youngsu: Sandor's (second) comment is what I had in mind. |
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Mar 24 |
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When $R/(f)$ is regular? The regular locus will be open by excellence, so a finite number of checks will suffice by quasi-compactness. However, this still involves looking at points... |
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Mar 22 |
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How many proofs of the Weil conjectures are there? Perhaps in connection to Kedlaya's proof, one should also mention Faltings' sketch of a crystalline version of Weil II (using convergent F-isocrystals, and logarithmic geometry) in the Grothendieck Festschrift. |
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Mar 17 |
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Regular subscheme of a projective limit of schemes I am glad it helped! |
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Mar 11 |
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Regular subscheme of a projective limit of schemes Here's a comment when $S$ and $S_i$ are local with rings $(A,m)$ and $(A_i,m_i)$ and all transition maps are local. In particular, $m/m^2 = \lim m_i/m_i^2$ (colimit). Then $Z$ is cut out by a regular sequence $\underline{f} = (f_1,...,f_r)$ that spans an $r$-dimensional subspace of $m/m^2$. Then $\underline{f}$ comes from a fixed $m_j$ and spans an $r$-dimensional subspace in $m_k/m_k^2$ for all $k \geq j$. The corresponding subschemes $Z_k = Z(\underline{f}) \subset S_k$ are then regular, and induce $Z$. Maybe the non-local case follows using the openness of regular locus (under qcqs hyp.) ? |
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Mar 10 |
awarded | ● Critic |
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Mar 5 |
awarded | ● Enthusiast |
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Feb 28 |
comment |
Comparison for formal local cohomology Actually, now it seems to me that this is always true for any noetherian ring $R$: each $H^i_m(R/I^n)$ is an artinian $R$-module, so the ML condition in the comment above is trivially verified (as any projective system of artinian $R$-modules is automatically ML). |
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Feb 28 |
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Comparison for formal local cohomology @kreck: I think the poster is completing along $I = 0$, so the topological space of $\widehat{X}$ is the same as that of $Spec(R/I)$. @Nick: The obstruction to the isomorphism you want lies in the $\lim^1$ of the projective system $\{ H^{j-1}_m(R/I^n) \}$. I don't see a reason why these should vanish in general. Have you tried looking at homogeneous ideals (so it becomes a question in projective geometry)? |
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Feb 23 |
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Betti numbers of Proper nonprojective varieties @LMN: No, I speak of the $\ell$-adic Betti numbers, defined via $\ell$-adic \'etale cohomology for any prime $\ell \neq p$, with $p$ being the characteristic of the field. Some standard facts are: these numbers are independent of the choice of $\ell$ (for proper smooth varieties), and agree with the classical ones if the variety comes via reduction modulo $p$ from something in characteristic $0$. (They do not, however, always equal the de Rham Betti numbers.) In particular, Junecue's result proves evenness in characteristic $0$ as well. |
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Feb 23 |
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Betti numbers of Proper nonprojective varieties The evenness of odd Betti numbers is true for proper smooth varieties over any algebraically closed field, including positive characteristic ones (by a recent paper by Junecue Suh). |
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Feb 22 |
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Is there excision for fppf cohomology? I don't think the second question has a positive answer. Take $X$ to be the spectrum of a dvr with uniformizer $t$, and set $Z \subset X$ to be the closed point defined by $t = 0$. Then the "kernel of multiplication of $t$" defines a sheaf F on the big fppf site of X-schemes. This sheaf also vanishes when restricted to the compliment of Z (I take this to mean "supported on Z"). However, I don't think $F$ is the pushforward of $F|_Z$ (by computing global sections). |
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Feb 21 |
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Is there excision for fppf cohomology? Assume X is noetherian. The hypothesis says that $\Omega^1_f$, whose formation commutes with base change, has vanishing fibres along all points in $Z'$. By Nakayama, $\Omega^1_f$ is supported on a closed subset that misses $Z'$ entirely. The \'etale locus $U \subset X$ of $f$ then contains $Z$. Since the cohomology supported along $Z$ is insensitive to passing to opens containing $Z$, we may replace $X$ with $U$ (and $X'$ by the preimage) to assume $f$ is \'etale. Now I didn't check this, but I believe the argument in Milne works for fppf cohomology. |
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Feb 4 |
