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We show that a directed graph
$E$
is a finite graph with no sinks if and only if, for each commutative unital ring
$R$
, the Leavitt path algebra
$L_{R}(E)$
is isomorphic to an algebraic Cuntz–Krieger algebra if and only if the
$C^{\ast }$
-algebra
$C^{\ast }(E)$
is unital and
$\text{rank}(K_{0}(C^{\ast }(E)))=\text{rank}(K_{1}(C^{\ast }(E)))$
. Let
$k$
be a field and
$k^{\times }$
be the group of units of
$k$
. When
$\text{rank}(k^{\times })<\infty$
, we show that the Leavitt path algebra
$L_{k}(E)$
is isomorphic to an algebraic Cuntz–Krieger algebra if and only if
$L_{k}(E)$
is unital and
$\text{rank}(K_{1}(L_{k}(E)))=(\text{rank}(k^{\times })+1)\text{rank}(K_{0}(L_{k}(E)))$
. We also show that any unital
$k$
-algebra which is Morita equivalent or stably isomorphic to an algebraic Cuntz–Krieger algebra, is isomorphic to an algebraic Cuntz–Krieger algebra. As a consequence, corners of algebraic Cuntz–Krieger algebras are algebraic Cuntz–Krieger algebras.
Keller proved in 1999 that the Gerstenhaber algebra structure on the Hochschild cohomology of an algebra is an invariant of the derived category. In this paper, we adapt his approach to show that the Gerstenhaber algebra structure on the Tate–Hochschild cohomology of an algebra is preserved under singular equivalences of Morita type with level, a notion introduced by the author in previous work.
Nakayama automorphisms play an important role in the fields of noncommutative algebraic geometry and noncommutative invariant theory. However, their computations are not easy in general. We compute the Nakayama automorphism ν of an Ore extension R[x; σ, δ] over a polynomial algebra R in n variables for an arbitrary n. The formula of ν is obtained explicitly. When σ is not the identity map, the invariant EG is also investigated in terms of Zhang’s twist, where G is a cyclic group sharing the same order with σ.
In this paper we consider the algebraic crossed product
${\mathcal{A}}:=C_{K}(X)\rtimes _{T}\mathbb{Z}$
induced by a homeomorphism
$T$
on the Cantor set
$X$
, where
$K$
is an arbitrary field with involution and
$C_{K}(X)$
denotes the
$K$
-algebra of locally constant
$K$
-valued functions on
$X$
. We investigate the possible Sylvester matrix rank functions that one can construct on
${\mathcal{A}}$
by means of full ergodic
$T$
-invariant probability measures
$\unicode[STIX]{x1D707}$
on
$X$
. To do so, we present a general construction of an approximating sequence of
$\ast$
-subalgebras
${\mathcal{A}}_{n}$
which are embeddable into a (possibly infinite) product of matrix algebras over
$K$
. This enables us to obtain a specific embedding of the whole
$\ast$
-algebra
${\mathcal{A}}$
into
${\mathcal{M}}_{K}$
, the well-known von Neumann continuous factor over
$K$
, thus obtaining a Sylvester matrix rank function on
${\mathcal{A}}$
by restricting the unique one defined on
${\mathcal{M}}_{K}$
. This process gives a way to obtain a Sylvester matrix rank function on
${\mathcal{A}}$
, unique with respect to a certain compatibility property concerning the measure
$\unicode[STIX]{x1D707}$
, namely that the rank of a characteristic function of a clopen subset
$U\subseteq X$
must equal the measure of
$U$
.
Let R→U be an associative ring epimorphism such that U is a flat left R-module. Assume that the related Gabriel topology
$\mathbb{G}$
of right ideals in R has a countable base. Then we show that the left R-module U has projective dimension at most 1. Furthermore, the abelian category of left contramodules over the completion of R at
$\mathbb{G}$
fully faithfully embeds into the Geigle–Lenzing right perpendicular subcategory to U in the category of left R-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an associative ring R, we consider the induced topology on every left R-module and, for a perfect Gabriel topology
$\mathbb{G}$
, compare the completion of a module with an appropriate Ext module. Finally, we characterize the U-strongly flat left R-modules by the two conditions of left positive-degree Ext-orthogonality to all left U-modules and all
$\mathbb{G}$
-separated
$\mathbb{G}$
-complete left R-modules.
