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1
CHRISTENSEN, J. DANIEL, and MARK HOVEY. "Quillen model structures for relative homological algebra." Mathematical Proceedings of the Cambridge Philosophical Society 133, no. 2 (September 2002): 261–93. http://dx.doi.org/10.1017/s0305004102006126.
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An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category [Ascr ] is exactly the information needed to do homological algebra in [Ascr ]. The main result is that, under weak hypotheses, the category of chain complexes of objects of [Ascr ] has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the ‘pure derived category’ of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and co-homology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories.
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NEHANIV, CHRYSTOPHER LEV. "ALGEBRAIC CONNECTIVITY." International Journal of Algebra and Computation 01, no. 04 (December 1991): 445–71. http://dx.doi.org/10.1142/s0218196791000316.
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Let [Formula: see text] be a type of algebra in the sense of universal algebra. By defining singular simplices in algebras and emulating singular [co] homology, we introduce for each variety, pseudo-variety, and divisional class V of type [Formula: see text], a homology and cohomology theory which measure the V-connectivity of type-[Formula: see text] algebras. Intuitively, if we were to think of an algebra as a space and subalgebras which lie in V as simplices, then V-connectivity describes the failure of subalgebras to lie in V, i.e., it describes the "holes" in this space. These [co]homologies are functorial on the class of type-[Formula: see text] algebras and are characterized by a natural topological interpretation. All these notions extend to subsets of algebras. One obtains for this algebraic connectivity, the long exact sequences, relative [co]homologies, and the analogues of the usual [co]homological notions of the algebraic topologists. In fact, we show that the [co]homologies are actually the same as the simplicial [co]homology of simplicial complexes that depend functorially on the algebras. Thus the connectivities in question have a natural geometric meaning. This allows the wholesale import into algebra of the concepts, results, and techniques of algebraic topology. In particular, functoriality implies that the [co]homology of a pair of algebras A ⊆ B is an invariant of the position of A in B. When one V contains another, we obtain relationships between the [co] homology theories in the form of long exact sequences. Furthermore for finite algebras, V-[co]homology is effectively computable if membership in V is. We obtain an analogue of the Poincaré lemma (stating that subsets of an algebra in V are V-homologically trivial), extremely general guarantees of the existence of subsets with non-trivial V-homology for algebras not in V, long exact V-homotopy sequences, as well as analogues of the powerful Eilenberg-Zilber theorems and Kunneth theorems in the setting of V-connectivity for V a variety or pseudo-variety. Also in the more general case of any divisionally closed V, we construct the long exact Mayer-Vietoris sequences for V-homology. Results for homomorphisms include an algebraic version of contiguity for homomorphisms (which implies they are V-homotopic) and a proof that V-surmorphisms are V-homotopy equivalences. If we allow the divisional classes to vary, then algebraic connectivity may be viewed as a functor from the category of pairs W ⊆ V of divisional classes of [Formula: see text]-algebras with inclusions as morphisms' to the category of functors from pairs of [Formula: see text]-algebras to pairs of simplicial complexes. Examples show the non-triviality of this theory (e.g. "associativity tori"), and two preliminary applications to semigroups are given: 1) a proof that the group connectivity of a torsion semigroup S is homotopy equivalent to a space whose points are the maximal subgroups of S, and 2) an aperiodic connectivity analogue of the fundamental lemma of complexity.
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Antosz, Jakub, and Stanislaw Betley. "Homological Algebra in the Category of Γ-Modules." Communications in Algebra 33, no. 6 (May 2005): 1913–36. http://dx.doi.org/10.1081/agb-200063342.
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Pasku, Elton. "On some homotopical and homological properties of monoid presentations." Semigroup Forum 76, no. 3 (January 3, 2008): 427–68. http://dx.doi.org/10.1007/s00233-007-9037-1.
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Rennemo, Jørgen Vold. "The homological projective dual of." Compositio Mathematica 156, no. 3 (January 17, 2020): 476–525. http://dx.doi.org/10.1112/s0010437x19007772.
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We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$. Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$. The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating $D^{b}(X)$ into a derived category of factorisations on a Landau–Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi–Yau 3-folds have equivalent derived categories.
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Betley, Stanisław. "Stable derived functors, the Steenrod algebra and homological algebra in the category of functors." Fundamenta Mathematicae 168, no. 3 (2001): 279–93. http://dx.doi.org/10.4064/fm168-3-4.
