The index in d-exact categories

Classifying exact categories via Wakamatsu tilting

Higher homological algebra was introduced by Iyama. It is also known as $n$-homological algebra where $n \geq 2$ is a fixed integer, and it deals with $n$-cluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequences with $n+2$ objects. This was recently formalised by Jasso in the theory of $n$-abelian categories. There is also a derived version of $n$-homological algebra, formalised by Geiss, Keller, and Oppermann in the theory of $( n+2 )$-angulated categories (the reason for the shift from $n$ to $n+2$ is that angulated categories have triangulated categories as the base case). We introduce torsion classes and t-structures into the theory of $n$-abelian and $( n+2 )$-angulated categories, and prove several results to motivate the definitions. Most of the results concern the $n$-abelian and $( n+2 )$-angulated categories ${\mathcal M}( \Lambda )$ and ${\mathcal C}( \Lambda )$ associated to an $n$-representation finite algebra $\Lambda$, as defined by Iyama and Oppermann. We characterise torsion classes in these categories in terms of closure under higher extensions, and give a bijection between torsion classes in ${\mathcal M}( \Lambda )$ and intermediate t-structures in ${\mathcal C}( \Lambda )$ which is a category one can reasonably view as the $n$-derived category of ${\mathcal M}( \Lambda )$. We hint at the link to $n$-homological tilting theory.

This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic [math] –theory. The main result goes as follows. Let [math] be an associative ring and let [math] be the localisation with respect to a set [math] of maps between finitely generated projective [math] –modules. Suppose that [math] vanishes for all [math] . View each map in [math] as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes [math] . Denote by [math] the thick subcategory generated by these complexes. Then the canonical functor [math] induces (up to direct factors) an equivalence [math] . As a consequence, one obtains a homotopy fibre sequence\n¶\n<math display="block">\n<mrow>\n<mi>K</mi>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>A</mi>\n<mo class="MathClass-punc">,</mo>\n<mi>σ</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mo class="MathClass-rel">→</mo>\n<mi>K</mi>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>A</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mo class="MathClass-rel">→</mo>\n<mi>K</mi>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>B</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n</mrow>\n</math>\n¶ (up to surjectivity of [math] ) of Waldhausen [math] –theory spectra.\n¶ In subsequent articles [?, ?] we will present the [math] – and [math] –theoretic consequences of the main theorem in a form more suitable for the applications to surgery. For example if, in addition to the vanishing of [math] , we also assume that every map in [math] is a monomorphism, then there is a description of the homotopy fiber of the map [math] as the Quillen [math] –theory of a suitable exact category of torsion modules.

In this paper, we study some important characteristics of the homological theory within infinity differential graded algebras (that denoted by DGA). Specifically, we explore definitions of Lie algebras and the simplicial homology theory of L_∞-algebras. Our primary objective is to improve the understanding of short exact sequence in simplicial homology of infinity algebras across specific classes of algebras. Additionally, we establish and prove the exact long sequence in simplicial homology of L_∞-algebras. Furthermore, we investigate the trace map and the inclusion map, clarifying their roles and connections within simplicial homology of L_∞-algebras. Notably, we show and apply the property of Morita equivalence in the context of simplicial homology of L_∞-algebras. Moreover, we establish that the trace map and the inclusion map are related inverse in simplicial homology of L_∞-algebras. Overall, this research improves our understanding of homological algebra within infinity differential graded algebras, focusing on their structures and practical applications.

As we now have tensor products in our armoury we can proceed with a definition of the homological invariants that are our basic object of study. The basic credit for their discovery, and equally for the creation of the powerful methods for investigating them (primarily the technique of resolutions and long exact sequences) is due to H. Cartan, Eilenberg and MacLane. The discipline they created, in a purely algebraic context, was given the name of “homological algebra” and is the subject of the widely known monographs [4, 5 and 19, Ch.I].

We show that categories of modules over a ring in homotopy type theory (HoTT) satisfy the internal versions of the AB axioms from homological algebra. The main subtlety lies in proving AB4, which is that coproducts indexed by arbitrary sets are left-exact. To prove this, we replace a set X with the strict category of lists of elements in X. From showing that the latter is filtered, we deduce left-exactness of the coproduct. More generally, we show that exactness of filtered colimits (AB5) implies AB4 for any abelian category in HoTT. Our approach is heavily inspired by Roswitha Harting’s construction of the internal coproduct of abelian groups in an elementary topos with a natural numbers object. To state the AB axioms, we define and study filtered (and sifted) precategories in HoTT. A key result needed is that filtered colimits commute with finite limits of sets. This is a familiar classical result but has not previously been checked in our setting. Finally, we interpret our most central results into an $\infty$ -topos $ {\mathscr{X}} $ . Given a ring R in $ {\tau_{\leq 0}({{\mathscr{X}}})} $ – for example, an ordinary sheaf of rings – we show that the internal category of R-modules in $ {\mathscr{X}} $ represents the presheaf which sends an object $ X \in {\mathscr{X}} $ to the category of $ (X{\times}R) $ -modules in ${\mathscr{X}} / X$ . In general, our results yield a product-preserving left adjoint to base change of modules over X. When X is 0-truncated, this left adjoint is the internal coproduct. By an internalisation procedure, we deduce left-exactness of the internal coproduct as an ordinary functor from its internal left-exactness coming from HoTT.

