On equivalence of derived categories

Let A be an Abelian category and B be a thick subcategory of A. Let D ^b(B) denote the derived category of cohomologically bounded chain complexes of objects in A and D _B ^b (A) denote the derived category of cohomologically bounded chain complexes of objects in A with cohomology in B. We give two if and only if conditions for equivalence of D(B) and D _B ^b (A), and we give an example where D ^b (B) and D _B ^b (A) are not equivalent.     

A note on bounded automorphisms of infinite graphs

LetX be a connected locally finite graph with vertex-transitive automorphism group. IfX has polynomial growth then the set of all bounded automorphisms of finite order is a locally finite, periodic normal subgroup ofAUT(X) and the action ofAUT(X) onV(X) is imprimitive ifX is not finite. IfX has infinitely many ends, the group of bounded automorphisms itself is locally finite and periodic.     

Derived Picard Groups of Finite-Dimensional Hereditary Algebras

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPic _k (A) is the group of triangle auto-equivalences of D> ^b( mod A) induced by two-sided tilting complexes. We study the group DPic _k (A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPic _k , as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic _k (A) on a certain infinite quiver ^irr. This representation is faithful when the quiver of A is a tree, and then DPic _k (A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic _k (A). When A is hereditary, DPic _k (A) coincides with the full group of k-linear triangle auto-equivalences of D^b( mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^b( mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.     

The picard group of a category of graded modules

For R a G-graded ring, we study Pic(R-gr), the group of isomorphism classes of autoequivalences of the category of graded left R-modules. For G infinite, this requires generalizing the classical sequences involving Pic(A), A a fc-algebra, to A a ring with local units. Then for G either finite or infinite, we characterize the inner automorphisms in some subgroups H of the automorphism group of the smash product R#PG and thus obtain some subgroups of Pic(R-gr).     

一类中心循环的有限p-群的自同构群的研究

本文研究了一类中心循环的有限p-群G的自同构群.利用在G的导群上作用平凡的自同构以及环上的辛群和正交群,确定了G的自同构群的结构,这推广了Bornand的相应结果.     

On the unipotence of autoequivalences of toric complete intersection Calabi–Yau categories

Consider the derived category of coherent sheaves, D ^b (X), on a compact Calabi–Yau complete intersection X in a toric variety. The scope of this work is to establish the (quasi-)unipotence of a class of elements in the group of autoequivalences, Aut(D ^b (X)). This is achieved by associating singularity categories, modelled by matrix factorizations, to the toric data. Each of these triangulated categories is equivalent to the derived category of coherent sheaves on X. The idea is then that, although the singularity categories share the group of autoequivalences, on each category there are elements in Aut(D ^b (X)), whose (quasi-)unipotence relations are easier to see than on the other categories.     

Categorical resolutions of a class of derived categories

We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D~b(A)and the subcategory K~b(P) of perfect complexes in D~b(A), by giving two classes of abelian categories A with enough projective objects such that D_(hf)~b(A) = K~b(P), and finding an example such that D_(hf)~b(A)≠K~b(P). We realize the bounded derived category D~b(A) as a Verdier quotient of the relative derived category D_C~b(A), where C is an arbitrary resolving contravariantly finite subcategory of A. Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension.We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dimT ∞ such that ~⊥T is finite, then D~b(modA) admits a categorical resolution; and that for a CM(Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant.     

Weakly power automorphisms of groups

An automorphism α of a group G is called a weakly power automorphism if it maps every non-periodic subgroup of G onto itself. The aim of this paper is to investigate the behavior of weakly power automorphisms. In particular, among other results, it is proved that all weakly power automorphisms of a soluble non-periodic group G of derived length at most 3 are power automorphisms, i.e. they fix all subgroups of G. This result is best possible, as there exists a soluble non-periodic group of derived length 4 admitting a weakly power automorphism, which is not a power automorphism.     

Exceptional Sequences Determined by their Cartan Matrix

We investigate complete exceptional sequences E=(E ₁,¨,E _n ) in the derived category D ^b of finite-dimensional modules over a canonical algebra, equivalently in the derived category D ^b X of coherent sheaves on a weighted projective line, and the associated Cartan matrices C(E)=( [E _i ],[E _j ]). As a consequence of the transitivity of the braid group action on such sequences we show that a given Cartan matrix has at most finitely many realizations by an exceptional sequence E, up to an automorphism and a multi-translation (E ₁,¨,E _n )(E ₁[i ₁],¨,E _n [i _n ]) of D ^b . Moreover, we determine a bound on the number of such realizations. Our results imply that a derived canonical algebra A is determined by its Cartan matrix up to isomorphism if and only if the Hochschild cohomology of A vanishes in nonzero degree, a condition satisfied if A is representation-finite.     

Automorphisms of -subshifts of finite type

Let (Σ, σ) be a ^d-subshift of finite type. Under a strong irreducibility condition (strong specification), we show that Aut(Σ) contains any finite group. For ^d-subshift of finite type without strong specification, examples show that topological mixing is not sufficient to give any finite group in the automorphism group in general: in particular, End(Σ) may be an abelian semigroup. For an example of a topologically mixing ²-subshift of finite type, the endomorphism semigroup and automorphism group are computed explicitly. This subshift has periodic-point permutations that do not extend to automorphisms.     

Singularities in derived categories

Let A be a finite dimensional algebra over an algebraically closed field k and let M and N be two complexes in the bounded derived category D^b(A) of finitely generated A-modules. Together with Alexander Zimmermann we have defined a notion of degeneration for derived categories. We say that M degenerates to N if there is a complex Z and an exact triangle N→M⊕Z→Z→N[1]. In this paper we define and study the type of singularity at every degeneration in the bounded derived categrory.     

