Global Dimension

Let R be a non-necessarily commutative ring and M an R-module. We use the category σ[M] to introduce the notion of I-module who is a generalization of I-ring. It is well known that every artinian object of σ[M] is co-hopfian but the converse is not true in general. The aim of this paper is to characterize for a fixed ring, the left (right) R-modules M for which every co-hopfian object of σ[M] is artinian. We obtain some characterization of finitely generated I-modules over a commutative ring, faithfully balanced finitely generated I-modules, and left serial finitely generated I-modules over a duo-ring.

In this note we study rings with ascending and descending chain conditions on small right ideals. We show that these classes are closed under finite direct sum and quotients for every non-zero small ideal, but they are not closed under extensions or ...

We investigate the rings over which every countably generated module is pure-projective and generalize the theory of rings of pure global dimension zero. This class of rings is studied in connection with Mittag-Leffler modules. We also give a characterization of Mittag-Leffler abelian groups.

In this note we study rings with ascending and descending chain conditions on small right ideals. We show that these classes are closed under finite direct sum and quotients for every non-zero small ideal, but they are not closed under extensions or ...

We show that Golod-Shafarevich algebras can be homomorphically mapped onto infinite dimensional algebras with polynomial growth when mild assumptions about the number of relations of given degrees are introduced. This answers a question by Zelmanov. In the case that these algebras are finitely presented, we show that they can be mapped onto infinite dimensional algebras with at most quadratic growth. We then use an elementary construction to show that any sufficiently regular function n log n may occur as the growth function of an algebra.

We show that over every countable algebraically closed field ‫ދ‬ there exists a finitely generated ‫-ދ‬algebra that is Jacobson radical, infinite-dimensional, generated by two elements, graded and has quadratic growth. We also propose a way of constructing examples of algebras with quadratic growth that satisfy special types of relations. 2010 Mathematics Subject Classification. 16N40, 16P90. Algebras with linear growth were described by Small et al. . Bergman [2, p. 18] proved that algebras with growth function smaller than f (n) = n(n+1) 2

A result of Bergman says that the Jacobson radical of a graded algebra is homogeneous. It is shown that while graded Jacobson radical algebras have homogeneous elements nilpotent, it is not the case that graded algebras all of whose homogeneous elements are nilpotent are Jacobson radical. To contrast this, the following result of the author is slightly extended. Let R be a graded algebra generated in the degree one. If for every n, the n × n matrix algebra over R has all homogeneous elements nilpotent, then R is Jacobson radical.

The polynomial closure in D of a subset E ~ [( is the largest subset F ~ [( containing E such that Int(E, Dj ; Int(F, Dj, and It Is denoted by clD(Ej. We study the polynomial closure of ideals in several classes of domains, including essential domains and domains of strong Krull-type, and we relate it with the t-closure. For domains of Krull-type we also compute the Krull dimension of Int(D). .. Partiaity Bupported by a" NATO Collaborative Research Grant No; 970140.

In this paper we consider a partial action α of a polycyclic by finite group G on a ring R. We prove that if R is right noetherian, then the partial skew group ring R α G is also right noetherian. Extending the methods of Passman in Passman (Trans Am Math Soc 301:737-759, 1987), we obtain a description of the prime spectrum of R α G. The results obtained are applied to get bounds for the Krull dimension and the classical Krull dimension of R α G. Partial actions • Partial skew group rings • Noetherian rings • Prime ideals Mathematics Subject Classifications (2000) 16D25 • 16P40 • 16S35 • 16W22 Presented by Donald Passman.

Let R be a ring with identity and M be a left R-module. The module M is called strongly injective if whenever M + K = N with M ⊆ N , there exists a submodule K of K such that M ⊕ K = N. In this paper, we provide the various properties of the class of these modules. In particular, we prove that M is strongly injective if and only if it is semisimple injective. Moreover, we give new characterizations of semisimple rings and left V-rings via strongly injective modules. Finally, we show that every strongly injective module is strongly noncosingular.

Assume that K is an algebraically closed field and denote by KG(R) the Krull-Gabriel dimension of R, where R is a locally bounded K-category (or a bound quiver K-algebra). Assume that C is a tilted K-algebra and C, Č, C are the associated repetitive category, cluster repetitive category and cluster-tilted algebra, respectively. Our first result states that KG( C) = KG( Č) ≤ KG( C). Since the Krull-Gabriel dimensions of tame locally support-finite repetitive categories are known, we further conclude that KG( C) = KG( Č) = KG( C) ∈ {0, 2, ∞}. Finally, in the Appendix Grzegorz Bobiński presents a different way of determining the Krull-Gabriel dimension of the cluster-tilted algebras, by applying results of Geigle.