A locally coherent exact category is a finitely accessible additive category endowed with an exact structure in which the admissible short exact sequences are the directed colimits of admissible short exact sequences of finitely presentable objects. We show that any exact structure on a small idempotent-complete additive category extends uniquely to a locally coherent exact structure on the category of ind-objects; in particular, any finitely accessible category has the unique maximal and the unique minimal locally coherent exact category structures. All locally coherent exact categories are of Grothendieck type in the sense of Št’ovíček. We also discuss the canonical embedding of a small exact category into the abelian category of additive sheaves in connection with the locally coherent exact structure on the ind-objects, and deduce two periodicity theorems as applications.

In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over $A$ is isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

We give a construction of triangulated categories as quotients of exact categories where the subclass of objects sent to zero is defined by a triple of functors. This includes the cases of homotopy and stable module categories. These categories naturally fit into a framework of relative derived categories, and once we prove that there are decent resolutions of complexes, we are able to prove many familiar results in homological algebra.

Given a torsion pair $\boldsymbol{t}=(\mathcal{T},\mathcal{F})$ in a module category $R\text{-}\mathrm{Mod}$ we give necessary and sufficient conditions for the associated Happel–Reiten–Smalo t-structure in $\mathcal{D}(R)$ to have a heart $\mathcal{H}_{\boldsymbol{t}}$ which is a module category. We also study when such a pair is given by a 2-term complex of projective modules in the way described by Hoshino–Kato–Miyachi ([HKM]). Among other consequences, we completely identify the hereditary torsion pairs $\boldsymbol{t}$ for which $\mathcal{H}_{\boldsymbol{t}}$ is a module category in the following cases: i) when $\boldsymbol{t}$ is the left constituent of a TTF triple, showing that $\boldsymbol{t}$ need not be HKM; ii) when $\boldsymbol{t}$ is faithful; iii) when $\boldsymbol{t}$ is arbitrary and the ring $R$ is either commutative, semi-hereditary, local, perfect or Artinian. We also give a systematic way of constructing non-tilting torsion pairs for which the heart is a module category generated by a stalk complex at zero.

The cyclic homology of an exact category

Sheaves on moment graphs and a localization of Verma flags

Let I be an ideal of a commutative ring R and M an R-module. It is well known that the I-adic completion functor ΛI defined by ΛI(M) = lim←tM/ItM is an additive exact covariant functor on the category of finitely generated R-modules, provided R is Noetherian. Unfortunately, even if R is Noetherian, ΛI is neither left nor right exact on the category of all R-modules. Nevertheless, we can consider the sequence of left derived functors {LIi} of ΛI, in which LI0 is right exact, but in general LI0 ≠ ΛI. Therefore the computation of these functors is in general very difficult. For the case that R is a local Noetherian ring with the maximal ideal [mfr ] and I is generated by a R-regular sequence, Matlis proved in [9, 10] thatwhere D(−) = HomR(−; E(R/[mfr ])) is the Matlis dual functor, and thatIn [18, 5] A.-M. Simon shows that LIo(M) = M and LIi(M) = 0 for i > 0, provided that M is complete with respect to the I-adic topology.Later, Greenlees and May [3] using the homotopy colimit, or telescope, of the cochain of Koszul complexes to define so-called local homology groups of a module M (over a commutative ring R) bywhere x is a finitely generated system of I. Then they showed, under some conditions on x which are satisfied when R is Noetherian, that HI[bull ](M) ≅ LI[bull ](M). Recently, Tarrío, López and Lipman [1] have presented a sheafified derived-category generalization of Greenlees–May results for a quasi-compact separated scheme. The purpose of this paper is to study, with elementary methods of homological and commutative algebra, local homology modules for the category of Artinian modules over Noetherian rings.

This is an introductory text on the theory of modules over rings or, equivalently, over algebras, addressed to beginning graduate students. It stresses the importance of the categorical perspective and the use of homological tools. Modules are studied both from a classical point of view and a categorical one. For this purpose, rudiments of category theory, homological algebra and representations of quivers are introduced and applied in the study of module categories. After introducing such fundamental tools as the Hom functor and the tensor product comes the study of projective, injective and flat modules. The classical results, such as the classification of modules over principal ideal domains, the Jordan-Hölder, Wedderburn-Artin, and Morita theorems, are proven, as well as the Eilenberg-Watts theorems. Radicals (of modules, algebras and categories) are thoroughly studied. Homological invariants, such as the extension and torsion modules or the homological dimensions are introduced using derived functors and are applied in module theory. The book stresses the construction and analysis of examples.