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Tokyo-Nagoya Algebra Seminar
Seminar information archive ～06/06｜Next seminar｜Future seminars 06/07～
| Organizer(s) | Noriyuki Abe, Aaron Chan, Osamu Iyama, Yasuaki Gyoda, Hiroyuki Nakaoka, Ryo Takahashi |
|---|
2026/05/27
16:00-17:30 Online
Yuki Mizuno (Utrecht University)
Bondal–Orlov’s reconstruction theorem in noncommutative projective geometry (Japanese)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Yuki Mizuno (Utrecht University)
Bondal–Orlov’s reconstruction theorem in noncommutative projective geometry (Japanese)
[ Abstract ]
The (derived) category of coherent sheaves on a scheme encodes rich
information about the underlying geometry. P. Gabriel showed that for
noetherian schemes X and Y, if Coh X and Coh Y are equivalent as abelian
categories, then X and Y are isomorphic. Furthermore, A. Bondal and D.
Orlov proved that for smooth projective schemes X and Y with
(anti-)ample canonical bundles, if D^b(Coh X) and D^b(Coh Y) are
equivalent as triangulated categories, then X and Y are isomorphic. On
the other hand, J.-P. Serre showed that the category of coherent sheaves
on a projective scheme can be described as the quotient category of
finitely generated graded modules over the homogeneous coordinate ring
by the subcategory of torsion modules. Motivated by the results of
Gabriel and Serre, the quotient category of finitely generated graded
modules over a (not necessarily commutative) graded ring by the
subcategory of torsion modules is called a noncommutative projective scheme.
In this talk, I will present an analogue of Bondal–Orlov’s
reconstruction theorem for noncommutative projective geometry.
Furthermore, if time permits, I will discuss recent progress on the
study of the derived autoequivalence groups of noncommutative projective
schemes. Specifically, I will mention a structure result for the derived
autoequivalence groups of certain noncommutative projective planes.
ミーティング ID: 828 6882 8074
パスコード: 131261
[ Reference URL ]The (derived) category of coherent sheaves on a scheme encodes rich
information about the underlying geometry. P. Gabriel showed that for
noetherian schemes X and Y, if Coh X and Coh Y are equivalent as abelian
categories, then X and Y are isomorphic. Furthermore, A. Bondal and D.
Orlov proved that for smooth projective schemes X and Y with
(anti-)ample canonical bundles, if D^b(Coh X) and D^b(Coh Y) are
equivalent as triangulated categories, then X and Y are isomorphic. On
the other hand, J.-P. Serre showed that the category of coherent sheaves
on a projective scheme can be described as the quotient category of
finitely generated graded modules over the homogeneous coordinate ring
by the subcategory of torsion modules. Motivated by the results of
Gabriel and Serre, the quotient category of finitely generated graded
modules over a (not necessarily commutative) graded ring by the
subcategory of torsion modules is called a noncommutative projective scheme.
In this talk, I will present an analogue of Bondal–Orlov’s
reconstruction theorem for noncommutative projective geometry.
Furthermore, if time permits, I will discuss recent progress on the
study of the derived autoequivalence groups of noncommutative projective
schemes. Specifically, I will mention a structure result for the derived
autoequivalence groups of certain noncommutative projective planes.
ミーティング ID: 828 6882 8074
パスコード: 131261
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