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Lekili, Yankı, and Alexander Polishchuk. "Homological mirror symmetry for higher-dimensional pairs of pants." Compositio Mathematica 156, no. 7 (June 18, 2020): 1310–47. http://dx.doi.org/10.1112/s0010437x20007150.
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Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $\mathbb{CP}^{n}$, for $k\geqslant n$, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $\mathbb{C}P^{n}$ ($n$-dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $x_{1}x_{2}\ldots x_{n+1}=0$. By localizing, we deduce that the (fully) wrapped Fukaya category of the $n$-dimensional pair of pants is equivalent to the derived category of $x_{1}x_{2}\ldots x_{n+1}=0$. We also prove similar equivalences for finite abelian covers of the $n$-dimensional pair of pants.
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Assem, Ibrahim, and Flávio Ulhoa Coelho. "Complete slices and homological properties of tilted algebras." Glasgow Mathematical Journal 36, no. 3 (September 1994): 347–54. http://dx.doi.org/10.1017/s0017089500030950.
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It is reasonable to expect that the representation theory of an algebra (finite dimensional over a field, basic and connected) can be used to study its homological properties. In particular, much is known about the structure of the Auslander-Reiten quiver of an algebra, which records most of the information we have on its module category. We ask whether one can predict the homological dimensions of a module from its position in the Auslander-Reiten quiver. We are particularly interested in the case where the algebra is a tilted algebra. This class of algebras of global dimension two, introduced by Happel and Ringel in [7], has since then been the subject of many investigations, and its representation theory is well understood by now (see, for instance, [1], [7], [8], [9], [11], [13]).In this case, the most striking feature of the Auslander-Reiten quiver is the existence of complete slices, which reproduce the quiver of the hereditary algebra from which the tilted algebra arises. It follows from well-known results that any indecomposable successor (or predecessor) of a complete slice has injective (or projective, respectively) dimension at most one, from which one deduces that a tilted algebra is representation-finite if and only if both the projective and the injective dimensions of almost all (that is, all but at most finitely many non-isomorphic) indecomposable modules equal two (see (3.1) and (3.2)). On the other hand, the authors have shown in [2, (3.4)] that a representation-infinite algebra is concealed if and only if both the projective and the injective dimensions of almost all indecomposable modules equal one (see also [14]). This leads us to consider, for tilted algebras which are not concealed, the case when the projective (or injective) dimension of almost all indecomposable successors (or predecessors, respectively) of a complete slice equal two. In order to answer this question, we define the notions of left and right type of a tilted algebra, then those of reduced left and right types (see (2.2) and (3.4) for the definitions).
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Song, Weiling, Tiwei Zhao, and Zhaoyong Huang. "Homological Dimensions Relative to Special Subcategories." Algebra Colloquium 28, no. 01 (January 20, 2021): 131–42. http://dx.doi.org/10.1142/s1005386721000122.
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Let [Formula: see text] be an abelian category, [Formula: see text] an additive, full and self-orthogonal subcategory of [Formula: see text] closed under direct summands, [Formula: see text] the right Gorenstein subcategory of [Formula: see text] relative to [Formula: see text], and [Formula: see text] the left orthogonal class of [Formula: see text]. For an object [Formula: see text] in [Formula: see text], we prove that if [Formula: see text] is in the right 1-orthogonal class of [Formula: see text], then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical; if the [Formula: see text]-projective dimension of [Formula: see text] is finite, then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical. We also prove that the supremum of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension and that of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension coincide. Then we apply these results to the category of modules.
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Thomas, Weichert. "A homological characterization of the category of socle projective modules." Communications in Algebra 18, no. 10 (January 1990): 3547–63. http://dx.doi.org/10.1080/00927879008824090.
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BARAKAT, MOHAMED, and MARKUS LANGE-HEGERMANN. "AN AXIOMATIC SETUP FOR ALGORITHMIC HOMOLOGICAL ALGEBRA AND AN ALTERNATIVE APPROACH TO LOCALIZATION." Journal of Algebra and Its Applications 10, no. 02 (April 2011): 269–93. http://dx.doi.org/10.1142/s0219498811004562.
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In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring R, i.e. a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over R. For a finitely generated maximal ideal 𝔪 in a commutative ring R, we show how solving (in)homogeneous linear systems over R𝔪 can be reduced to solving associated systems over R. Hence, the computability of R implies that of R𝔪. As a corollary, we obtain the computability of the category of finitely presented R𝔪-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a byproduct, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings, we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm.
