Thesis of Hoàng Xuân Sính

"Gr-catégories"

Scans (pdf)

Chapter I
Chapter II
Chapter III (part a)
Chapter III (part b)



"Gr-catégories strictes", Acta Mathematica Vietnamica 3 (2), p. 47-59, 1978, scans (pdf)



"Catégories de Picard restreintes", Acta Mathematica Vietnamica 7 (1), p. 117-122, 1982, scans (pdf)



Grothendieck comments in a letter to R. Brown (5.5.82) :

"Also he [Quillen] had a promising approach to higher
K-invariants, which, he told me, was more or less
equivalent to a more computational transcription
of a somewhat abstract definition I had in mind
in terms of "enveloping n-Picard categories"
of a given additive category C, say, whose invariants
pi_i should yield the invariants K^i(C). (The
case n = 1 was worked out by a Vietnamese woman
student of mine around that time, Mme Sinh...)"

In "Récoltes et Semailles", Grothendieck writes :

"Un autre cas assez à part est celui de Mme Sinh,
que j'avais d'abord rencontrée à Hanoi en décembre
1967, à l'occasion d'un cours-séminaire d'un mois
que j'ai donné à l'université évacuée de Hanoi.
Je lui ai proposé l'année suivante son sujet de thèse.
Elle a travaillé dans les conditions particulièrement
difficiles des temps de guerre, son contact avec moi
se bornant à une correspondance épisodique.
Elle a pu venir en France en 1974/75 (à l'occasion
du congrès international de mathématiciens à
Vancouver), et passer alors sa thèse à Paris (devant
un jury présidé par Cartan, et comprenant de plus
Schwartz, Deny, Zisman et moi)."



Sinh's approach is also mentioned in a letter
of Grothendieck to Knudsen of 1973, see here, p. 41.



A subtle problem with traces and determinants in triangulated categories
has originally been discovered by Daniel Ferrand; cf. arxiv.



Cf. also P. Deligne, "Le déterminant de la cohomologie", §4,
in: Current trends in arithmetical algebraic geometry
(Arcata, Calif., 1985), AMS Contemp. Math. 67, p. 93-177, 1987.



Grothendieck reports on his stay in Vietnam, Dec. 1967, scans (pdf)



Illusie relates the following (Notices AMS, Oct. 2010, p. 1111 f.).

"Once, Grothendieck told me, it must have been in 1969:
«We have the K-groups defined by vector bundles,
but we could take vector bundles with a filtration of
length one (with quotient a vector bundle), vector bundles
with filtrations of length 2, length n, with associated
graded still vector bundles... Then you have operations
such as forgetting a step of the filtration, or taking
a quotient by one step. This way you get some simplicial
structure, which should deserve to be studied and
could yield interesting homotopy invariants.»"