On t-Structures and Tilting Theory

Abstract

Let A be a right coherent associative ring with unit. We introduce the notion of coendofinite complex and we associate to such a complex a t-structure in D ^b (mod A). We give conditions for the heart of that t-structure to be a module category. We also give some applications in connection with derived equivalent rings and tilting theory. In particular for a tilting module over a finite dimensional k-algebra, we get a reformulation of Brenner-Butler's theorem in terms of t-structures.     

Compact DG modules and Gorenstein DG algebras

When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras. When A is a regular DG algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul DG algebra. For a general regular DG algebra A, we give some equivalent conditions for the Gorensteiness. For a finite connected DG algebra A, we prove that D^c(A) and D^c(A ^op) admit Auslander-Reiten triangles if and only if A and A ^op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, D^c(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D_lf^b(A) and D_lf^b (A ^op) instead, when A is a regular DG algebra. This work was supported by the National Natural Science Foundation of China (Grant No. 10731070) and the Doctorate Foundation of Ministry of Education of China (Grant No. 20060246003)     

On derived equivalences of categories of sheaves over finite posets

A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D^b(X) the bounded derived category of sheaves of finite dimensional k-vector spaces over X. Two posets X and Y are said to be derived equivalent if D^b(X) and D^b(Y) are equivalent as triangulated categories.We give explicit combinatorial properties of X which are invariant under derived equivalence; among them are the number of points, the Z-congruency class of the incidence matrix, and the Betti numbers. We also show that taking opposites and products preserves derived equivalence.For any closed subset Y⊆X, we construct a strongly exceptional collection in D^b(X) and use it to show an equivalence D^b(X)?D^b(A) for a finite dimensional algebra A (depending on Y). We give conditions on X and Y under which A becomes an incidence algebra of a poset.We deduce that a lexicographic sum of a collection of posets along a bipartite graph S is derived equivalent to the lexicographic sum of the same collection along the opposite .This construction produces many new derived equivalences of posets and generalizes other well-known ones.As a corollary we show that the derived equivalence class of an ordinal sum of two posets does not depend on the order of summands. We give an example that this is not true for three summands.     

Automorphisms of hyperbolic domains inR n

The main result of this paper is a fixed-point theorem for projective automorphisms of a bounded strongly convex domain inR ⁿ . Several corollaries and applications are derived, especially on the dimension of the full automorphism group in the smooth case.     

Folding derived categories with Frobenius functors

Following the work [B. Deng, J. Du, Frobenius morphisms and representations of algebras, Trans. Amer. Math. Soc. 358 (2006) 3591-3622], we show that a Frobenius morphism F on an algebra A induces naturally a functor F on the (bounded) derived category D^b(A) of , and we further prove that the derived category D^b(A^F) of for the F-fixed point algebra A^F is naturally embedded as the triangulated subcategory D^b(A)^F of F-stable objects in D^b(A). When applying the theory to an algebra with finite global dimension, we discover a folding relation between the Auslander-Reiten triangles in D^b(A^F) and those in D^b(A). Thus, the AR-quiver of D^b(A^F) can be obtained by folding the AR-quiver of D^b(A). Finally, we further extend this relation to the root categories ?(A^F) of A^F and ?(A) of A, and show that, when A is hereditary, this folding relation over the indecomposable objects in ?(A^F) and ?(A) results in the same relation on the associated root systems as induced from the graph folding relation.     

Fatou–Bieberbach Domains as Basins of Attraction of Automorphisms Tangent to the Identity

We prove that there exists an automorphism of ℂ² tangent to the identity with a domain of attraction D to the origin, biholomorphic to ℂ², along a degenerate characteristic direction. Our automorphism of ℂ² is conjugate to a translation in D. We also prove the existence of a curve Γ, a biholomorphic copy of ℂ, entirely contained in the boundary of D. In our construction Γ is tangent to the z-axis in a neighborhood of the origin. The automorphisms we construct also fix the w-axis; therefore we obtain D, a Fatou–Bieberbach domain that does not intersect two biholomorphic copies of ℂ locally transversal at the origin.     

Power automorphisms of finitep-groups

For a finite groupG letA(G) denote the group of power automorphisms, i.e. automorphisms normalizing every subgroup ofG. IfG is ap-group of class at mostp, the structure ofA (G) is shown to be rather restricted, generalizing a result of Cooper ([2]). The existence of nontrivial power automorphisms, however, seems to impose restrictions on thep-groupG itself. It is proved that the nilpotence class of a metabelianp-group of exponentp ² possessing a nontrival power automorphism is bounded by a function ofp. The “nicer” the automorphism—the lower the bound for the class. Therefore a “type” for power automorphisms is introduced. Several examples ofp-groups having large power automorphism groups are given.     

On Generators of the Tame Automorphism Group of Free Metabelian Lie Algebras

We prove that the tame automorphism group TAut(M _n ) of a free metabelian Lie algebra M _n in n variables over a field k is generated by a single nonlinear automorphism modulo all linear automorphisms if n ≥ 4 except the case when n = 4 and char(k) ≠ 3. If char(k) = 3, then TAut(M ₄) is generated by two automorphisms modulo all linear automorphisms. We also prove that the tame automorphism group TAut(M ₃) cannot be generated by any finite number of automorphisms modulo all linear automorphisms.