We show that a
$\mathbb{P}$
-object and simple configurations of
$\mathbb{P}$
-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient triangulated category. We also observe that the category generated by the structure sheaf of a smooth projective variety over the complex numbers only depends on its graded cohomology algebra.
We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the differential graded (DG) algebra
$\operatorname{RHom}^{\bullet }(F,F)$
is formal for any sheaf
$F$
polystable with respect to an ample line bundle. Our main tool is the uniqueness of the DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.
In order to better unify the tilting theory and the Auslander–Reiten theory, Xi introduced a general transpose called the relative transpose. Originating from this, we introduce and study the cotranspose of modules with respect to a left A-module T called n-T-cotorsion-free modules. Also, we give many properties and characteristics of n-T-cotorsion-free modules under the help of semi-Wakamatsu-tilting modules AT.
We introduce the class of partially invertible modules and show that it is an inverse category which we call the Picard inverse category. We use this category to generalize the classical construction of crossed products to, what we call, generalized epsilon-crossed products and show that these coincide with the class of epsilon-strongly groupoid-graded rings. We then use generalized epsilon-crossed groupoid products to obtain a generalization, from the group-graded situation to the groupoid-graded case, of the bijection from a certain second cohomology group, defined by the grading and the functor from the groupoid in question to the Picard inverse category, to the collection of equivalence classes of rings epsilon-strongly graded by the groupoid.
We study Van den Bergh's non-commutative symmetric algebra 𝕊nc(M) (over division rings) via Minamoto's theory of Fano algebras. In particular, we show that 𝕊nc(M) is coherent, and its proj category ℙnc(M) is derived equivalent to the corresponding bimodule species. This generalizes the main theorem of [8], which in turn is a generalization of Beilinson's derived equivalence. As corollaries, we show that ℙnc(M) is hereditary and there is a structure theorem for sheaves on ℙnc(M) analogous to that for ℙ1.
We prove formulas of different types that allow us to calculate the Gerstenhaber bracket on the Hochschild cohomology of an algebra using some arbitrary projective bimodule resolution for it. Using one of these formulas, we give a new short proof of the derived invariance of the Gerstenhaber algebra structure on Hochschild cohomology. We also give some new formulas for the Connes differential on the Hochschild homology that lead to formulas for the Batalin–Vilkovisky (BV) differential on the Hochschild cohomology in the case of symmetric algebras. Finally, we use one of the obtained formulas to provide a full description of the BV structure and, correspondingly, the Gerstenhaber algebra structure on the Hochschild cohomology of a class of symmetric algebras.
We apply the Auslander–Buchweitz approximation theory to show that the Iyama and Yoshino's subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitz's triangle equivalence from Iwanaga–Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitz's triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga–Gorenstein rings and Gorenstein algebras.
We show that silting modules are closely related with localizations of rings. More precisely, every partial silting module gives rise to a localization at a set of maps between countably generated projective modules and, conversely, every universal localization, in the sense of Cohn and Schofield, arises in this way. To establish these results, we further explore the finite-type classification of tilting classes and we use the morphism category to translate silting modules into tilting objects. In particular, we prove that silting modules are of finite type.
Let
$k$
be a commutative ring, let
${\mathcal{C}}$
be a small,
$k$
-linear, Hom-finite, locally bounded category, and let
${\mathcal{B}}$
be a
$k$
-linear abelian category. We construct a Frobenius exact subcategory
${\mathcal{G}}{\mathcal{P}}({\mathcal{G}}{\mathcal{P}}_{P}({\mathcal{B}}^{{\mathcal{C}}}))$
of the functor category
${\mathcal{B}}^{{\mathcal{C}}}$
, and we show that it is a subcategory of the Gorenstein projective objects
${\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$
in
${\mathcal{B}}^{{\mathcal{C}}}$
. Furthermore, we obtain criteria for when
${\mathcal{G}}{\mathcal{P}}({\mathcal{G}}{\mathcal{P}}_{P}({\mathcal{B}}^{{\mathcal{C}}}))={\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$
. We show in examples that this can be used to compute
${\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$
explicitly.