Assume that K is an algebraically closed field, R a locally support-finite locally bounded K-category, G a torsion-free admissible group of K-linear automorphisms of R and A = R/G. We show that the Krull-Gabriel dimension KG(R) of R is finite if and only if the Krull-Gabriel dimension KG(A) of A is finite. In these cases KG(R) = KG(A). We apply this result to determine the Krull-Gabriel dimension of standard selfinjective algebras of polynomial growth. Finally, we show that there are no super-decomposable pure-injective modules over standard selfinjective algebras of domestic type.

We prove that the finitistic dimension conjecture, the Gorenstein Symmetry Conjecture, the Wakamatsu-tilting conjecture and the generalized Nakayama conjecture hold for artin algebras which can be realized as endomorphism algebras of modules over representation-finite algebras. Note it is a question whether or not all artin algebras have such realizations. It was also shown that if every quasi-hereditary algebras has a

We investigate the relations between finitistic dimensions and restricted flat dimensions (introduced by Foxby [L.W. Christensen, H.-B. Foxby, A. Frankild, Restricted homological dimensions and Cohen-Macaulayness, J. Algebra 251 (1) (2002) 479-502]). In particular, we show the following result. (1) If T is a selforthogonal left module over a left noetherian ring R with the endomorphism ring A, then rfd(T A ) R are Artin algebras with the same identity such that, for each 0 i m -1, rad A i is a right ideal in A i+1 and rfd(A i+1 A i ) < ∞ (e.g., A i+1 A i is of finite projective dimension, or finite Gorenstein projective dimension, or finite Tor-bound dimension), then fdim( R R) < ∞ implies fdim( A A) < ∞. As applications, we disprove Foxby's conjecture [H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004) 167-193] on restricted flat dimensions by providing a counterexample and give a partial answer to a question posed by Mazorchuk [V. Mazorchuk, On finitistic dimension of stratified algebras, arXiv:math.RT/0603179, 6.4].

We introduce the notion of Krull super-dimension of supermodules over certain super-commutative Noetherian super-rings. We investigate how this notion relates to the notion of odd regular sequence from and how it behaves with respect to the transition to the graded and bigraded supermodules and super-rings associated with the original ones. We also apply these results to the super-dimension theory of superschemes of finite type and their morphisms. The notion of Krull dimension plays crucial role in the theory of commutative rings and in the algebraic geometry. Recall that the Krull dimension Kdim(R) of a commutative ring R is the supremum of heights of the prime ideals of R. If Kdim(R) < ∞, then Kdim(R) is just the largest nonnegative integer n such that there is a chain p 0 ⊆ . . . ⊆ p n of pairwise different prime ideals. Geometrically, Kdim(R) is nothing else but the dimension of the affine scheme Spec(R) (cf. [2], Chapter II, Example 3.2. In what follows all super-rings are supposed to be super-commutative, unless stated otherwise. If R is a super-ring, then each prime superideal P of R has a form p⊕R 1 , where p = P 0 is a prime ideal of the ring R 0 . In other words, the prime spectrum of R coincides with the prime spectrum of R 0 , and it does not detect the odd dimension of the affine superscheme SSpec(R). Therefore, a relevant notion of Krull super-dimension of a super-ring has to include an odd component also. Such a notion has been recently suggested in . Let R be a super-ring with Kdim(R 0 ) to be finite. Then a collection of odd elements y 1 , . . . , y s is called a system of odd parameters of R if Kdim(R 0 /Ann R0 (y 1 . . . y s )) = Kdim(R 0 ).