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ALB, ALINA, and MIHAIL URSUL. "A FEW HOMOLOGICAL CHARACTERIZATIONS OF COMPACT SEMISIMPLE RINGS." Journal of Algebra and Its Applications 04, no. 05 (October 2005): 539–49. http://dx.doi.org/10.1142/s0219498805001411.
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Fix any compact ring R with identity. We associate to R the following categories of topological R-modules: (i) R𝔇 (𝔇R) the category of all discrete topological left (right) R-modules; (ii) Rℭ (ℭR) the category of all compact left (right) R-modules. We have introduced the following notions (analogous with classical notions of module theory): (i) the tensor product [Formula: see text] of A ∈ ℭR and B ∈Rℭ ([Formula: see text] has a structure of a compact Abelian group); (ii) a topologically semisimple module; (iii) a compact topologically flat module. We give a characterization of compact semisimple rings by using of flat modules.
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Pascaleff, James, and Nicolò Sibilla. "Topological Fukaya category and mirror symmetry for punctured surfaces." Compositio Mathematica 155, no. 3 (March 2019): 599–644. http://dx.doi.org/10.1112/s0010437x19007073.
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In this paper we establish a version of homological mirror symmetry for punctured Riemann surfaces. Following a proposal of Kontsevich we model A-branes on a punctured surface$\unicode[STIX]{x1D6F4}$via the topological Fukaya category. We prove that the topological Fukaya category of$\unicode[STIX]{x1D6F4}$is equivalent to the category of matrix factorizations of a certain mirror LG model$(X,W)$. Along the way we establish new gluing results for the topological Fukaya category of punctured surfaces which are of independent interest.
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WEI, REN, and ZHONGKUI LIU. "A QUILLEN MODEL STRUCTURE APPROACH TO HOMOLOGICAL DIMENSIONS OF COMPLEXES." Journal of Algebra and Its Applications 13, no. 03 (October 31, 2013): 1350106. http://dx.doi.org/10.1142/s0219498813501065.
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In this paper, we first give an alternative characterization of the derived functor Ext via the Quillen model structure on the category of complexes induced by a given cotorsion pair [Formula: see text] in the category of modules, then based on this, we consider homological dimensions of complexes related to [Formula: see text]. As applications, we extend Gorenstein projective dimension of homologically bounded below complexes (in the sense of Christensen and coauthors) to unbounded complexes whenever R is Gorenstein. Moreover, we extend Stenström's FP-injective dimension from modules to complexes, define FP-projective dimension for complexes, and characterize Noetherian and von Neumann regular rings by these dimensions.
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EICK, BETTINA, and DAVID J. GREEN. "COCHAIN SEQUENCES AND THE QUILLEN CATEGORY OF A COCLASS FAMILY." Journal of the Australian Mathematical Society 102, no. 2 (May 12, 2016): 185–204. http://dx.doi.org/10.1017/s1446788716000185.
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We introduce the concept of infinite cochain sequences and initiate a theory of homological algebra for them. We show how these sequences simplify and improve the construction of infinite coclass families (as introduced by Eick and Leedham-Green) and also how they can be applied to prove that almost all groups in such a family have equivalent Quillen categories. We also include some examples of infinite families of$p$-groups from different coclass families that have equivalent Quillen categories.
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SANTIAGO, VALENTE. "STRATIFYING SYSTEMS FOR EXACT CATEGORIES." Glasgow Mathematical Journal 61, no. 03 (July 23, 2018): 501–21. http://dx.doi.org/10.1017/s0017089518000320.
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AbstractIn this paper, we develop the theory of stratifying systems in the context of exact categories as a generalisation of the notion of stratifying systems in module categories, which have been studied by different authors. We prove that attached to a stratifying system in an exact category $(\mathcal{A},\mathcal{E})$ there is an standardly stratified algebra B such that the category $\mathscr{F}$F(Θ), of F-filtered objects in the exact category $(\mathcal{A},\mathcal{E})$ is equivalent to the category $\mathscr{F}$(Δ) of Δ-good modules associated to B. The theory we develop in exact categories, give us a way to produce standardly stratified algebras from module categories by just changing the exact structure on it. In this way, we can construct exact categories whose bounded derived category is equivalent to the bounded derived category of an standardly stratified algebra. Finally, applying the relative homological algebra developed by Auslander–Solberg, we can construct examples of stratifying systems that are not a stratifying system in the classical sense, so our approach really produces new stratifying systems.