We consider the unital Banach algebra
$\ell ^{1}(\mathbb{Z}_{+})$
and prove directly, without using cyclic cohomology, that the simplicial cohomology groups
${\mathcal{H}}^{n}(\ell ^{1}(\mathbb{Z}_{+}),\ell ^{1}(\mathbb{Z}_{+})^{\ast })$
vanish for all
$n\geqslant 2$
. This proceeds via the introduction of an explicit bounded linear operator which produces a contracting homotopy for
$n\geqslant 2$
. This construction is generalised to unital Banach algebras
$\ell ^{1}({\mathcal{S}})$
, where
${\mathcal{S}}={\mathcal{G}}\cap \mathbb{R}_{+}$
and
${\mathcal{G}}$
is a subgroup of
$\mathbb{R}_{+}$
.
If H is a monoid and a = u1 ··· uk ∈ H with atoms (irreducible elements) u1, … , uk, then k is a length of a, the set of lengths of a is denoted by Ⅼ(a), and ℒ(H) = {Ⅼ(a) | a ∈ H} is the system of sets of lengths of H. Let R be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors R• can be written as a product of atoms. We show that if R is bounded and every stably free right R-ideal is free, then there exists a transfer homomorphism from R• to the monoid B of zero-sum sequences over a subset Gmax(R) of the ideal class group G(R). This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoids R• and B coincide. It is well known that commutative Dedekind domains allow transfer homomorphisms to monoids of zero-sum sequences, and the arithmetic of the latter has been the object of much research. Our approach is based on the structure theory of finitely generated projective modules over HNP rings, as established in the recent monograph by Levy and Robson. We complement our results by giving an example of a non-bounded HNP ring in which every stably free right R-ideal is free but which does not allow a transfer homomorphism to a monoid of zero-sum sequences over any subset of its ideal class group.
We solve two problems in representation theory for the periplectic Lie superalgebra
$\mathfrak{p}\mathfrak{e}(n)$
, namely, the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category
${\mathcal{O}}$
into indecomposable blocks.
To solve the first problem, we establish a new type of equivalence between category
${\mathcal{O}}$
for all (not just simple or basic) classical Lie superalgebras and a category of Harish-Chandra bimodules. The latter bimodules have a left action of the Lie superalgebra but a right action of the underlying Lie algebra. To solve the second problem, we establish a BGG reciprocity result for the periplectic Lie superalgebra.
We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring that is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo–Gillespie–Hovey. It is also compatible with the notion of
$n$
-coherent rings introduced by Bravo–Perez. So a
$0$
-coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a
$1$
-coherent AC-Gorenstein ring is precisely a Ding–Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of two Quillen equivalent abelian model category structures. One is projective with cofibrant objects that are Gorenstein AC-projective modules while the other is an injective model structure with fibrant objects that are Gorenstein AC-injectives.
We introduce the notion of a perfect path for a monomial algebra. We classify indecomposable non-projective Gorenstein-projective modules over the given monomial algebra via perfect paths. We apply the classification to a quadratic monomial algebra and describe explicitly the stable category of its Gorenstein-projective modules.
We introduce properties of metric spaces and, specifically, finitely
generated groups with word metrics, which we call coarse
coherence and coarse regular coherence. They are
geometric counterparts of the classical algebraic notion of coherence and
the regular coherence property of groups defined and studied by Waldhausen.
The new properties can be defined in the general context of coarse metric
geometry and are coarse invariants. In particular, they are quasi-isometry
invariants of spaces and groups. The new framework allows us to prove
structural results by developing permanence properties, including the
particularly important fibering permanence property, for coarse regular
coherence.