Let B be a generalized Koszul algebra over a finite dimensional algebra S. We construct a bimodule Koszul resolution of B when the projective dimension of S B equals 2. Using this we prove a Poincaré-Birkhoff-Witt (PBW) type theorem for a deformation of a generalized Koszul algebra. When the projective dimension of S B is greater than 2, we construct bimodule Koszul resolutions for generalized smash product algebras obtained from braidings between finite dimensional algebras and Koszul algebras, and then prove the PBW type theorem. The results obtained can be applied to standard Koszul Artin-Schelter Gorenstein algebras in the sense of Minamoto and Mori [MM]. However, if the projective dimension of the base ring, viewed as a graded module over the generalized Koszul algebra, is not larger than 2, then, as we will see in Section 1, the cohomology Key words and phrases. Generalized Koszul algebra, PBW deformation, Bimodule Koszul resolution, Artin-Schelter Gorenstein algebra. of the base ring will not complicate the cohomology of the generalized Koszul algebra too horribly. In this case, we can construct a bimodule projective resolution for the generalized Koszul algebra, and are able to prove the PBW theorem. Let S be a finite dimensional algebra, S M S a finite dimensional S-bimodule. Let T S (M ) be the tensor algebra over S. Assume that B = T S (M )/(R) is a generalized Koszul algebra where R ⊆ M ⊗ S M is a sub-S-bimodule. Take a linear map θ : R → S. Then U is a filtered algebra with the natural ascending filtration. If the projective dimension of S B is 2, we provide a method for constructing the bimodule Koszul resolution of B in Section 2. We then prove the following result (cf. Theorem 2.5.1). Theorem A. U is a PBW deformation of B if and only if θ is an S-bimodule morphism. Note that in the theorem above the generating relations of the algebra U do not involve the degree 1 part of the tensor algebra. We remind the reader that the PBW deformations appeared in [BG, PP, SW1, SW2, WW] are more general in terms of generating relations. However, the situation in the theorem above occurs very common, and it applies to many interesting algebras, such as symplectic reflection algebras (cf. [CBH, EG, No]), graded Hecke algebras (cf. [EG, RS, NW]) for arbitrary finite group rings, and Lusztig-type algebras in positive characteristic (cf. [Lu], also [SW2]),...,etc. Our result provides an alternative approach to those algebras. If the projective dimension of S B is greater than 2, then the method used in Section 2 no longer works. In what follows, we write B = ⊕ i≥0 B i for a generalized Koszul algebra. In order to construct the bimodule Koszul resolution for a Koszul algebra B with pdimS B > 2, we assume further that B i is free both as a left S-module and as a right S-module for all i ≥ 1. Note that B is generated by M over S. Since M is free as a right S-module, we may assume M = V ⊗ S as a free right S-module for some finite dimensional vector space V . The left action of S on M induces a braiding Ψ : S ⊗ V → V ⊗ S (for details, see Subsection 3.1). This yields a generalized smash product T (V )#S such that B is a quotient algebra of T (V )#S. Hence in Section 3, we consider the PBW deformations of a generalized smash product algebra obtained from a braiding between S and a classical Koszul algebra A. We provide a method for constructing the bimodule Koszul resolution of the generalized smash product algebra A#S in Subsection 3.3 (cf. Theorem 3.3.3), and then prove the following result (cf. Theorem 3.6.1, for the notions see Subsection 3.6), generalizing the corresponding results in [GGV, SW1, SW2, WW]. and only if the following conditions are satisfied: In [MM] Minamoto and Mori introduced the notion of Artin-Schelter Gorenstein algebra over nonsemisimple algebras. We study the PBW deformation of Koszul Artin-Schelter Gorenstein algebras in Section 4. We show that if B is a standard Artin-Schelter Gorenstein algebra, then B 0 is selfinjective and the projective dimension of B 0 viewed as a graded B-module is finite (cf. Proposition 4.1.2). This generalizes a similar result in [DW] for classical Koszul algebras. As the main result of Section 4, we prove the PBW theorem for a standard Koszul Artin-Schelter Gorenstein algebra of graded injective dimension 2 (cf. Theorem 4.2.2). Throughout the paper, is a fixed field. Unadorned ⊗ means ⊗ , and Hom means Hom . All the vector spaces, algebras and modules are defined over . Let B = B 0 ⊕ B 1 ⊕ • • • be a positively graded algebra assumed to be locally finite, that is, dim B i < ∞ for all i ≥ 0. The definition of the classic Koszulity of B requires B 0 to be semisimple [P, BGS]. This requirement is sometimes too strong. Many interesting algebras with properties similar to a Koszul algebra do not satisfy this requirement. For example, the recent study of the structure of Artin-Schelter regular algebras requires B 0 not to be a semisimple algebra [MM]. A generalization of the Koszulity is necessary. In fact, there exist several versions of a generalized Koszulity, see [W, GRS, Li, M]. In this paper, we use Woodcock's definition [W], that is, a positively graded algebra B is a right Koszul algebra if B i is projective both as a left S-module and as a right S-module for all i ≥ 1, and B 0 viewed as a right graded B-module has a graded projective resolution such that P n is generated in degree n for all n ≥ 0. If B 0 = , we say that B is classically Koszul. A left Koszul algebra is defined similarly. Woodcock proved in [W] that B is left Koszul if and only if it is right Koszul. Henceforth, we omit the prefix "left" or "right".

