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seen Jun 22 '13 at 6:39

Aug
20
awarded  Yearling
Jun
15
comment Schemes over a noetherian local ring and its completion
@ayanta: Doesn't the descent datum, at least a priori, entail specifying additional data over $\hat{R} \otimes_R \hat{R}$ (satisfying a cocycle condition) in addition to the isomorphism over $K \otimes_R \hat{R}$?
Jun
15
comment Schemes over a noetherian local ring and its completion
@ayanta: Thanks; of course, I meant (and should have said) to keep track of the ample line bundle. How does one get around having to worry about $\hat{R} \otimes_R \hat{R}$ while directly applying fpqc descent?
Jun
15
comment Schemes over a noetherian local ring and its completion
stacks.math.columbia.edu/tag/05E5
Jun
15
comment Schemes over a noetherian local ring and its completion
For any noetherian pair $(R,\mathfrak{m})$, one knows: giving a quasicoherent (qc) sheaf $F$ on $S := \mathrm{Spec}(R)$ is the same as giving a qc sheaf $\hat{F}$ on $\hat{S} := \mathrm{Spec}(\hat{R})$, a qc sheaf $F_U$ on $U = S - V(\mathfrak{m})$, and an isomorphism between the two over $U \times_S \hat{S}$. The same comment applies to "quasicoherent sheaves" replaced by "affine/quasiprojective morphisms."
Jun
2
comment are these functors exact?
$j_!$ does not preserve quasicoherent sheaves...
May
25
comment 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 $\mathbf{Z}^n$.
Apr
21
awarded  Enlightened
Apr
21
awarded  Nice Answer
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
Apr
20
comment 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.
Apr
19
revised Is the derived double dual of a quasi-coherent sheaf a sheaf?
more clarification
Apr
19
revised Is the derived double dual of a quasi-coherent sheaf a sheaf?
Added clarification about missing details
Apr
19
answered Is the derived double dual of a quasi-coherent sheaf a sheaf?
Apr
11
comment $k[[x]]$ 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.
Apr
10
answered $k[[x]]$ as a $(k[[x]])^p$ module for ugly fields
Apr
10
comment $k[[x]]$ 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 $m_n$).
Apr
10
comment $k[[x]]$ 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.
Apr
6
comment 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 $\mathbf{F}_p ((t))$.
Mar
28
awarded  Editor