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Noncommutative localisation in algebraic K-theory II
No full textIn [Amnon Neeman, Andrew Ranicki, Noncommutative localisation in algebraic K-theory I, Geom. Topol. 8 (2004) 1385–1425] we proved a localisation theorem in the algebraic K-theory of noncommutative rings. The main purpose of the current article is to express the general theorem of the previous paper in a more user-friendly fashion, in a way more suitable for applications. In the process we compare our result to the existing theorems in the literature, showing how the previous paper improves all the existing results. It should be pointed out that there have been two very interesting recent preprints on related topics. The reader is referred to the beautiful papers of Krause [Henning Krause, Cohomological quotients and smashing localizations
Strong Generators in Dperf(X) for Schemes with a Separator
No full textThis work extends the result from Amnon Neeman regarding strong generators in Dperf(X), from X being a quasicompact, separated scheme to X being quasicompact, quasiseparated
scheme that admits a separator with some extra conditions. Neeman's result states a necessary and su cient condition for Dperf(X) to be regular.
Together with being proper over a noetherian commutative ring, the conditions above give an interesting description for when an R-linear functor H is representable
A weak GAGA statement for arbitrary morphisms
No full textLet
f
:
X
→
Y
f:X \to Y
be an arbitrary morphism of schemes of finite type over
C
{\mathbf {C}}
, and let
f
an
{f^{{\text {an}}}}
be the associated map of analytic spaces. Let
S
\mathcal {S}
be a coherent sheaf on
X
X
. Then
(
f
∗
S
)
an
→
f
∗
an
(
S
an
)
{({f_*}\mathcal {S})^{{\text {an}}}} \to f_*^{{\text {an}}}({\mathcal {S}^{{\text {an}}}})
is injective.</p
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