We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if Ext 1 R (M, T ) = 0 for all torsion modules T , and We show that a module M is Baer iff M is p-filtered where p is the preprojective component of the tame hereditary algebra R. We apply this to prove that the universal localization of a Baer module is projective in case we localize with respect to a complete tube. Using infinite dimensional tilting theory we then obtain a structure result showing that Baer modules are more complex then the (infinite dimensional) preprojective modules. In the final section, we give a complete classification of the Mittag-Leffler modules.

We calculate the Grothendieck group K0(A ), where A is an additive category, locally finite over a Dedekind ring and satisfying some additional conditions. The main examples are categories of modules over finite algebras and the stable homotopy category SW of finite CW-complexes. We show that this group is a direct sum of a free group arising from localizations of the category A and a group analogous to the groups of ideal classes of maximal orders. As a corollary, we obtain a new simple proof of the Freyd's theorem describing the group K0(SW). Introduction 1 1. Generalities 2 2. Localization and genera 5 2.1. Arithmetic case 10 3. K 0 , local case. 11 4. K 0 , global case 15 References 20 11 Lemma 1.1. Let A be a fully additive category, C be an object of A and Λ = End A C. The map A → A (C, A) induces an equivalence add C ∼ → proj-Λ, the category of finitely generated projective right Λ-modules. Proof. Note that every functor F : add C → C , where C is a fully additive category, is completely determined (up to isomorphism) by its values on C and on endomorphisms of C. As the functor A (C, ) maps C to Λ and induces an isomorphism End A C ∼ → End Λ Λ ≃ Λ, it only remains to apply the Yoneda's lemma.

We calculate the Grothendieck group $K_0(\cal A)$, where $\cal A$ is an additive category, locally finite over a Dedekind ring and satisfying some additional conditions. The main examples are categories of modules over finite algebras and the stable homotopy category $\mathsf{SW}$ of finite CW-complexes. We show that this group is a direct sum of a free group arising from localizations of the category $\cal A$ and a group analogous to the groups of ideal classes of maximal orders. As a corollary, we obtain a new simple proof of the Freyd&#39;s theorem describing the group $K_0(\mathsf{SW})$.

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This chapter illustrates a method of co-designing curricula for twentyfirst-century teaching and learning in higher education. The main goal of SLP is to put learners at the center of active collaboration in the decision-making processes of curriculum design. The SLP approach is an opportunity to create a facilitated space for teaching staff to reflect and generate collaborative solutions to think about teaching and learning in higher education in new ways.

Let f : A → B and g : A → C be two ring homomorphisms and let J and J be two ideals of B and C, respectively, such that f -1 (J) = g -1 (J ). The bi-amalgamation of A with (B, C) along (J, J ) with respect to (f, g) is the subring of B × C given by The aim of this paper is to characterize the global dimension of bi-amalgamated algebras over pure ideals.

The nth Weyl "algebra" over a ring R is defined as A,(R) =I+,, e1,x2, e2,...4,, enI, where the xi commute among themselves, the 0, commute among themselves, and Bixj-xjei = 6,. In the case where R is a field of characteristic zero, these algebras are intimately connected with the representation theory of enveloping algebras of solvable Lie algebras [ 1, 31, while in the case where R is a right noetherian ring they provide a very interesting class of examples of right noetherian rings. Scattered throughout the literature and folklore of noncommutative ring theory are various results concerning the Krull and global dimensions of Weyl algebras, usually over fields, but occasionally over more general coefficient rings. The earliest results appeared before the introduction of noncommutative Krull dimension and so concern global dimension. The first result in the literature was apparently an observation of Rinehart [20] that a more general theorem due to Hochschild, Kostant, and Rosenberg [ 1 l] established the extreme bounds IZ < gl.dim(A,(F)) < 2n, for any field F. Rinehart showed that gl.dim(A,(l;)) = 2n for any field F of positive characteristic, and that gl.dim(A r(F)) = 1 for any field F of characteristic zero. In the latter case, he also improved the upper bound for gl.dim(A,(F)) to 2n-1. The obvious conjecture, that gl.dim(A,(F)) = n for any field F of characteristic zero, was verified by Roos [22]. In the meantime, Hart [ 101 had shown that gl.dim(A r(Di)) = 2, where D, is the quotient division ring of the first Weyl algebra over a field of characteristic 334