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Zhang, Chao. "On the representation type of subcategories of derived categories." Journal of Algebra and Its Applications 18, no. 05 (May 2019): 2050032. http://dx.doi.org/10.1142/s0219498820500322.
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Let [Formula: see text] be a finite-dimensional [Formula: see text]-algebra. In this paper, we mainly study the representation type of subcategories of the bounded derived category [Formula: see text]. First, we define the representation type and some homological invariants including cohomological length, width, range for subcategories. In this framework, we provide a characterization for derived discrete algebras. Moreover, for a finite-dimensional algebra [Formula: see text], we establish the first Brauer–Thrall type theorem of certain contravariantly finite subcategories [Formula: see text] of [Formula: see text], that is, [Formula: see text] is of finite type if and only if its cohomological range is finite.
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Huebschmann, J. "Minimal Free Multi-Models for Chain Algebras." gmj 11, no. 4 (December 2004): 733–52. http://dx.doi.org/10.1515/gmj.2004.733.
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Abstract Let 𝑅 be a local ring and 𝐴 a connected differential graded algebra over 𝑅 which is free as a graded 𝑅-module. Using homological perturbation theory techniques, we construct a minimal free multi-model for 𝐴 having properties similar to those of an ordinary minimal model over a field; in particular the model is unique up to isomorphism of multialgebras. The attribute ‘multi’ refers to the category of multicomplexes.
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Sheridan, Nick. "Formulae in noncommutative Hodge theory." Journal of Homotopy and Related Structures 15, no. 1 (November 21, 2019): 249–99. http://dx.doi.org/10.1007/s40062-019-00251-2.
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AbstractWe prove that the cyclic homology of a saturated $$A_\infty $$A∞ category admits the structure of a ‘polarized variation of Hodge structures’, building heavily on the work of many authors: the main point of the paper is to present complete proofs, and also explicit formulae for all of the relevant structures. This forms part of a project of Ganatra, Perutz and the author, to prove that homological mirror symmetry implies enumerative mirror symmetry.
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Cristian Iovanov, Miodrag. "Generalized Frobenius Algebras and Hopf Algebras." Canadian Journal of Mathematics 66, no. 1 (February 2014): 205–40. http://dx.doi.org/10.4153/cjm-2012-060-7.
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Abstract“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius (QcF) coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or right) rational dual Rat(C*) in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, QcF, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra A that is isomorphic to its complete topological dual Aν. We show that A is a (quasi)Frobenius algebra if and only if A is the dual C* of a (quasi)co-Frobenius coalgebra C. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer q-homological algebra, topology or graph theory, showing the importance of the concept.
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Gillespie, James, and Mark Hovey. "Gorenstein model structures and generalized derived categories." Proceedings of the Edinburgh Mathematical Society 53, no. 3 (August 5, 2010): 675–96. http://dx.doi.org/10.1017/s0013091508000709.
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AbstractIn a paper from 2002, Hovey introduced the Gorenstein projective and Gorenstein injective model structures on R-Mod, the category of R-modules, where R is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce: the Gorenstein flat model structure. The homotopy category with respect to each of these is called the stable module category of R. If such a ring R has finite global dimension, the graded ring R[x]/(x2) is Gorenstein and the three associated Gorenstein model structures on R[x]/(x2)-Mod, the category of graded R[x]/(x2)-modules, are nothing more than the usual projective, injective and flat model structures on Ch(R), the category of chain complexes of R-modules. Although these correspondences only recover these model structures on Ch(R) when R has finite global dimension, we can set R = ℤ and use general techniques from model category theory to lift the projective model structure from Ch(ℤ) to Ch(R) for an arbitrary ring R. This shows that homological algebra is a special case of Gorenstein homological algebra. Moreover, this method of constructing and lifting model structures carries through when ℤ[x]/(x2) is replaced by many other graded Gorenstein rings (or Hopf algebras, which lead to monoidal model structures). This gives us a natural way to generalize both chain complexes over a ring R and the derived category of R and we give some examples of such generalizations.
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Tu, Junwu. "Matrix factorizations via Koszul duality." Compositio Mathematica 150, no. 9 (July 17, 2014): 1549–78. http://dx.doi.org/10.1112/s0010437x14007295.