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purity for finite flat group schemes Doesn't the $p$-power map $\mathbb{G}_a \to \mathbb{G}_a$ induce the Frobenius on $H^1(U,\mathbb{G}_a)$ (which is injective)? Also, aren't the categories of finite flat schemes over $U$ and $X$ equivalent in this example (via Hartog's)? |
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Jan 24 |
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deformation of stable curve Ah, yes, I missed the potential non-completeness, thanks! Is the claim clear in the henselian case without excellence assumption? |
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Jan 24 |
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deformation of stable curve If by a stable model you mean a flat family of stable curves, then the answer is "yes" as stable curves have unobstructed deformations (and all formal deformations are algebraic). Probably you can also make the generic fibre smooth by choosing the deformations carefully... |
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Jan 23 |
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showing that abelian varieties are de Rham *without* showing that they are crystalline In the case of abelian varieties with good reduction (or even p-divisible groups), Faltings give a direct proof (avoiding the machinery of almost etale extensions, close in spirit to Fontaine's proof) in his "Integral crystalline cohomology over very ramified rings" paper. |
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Jan 22 |
awarded | ● Scholar |
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Jan 22 |
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Automorphism groups of general type varieties Perfect, thank you! |
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Jan 21 |
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Automorphism groups of general type varieties Thanks, this is a nice construction! |
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Jan 21 |
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Automorphism groups of general type varieties Ah, thanks. I don't have access to Lang's paper, so I just wanted to check: do Lang's examples also have $K_X$ ample (as opposed to say big)? |
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Jan 21 |
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Automorphism groups of general type varieties I think the automorphism functor is representable by a finite type group scheme since $K_X$ is ample (look at the action on sections of $K_X^n$). Also, $\mathrm{Aut}(X)$ is always represented by a locally finitely presented group scheme for $X$ projective (look inside the Hilbert scheme of $X \times X$), so "reducedness" makes sense. However, I don't want to worry about these issues too much here, so I'm happy to take reducedness to mean that $H^0(X,T_X) = 0$. |
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Jan 21 |
asked | Automorphism groups of general type varieties |
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Jan 13 |
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A more general form of Grauert’s Theorem on Higher Direct Image Sheaves? @Piotr: What you suggest fails for $X = A \times A^t$, $Y = A^t$, $f$ the natural projection, and $F$ being the Poincare line bundle for an abelian variety $A$ of dimension $g$. Indeed, $R^i f_*(F) = 0$ for $i < g$, but the same is not true for the fibre over $0$. |
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Jan 12 |
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Applications for knowing the singularities parametrized by the boundary of a moduli space The output of de Jong's work on alterations of singularities (and subsequent work by various authors, such as Temkin and Gabber) is unrelated to moduli spaces. However, the method itself crucially relies on the ability to compactify the moduli space of smooth curves by adding (the very slightly singular) stable curves. |
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Jan 9 |
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Does the Riemann hypothesis for liftable varieties over a finite field imply the Riemann hypothesis for all varieties over a finite field There's a recent paper by Scholl explaining how to deduce general RH from the corresponding statement for smooth projective hypersurfaces in projective space (which are clearly liftable). His argument does not use Lefschetz pencils or the Fourier transform (but it does uses a theorem of Deligne from Weil II on local monodromy). |
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Jan 8 |
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Comparison of etale and formal etale cohomologies for l=p The cohomology (with constant coefficients) of $X_{et}$ and $\mathcal{X}_{et}$ coincide, and they both coincide with the cohomology of the special fibre $X_{0,et}$: by the topological invariance of the etale site for $\mathcal{X}$, and the proper base change theorem for $X$ (since $O_K$ is henselian with respect to its maximal ideal). The semistability assumption has no bearing on this discussion. |
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Jan 1 |
awarded | ● Commentator |