Introduction. Labor potential constitutes an integral, important part of the potential of the company, being an attribute parameter, which predetermines the competitive success of the company either by accelerating it or leading to a complete failure in case of poor management. Purpose. to redefine the definition and role of the labor potential in the company, their characteristics, and the main principles of development. Clarify the most essential characteristics of the employees, who are the key elements of labor potential for the employer, and preferences of the management to different methods of treating them in case of poor performance. Results. Major characteristics of the labor potential were outlined. According to the survey provided at SMEs in Ukraine, employers put higher rank to professional skills, experience in the company, loyalty and dedication, prior experience in the sphere, qualifications obtained through formal education, ability to work in a team and cooperate. Less importance was assigned to psychological traits, temper, physical characteristics, health, physical strength, and marital status, family, moral principles. In the case of poor performance, 16% of the managers declared their desire to fire the worker but the real rate of termination of the employment amounted to 31.6% in 2018. Conclusions. Labor potential of the enterprise is a critical element of the company's potential due to the unique role in the economic activity, it can easily facilitate the increase of profit or otherwise lead a company to bankruptcy. It depends on a range of objective and subjective factors which include personal characteristics of the workers and their ability to cooperate and are strongly connected with material and information supply. Many of the components can be enhanced without the change of personnel composition through the training and self-improvement. But this will only happen if the company follows the correct principles and set distinct goals. Nevertheless, in the case of poor performance managers will consider layoffs with subsequent recruitment as a possible option with more than 30% of employees being fired during the year on average. Employee retention should be considered as strategic imperative and resources should be allocated for that.

We consider $R$-modules as module functors in the following way: if $M$ is a (left) $R$-module, let $\mathcal M$ be the functor of $\mathcal R$-modules defined by $\mathcal M(S) := S\otimes_R M$ for every $R$-algebra $S$. With the corresponding notion of dual functor, we prove that the natural morphism of functors $\mathcal M\to \mathcal M^{**}$ is an isomorphism. We give functorial characterizations of finitely generated projective modules, flat modules and flat Mittag-Leffler modules.

We determine the universal localization R Γ(S) at certain semiprime ideals S of a Noetherian ring R that is a finite extension of its center. For this class of semiprime ideals, R Γ(S) is Noetherian, which need not be true in the general case.

Elan Bhuyan, N. Jagannadham Research Scholar, Department of Mathematics, GIET University, Gunupur 2 Assistant Professor, Department of Mathematics, GIET University, Gunupur Abstract The purpose of this work is to build a graded algebra A = A(q1, q2, q3) with three shift parameters, q1, q2, and q3. By introducing a specific filtration connected with the dominance ordering among partitions, we verify the basic features of the algebra A, including commutativity and the Poincar e series. The Gordon filtration is a stratification that is characterised by a series of null conditions related with the partitions and the shift parameter qi. The elliptic algebra [EO] can be thought of as a smooth limit of our algebra. Specifically, the original algebra is built over an elliptic curve, whereas our algebra A is built over a degenerate CP1. Keyword: Algebra, Poincar. Elliptic Algebra

The fundamental theorem of projective geometry gives an algebraic representation of isomorphisms between projective geometries of dimension at least 3 over vector spaces and has been generalized in different ways. This note briefly presents some further generalizations which will be proved in the author's thesis. We introduce the notion of global-affine morphisms between projective lattice geometries. Our investigations result in a general partial representation of global-affine morphisms which yields a complete representation of global-affine homomorphisms between large classes of module-induced projective geometries by semilinear mappings between the underlying modules.

The study of rings and modules with homological criteria is fundamental to commutative algebra. Consider a commutative Noetherian ring R with identity (which need not be local) and a proper ideal a of R. In this paper, we develop a relative analogue of the theory of homological rings and modules. Specifically, we introduce the notions of a-relative regular, a-relative complete intersection, and a-relative Gorenstein rings and modules. By demonstrating some interactions between these types of rings and modules, we extend some classical results.

