151,407 research outputs found
Symmetric approximation sequences, Beilinson-Green algebras and derived equivalences
In this paper, we will consider a class of locally -Beilinson-Green
algebras, where is an infinite admissible set of the integers, and show
that symmetric approximation sequences in -exangulated categories give rise
to derived equivalences between quotient algebras of locally
-Beilinson-Green algebras in the principal diagonals modulo some
factorizable ghost and coghost ideals by the locally finite tilting sets of
complexes. Then we get a class of derived equivalent algebras that have not
been obtained by using previous techniques. From higher exact sequences which
are generalizations of higher split sequences, we obtain derived equivalences
between subalgebras of endomorphism algebras by constructing tilting complexes,
which generalizes Chen and Xi's result for exact sequences. From a given
derived equivalence, we get derived equivalences between locally
-Beilinson-Green algebras of semi-Gorenstein modules. Finally, from given
graded derived equivalences of group graded algebras, we get derived
equivalences between associated Beilinson-Green algebras of group graded
algebras.Comment: Welcome to give comments. This is a revised version, some errors are
corrected and n-extriangulated categories are considere
Higher order derived functors and the Adams spectral sequence
Classical homological algebra considers chain complexes, resolutions, and
derived functors in additive categories. We describe "track algebras in
dimension n", which generalize additive categories, and we define higher order
chain complexes, resolutions, and derived functors. We show that higher order
resolutions exist in higher track categories, and that they determine higher
order Ext-groups. In particular, the E_m-term of the Adams spectral sequence
(m<n+3) is a higher order Ext-group, which is determined by the track algebra
of higher cohomology operations.Comment: To appear in J. Pure & Appl. Algebr
Little finitistic dimensions and generalized derived categories
In this paper, we introduced a generalization of a derived category, which is
called -derived category and denoted by , of a given ring for
each . The -derived category of a ring is
proved to be very closely connected with its left little finitistic dimension.
We also introduce and investigate the notions of -exact sequences,
-projective (resp., -injective) modules and -exact complexes. In
particular, we characterize the left little finitistic dimensions in terms of
all above notions. Besides, the -global dimension -gldim of is
introduced and investigated. Finally, we build a connection of the classical
derived categories and -derived categories
Triangulated categories of periodic complexes and orbit categories
We investigate the triangulated hull of the orbit categories of the perfect
derived category and the bounded derived category of a ring concerning the
power of the suspension functor. It turns out that the triangulated hull will
correspond to the full subcategory of compact objects of certain triangulated
categories of periodic complexes. This specializes to Stai and Zhao's result
when the ring is a finite dimensional algebra with finite global dimension over
a field. As the first application, if are flat algebras over a
commutative ring and they are derived equivalent, then the corresponding
derived categories of -periodic complexes are triangle equivalent. As the
second application, we get the periodic version of the Koszul duality.Comment: 22 page
Derived sections of Grothendieck fibrations and the problems of homotopical algebra
The description of algebraic structure of n-fold loop spaces can be done
either using the formalism of topological operads, or using variations of
Segal's -spaces. The formalism of topological operads generalises well
to different categories yielding such notions as -algebras in
chain complexes, while the -space approach faces difficulties.
In this paper we discuss how, by attempting to extend the Segal approach to
arbitrary categoires, one arrives to the problem of understanding "weak"
sections of a homotopical Grothendieck fibration. We propose a model for such
sections, called derived sections, and study the behaviour of homotopical
categories of derived sections under the base change functors. The technology
developed for the base-change situation is then applied to a specific class of
"resolution" base functors, which are inspired by cellular decompositions of
classifying spaces. For resolutions, we prove that the inverse image functor on
derived sections is homotopically full and faithful.Comment: 50 pages, improved in line with referee remark
Noncommutative crepant resolutions of singularities via Fukaya categories
We compute the wrapped Fukaya category of a cylinder
relative to a divisor of points, proving a mirror
equivalence with the category of perfect complexes on a crepant resolution
(over ) of the singularity . Upon
making the base-change , we obtain the derived category of any
crepant resolution of the singularity given by the equation . These categories inherit braid group actions via the action on
of the mapping class group of fixing .Comment: 22 pages, 7 figure
Boij-S\"oderberg Conjectures for Differential Modules
Boij-S\"oderberg theory gives a combinatorial description of the set of Betti
tables belonging to finite length modules over the polynomial ring . We posit that a similar combinatorial description can be given
for analogous numerical invariants of graded differential -modules, which
are natural generalizations of chain complexes. We prove several results that
lend evidence in support of this conjecture, including a categorical pairing
between the derived categories of graded differential -modules and coherent
sheaves on and a proof of the conjecture in the case where
.Comment: 21 page
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