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From: Guillermo Cortinas
To: marco.schlichting@uniessen.de
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Subject: HN and blowing up squares
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Marco: When still in Valladolid, I looked at Thomason's and
GilletSoul=E9's papers and got fairly convinced that any theory of
Waldhausen categories satisfying additivity, fibration and
approximation (Thomason's variants) has descent for blowups along
regular immersions. Then I turned to Keller's papers, which prove
that the cyclic theories enjoy similar properties; however the setup
is different: instead of cofibrations and weak equivalences between
objects you have a DG exact category A_0 and a DG exact subcategory A_1
whose objects which are equivalent to zero. The philosophy, I suppose, is
that replacing the Frobenius pair Z^0A=3D(Z^0A_0,Z^0A_1) by the
corresponding Waldhausen category WZ^0A of def. 11.1 of your paper (which
by the way came to my attention only today; why didn't you tell me to look
at it in connexion with this!), every needed result comes as a consequence =
of
what Keller proves. Now the problem is, if I know that the hypothesis on
the Thomason/Waldhausen side of the problem are satisfied,
does it follow that those of the corresponding Keller/Schlichting theorem
are too? For example, if $vA\subset wA$ are two choices of weak
equivalences for the exact DGFrobenius pair $A$, is it true that=20
$TvA^w\to TvA\to TwA$ is an exact sequence of triangulated categories?
It would appear so, but I haven't found this stated
explicitly anywhere, (probably because they are too obvious?).
Similar questions apply to the remaining needed results. It would be nice
to have a precise general statement which formalizes the "anything true
for the Waldhausen/Thomason setting remains true in the
Keller/Schilichting setting" philosophy.=20
Also: I wanted to check that HC has covariant functoriality for=20
perfectprojective maps, but I do not recall how this is defined in
Thomason's setting (volume III of the Grothendieck Festschrift, where the
ThomasonTrobaugh paper is, is missing from the BA library). I suppose
that for the cyclic theories one should somehow use the Extended
Functoriality of 1.14, but do not see exactly how.
Finally: I looked at FriedlanderSuslinVoevodsky's book, because I had=20
understood from what you had said that the "same topos" mystery was solved
there. The only thing I found is this statement that any abstract blowup
f with smooth target is of the form g=3Dg'f where g is the composite of a
finite sequence of blowups along smooth centers.
Is that it? I mean, as far as I understand what we want is that a
Nisnevich sheaf on Sch/k which has descent for blowups along regular
immersions, is a cdh sheaf on Sm/k.
Did Cisinski say that the latter assertion follows from the previous one?
Does it? =20
Willie.
=

Guillermo Corti\~nas
Filiations: 1. Departamento de Matem\'atica
Facultad de Cs. Exactas y Naturales
Universidad de Buenos Aires
=20
2. Consejo Nacional de Investigaciones
Cient\'\i ficas y T\'ecnicas.=20
Address: Departamento de Matem\'atica
Ciudad Universitaria, Pabell\'on 1
(1428) Buenos Aires
Argentina =20
email: gcorti@dm.uba.ar
tel/fax: 541145763335
webpage: mate.dm.uba.ar/~gcorti
=20