Let R be a commutative Noetherian ring with identity (not necessarily local) and a a proper ideal of R. We study the invariance of some classes of a-relative Cohen-Macaulay modules under pure ring homomorphisms and ring homomorphisms of finite flat dimension. Our results extend several results in the existing literature on homological modules.

In this paper, we establish the global analogues of some dualities and equivalences in local algebra by developing the theory of relative Cohen-Macaulay modules. Let R be a commutative Noetherian ring (not necessarily local) with identity and a a proper ideal of R. The notions of a-relative dualizing modules and a-relative big Cohen-Macaulay modules are introduced. With the help of a-relative dualizing modules, we establish the global analogue of the duality on the subcategory of Cohen-Macaulay modules in local algebra. Lastly, we investigate the behavior of the subcategory of a-relative Cohen-Macaulay modules and a-relative generalized Cohen-Macaulay modules under Foxby equivalence.

A new proof of variance-optimal hedging in incomplete time discrete markets Part I: Valuation Model Börge Thiel 1 in cooperation with Ulm University This article presents a new proof for the existence and uniqueness of a variance-optimal trading strategy to hedge square integrable claims in incomplete and time discrete markets provided the underlying price process fullls the mean-variance-tradeo condition (nondegeneracy condition) of Martin Schweizer. The variance-optimal hedging is described as the minimum problem of an energy functional and the existence and uniqueness of the variance-optimal trading strategy is done by the theorem of Lax-Milgram.

Abstract: In this paper we prove; If R is a left quasi-Noetherian ring,then every nil subring is nilpotent). Next we show that a commutative semi-prime quasi-Noetherian ring is Noetherian. Then we study the relationship between left Quasi-Noetherian and left Quasi-Artinian, in particular we prove that If R is a non-nilpotent left Quasi-Artinian ring. Then any left R-module is left Quasi-Artinian if and only if it is left Quasi-Noetherian. Finally we show that a commutative ring R is Quasi-Artinian if and only ifR is Quasi-Noetherian and every proper prime ideal of R is maximal.

One of the challenges in science teacher education program in universities and colleges of education around the world is preparing pre-service teachers to teach effectively from a global perspective. Adding a global dimension in a science classroom helps students to develop global knowledge and understanding, and a cosmopolitan spirit needed for the twenty-first century global citizenship. This paper discusses how Department of Education at Federal University Kashere, Nigeria, attempts to prepare its pre-service science teachers to add a global dimension in the classroom to prepare students for global competitiveness and global citizenship. The teacher education program adheres strictly to national and international standards and to best instructional practices.

Let D be a 2-dimensional regular local ring and let Q(D) denote the quadratic tree of 2-dimensional regular local overrings of D. We examine the Noetherian rings that are intersections of rings in Q(D). To do so, we describe the desingularization of projective models over D both algebraically in terms of the saturation of complete ideals and order-theoretically in terms of the quadratic tree Q(D).

Abstract. Skew polynomial rings have invited attention of mathematicians and various properties of these rings have been discussed. The nature of ideals (in particular prime ideals, minimal prime ideals, associated prime ideals), primary decomposition and Krull dimension have ...

Let R be a commutative noetherian ring. In this paper, we study specialization-closed subsets of Spec R. More precisely, we first characterize the specialization-closed subsets in terms of various closure properties of subcategories of modules. Then, for each nonnegative integer n we introduce the notion of n-wide subcategories of R-modules to consider the question asking when a given specialization-closed subset has cohomological dimension at most n.

In this article, we prove that every triangulated category with a heart (of a bounded t-structure) of homological dimension at most 1 is uniquely determined up to exact equivalence. The last section describes how to obtain a t-structure from a semiorthogonal decomposition with compatible t-structures. This will provide an associated heart of homological dimension at most 1 for any full strong exceptional sequence $E_1 , E_2$ with finite dimensional $\hom(E_1, E_2)$. As a consequence, a triangulated category admitting such an exceptional sequence is determined by $\dim \hom (E_1, E_2)$.

We associate a t t -structure to a family of objects in D ( A ) \boldsymbol {\mathsf {D}}(\mathcal {A}) , the derived category of a Grothendieck category A \mathcal {A} . Using general results on t t -structures, we give a new proof of Rickard’s theorem on equivalence of bounded derived categories of modules. Also, we extend this result to bounded derived categories of quasi-coherent sheaves on separated divisorial schemes obtaining, in particular, Beĭlinson’s equivalences.