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AbstractIn this paper we prove a version of curved Koszul duality for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Z}/2\mathbb{Z}$-graded curved coalgebras and their cobar differential graded algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the category of matrix factorizations $\mathsf{MF}(R,W)$. We show how Dyckerhoff’s generating results fit into the framework of curved Koszul duality theory. This enables us to clarify the relationship between the Borel–Moore Hochschild homology of curved (co)algebras and the ordinary Hochschild homology of the category $\mathsf{MF}(R,W)$. Similar results are also obtained in the orbifold case and in the graded case.
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KIM, BUMSIG, and HWAYOUNG LEE. "WALL-CROSSINGS FOR TWISTED QUIVER BUNDLES." International Journal of Mathematics 24, no. 05 (May 2013): 1350038. http://dx.doi.org/10.1142/s0129167x13500389.
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Given a double quiver, we study homological algebra of twisted quiver sheaves with the moment map relation using the short exact sequence of Crawley-Boevey, Holland, Gothen, and King. Then in a certain one-parameter space of the stability conditions, we obtain a wall-crossing formula for the generalized Donaldson–Thomas invariants of the abelian category of framed twisted quiver sheaves on a smooth projective curve. To do so, we closely follow the approach of Chuang, Diaconescu and Pan in the ADHM quiver case, which makes use of the theory of Joyce and Song. The invariants virtually count framed twisted quiver sheaves with the moment map relation and directly generalize the ADHM invariants of Diaconescu.
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Martsinkovsky, Alex, and Jeremy Russell. "Injective stabilization of additive functors, III. Asymptotic stabilization of the tensor product." Algebra and Discrete Mathematics 31, no. 1 (2021): 120–51. http://dx.doi.org/10.12958/adm1728.
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The injective stabilization of the tensor product is subjected to an iterative procedure that utilizes its bifunctor property. The limit of this procedure, called the asymptotic stabilization of the tensor product, provides a homological counterpart of Buchweitz's asymptotic construction of stable cohomology. The resulting connected sequence of functors is isomorphic to Triulzi's J-completion of the Tor functor. A comparison map from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be always epic. The category of finitely presented functors is shown to be complete and cocomplete. As a consequence, the inert injective stabilization of the tensor product with fixed variable a finitely generated module over an artin algebra is shown to be finitely presented. Its defect and consequently all right-derived functors are determined. New notions of asymptotic torsion and cotorsion are introduced and are related to each other.
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ANGELES, DAVID, JASON LO, and COURTNEY M. VAN DER LINDEN. "A FRAMEWORK FOR TORSION THEORY COMPUTATIONS ON ELLIPTIC THREEFOLDS." Journal of the Australian Mathematical Society, May 14, 2020, 1–19. http://dx.doi.org/10.1017/s144678872000018x.
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We give a list of statements on the geometry of elliptic threefolds phrased only in the language of topology and homological algebra. Using only notions from topology and homological algebra, we recover existing results and prove new results on torsion pairs in the category of coherent sheaves on an elliptic threefold.
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Auroux, Denis, and Ivan Smith. "Fukaya categories of surfaces, spherical objects and mapping class groups." Forum of Mathematics, Sigma 9 (2021). http://dx.doi.org/10.1017/fms.2021.21.
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Abstract We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least $2$ whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank $1$ local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included.
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Gao, Zenghui, and Wan Wu. "Gorenstein flat modules relative to injectively resolving subcategories." Journal of Algebra and Its Applications, February 24, 2021, 2250099. http://dx.doi.org/10.1142/s0219498822500992.
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Let [Formula: see text] be an injectively resolving subcategory of left [Formula: see text]-modules. We introduce and study [Formula: see text]-Gorenstein flat modules as a common generalization of some known modules such as Gorenstein flat modules (Enochs, Jenda and Torrecillas, 1993), Gorenstein AC-flat modules (Bravo, Estrada and Iacob, 2018). Then we define a resolution dimension relative to the [Formula: see text]-Gorensteinflat modules, investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, stability of the category of [Formula: see text]-Gorensteinflat modules is discussed, and some known results are obtained as applications.
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Becerril, Víctor. "Relative global Gorenstein dimensions." Journal of Algebra and Its Applications, July 6, 2021, 2250208. http://dx.doi.org/10.1142/s0219498822502085.
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Let [Formula: see text] be an abelian category. In this paper, we investigate the global [Formula: see text]-Gorenstein projective dimension [Formula: see text], associated to a GP-admissible pair [Formula: see text]. We give homological conditions over [Formula: see text] that characterize it. Moreover, given a GI-admissible pair [Formula: see text], we study conditions under which [Formula: see text] and [Formula: see text] are the same.