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We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $q$.
We prove that there is a natural plectic weight filtration on the cohomology of Hilbert modular varieties in the spirit of Nekovář and Scholl. This is achieved with the help of Morel’s work on weight t-structures and a detailed study of partial Frobenius. We prove in particular that the partial Frobenius extends to toroidal and minimal compactifications.
A module M is called a D4-module if, whenever A and B are submodules of M with M = A ⊕ B and f : A → B is a homomorphism with Imf a direct summand of B, then Kerf is a direct summand of A. The class of D4-modules contains the class of D3-modules, and hence the class of semi-projective modules, and so the class of Rickart modules. In this paper we prove that, over a commutative Dedekind domain R, for an R-module M which is a direct sum of cyclic submodules, M is direct projective (equivalently, it is semi-projective) iff M is D3 iff M is D4. Also we prove that, over a prime PI-ring, for a divisible R-module X, X is direct projective (equivalently, it is Rickart) iff X ⊕ X is D4. We determine some D3-modules and D4-modules over a discrete valuation ring, as well. We give some relevant examples. We also provide several examples on D3-modules and D4-modules via quivers.
Linckelmann and Murphy have classified the Morita equivalence classes of p-blocks of finite groups whose basic algebra has dimension at most
$12$
. We extend their classification to dimension
$13$
and
$14$
. As predicted by Donovan’s conjecture, we obtain only finitely many such Morita equivalence classes.
For a field K, let
$\mathcal {R}$
denote the Jacobson algebra
$K\langle X, Y \ | \ XY=1\rangle $
. We give an explicit construction of the injective envelope of each of the (infinitely many) simple left
$\mathcal {R}$
-modules. Consequently, we obtain an explicit description of a minimal injective cogenerator for
$\mathcal {R}$
. Our approach involves realizing
$\mathcal {R}$
up to isomorphism as the Leavitt path K-algebra of an appropriate graph
$\mathcal {T}$
, which thereby allows us to utilize important machinery developed for that class of algebras.
By using a representation of a Lie algebra on the second Hochschild cohomology group, we construct an obstruction class to extensibility of derivations and a short exact sequence of Wells type for an abelian extension of an associative algebra.
A ring $\unicode[STIX]{x1D6EC}$ is called right Köthe if every right $\unicode[STIX]{x1D6EC}$-module is a direct sum of cyclic modules. In this paper, we give a characterization of basic hereditary right Köthe rings in terms of their Coxeter valued quivers. We also give a characterization of basic right Köthe rings with radical square zero. Therefore, we give a solution to Köthe’s problem in these two cases.
We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density
$\unicode[STIX]{x1D70C}$
and inverse temperature
$\unicode[STIX]{x1D6FD}$
differs from the one of the noninteracting system by the correction term
$4\unicode[STIX]{x1D70B}\unicode[STIX]{x1D70C}^{2}|\ln \,a^{2}\unicode[STIX]{x1D70C}|^{-1}(2-[1-\unicode[STIX]{x1D6FD}_{\text{c}}/\unicode[STIX]{x1D6FD}]_{+}^{2})$
. Here,
$a$
is the scattering length of the interaction potential,
$[\cdot ]_{+}=\max \{0,\cdot \}$
and
$\unicode[STIX]{x1D6FD}_{\text{c}}$
is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. The result is valid in the dilute limit
$a^{2}\unicode[STIX]{x1D70C}\ll 1$
and if
$\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70C}\gtrsim 1$
.
The Torelli group of
$W_g = \#^g S^n \times S^n$
is the group of diffeomorphisms of
$W_g$
fixing a disc that act trivially on
$H_n(W_g;\mathbb{Z} )$
. The rational cohomology groups of the Torelli group are representations of an arithmetic subgroup of
$\text{Sp}_{2g}(\mathbb{Z} )$
or
$\text{O}_{g,g}(\mathbb{Z} )$
. In this article we prove that for
$2n \geq 6$
and
$g \geq 2$
, they are in fact algebraic representations. Combined with previous work, this determines the rational cohomology of the Torelli group in a stable range. We further prove that the classifying space of the Torelli group is nilpotent.
Let
$R$
be a ring and
$T$
be a good Wakamatsu-tilting module with
$S=\text{End}(T_{R})^{op}$
. We prove that
$T$
induces an equivalence between stable repetitive categories of
$R$
and
$S$
(i.e., stable module categories of repetitive algebras
$\hat{R}$
and
${\hat{S}}$
). This shows that good Wakamatsu-tilting modules seem to behave in Morita theory of stable repetitive categories as that tilting modules of finite projective dimension behave in Morita theory of derived categories.
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.
Kaplansky introduced the notions of CCR and GCR
$C^{\ast }$
-algebras, because they have a tractable representation theory. Many years later, he introduced the notions of CCR and GCR rings. In this paper we characterize when the algebra of an ample groupoid over a field is CCR and GCR. The results turn out to be exact analogues of the corresponding characterization of locally compact groupoids with CCR and GCR
$C^{\ast }$
-algebras. As a consequence, we classify the CCR and GCR Leavitt path algebras.
Let 𝔟 be the Borel subalgebra of the Lie algebra 𝔰𝔩2 and V2 be the simple two-dimensional 𝔰𝔩2-module. For the universal enveloping algebra $\[{\cal A}: = U(\gb \ltimes {V_2})\]$ of the semi-direct product 𝔟⋉V2 of Lie algebras, the prime, primitive and maximal spectra are classified. Please approve edit to the sentence “The sets of completely prime…”.The sets of completely prime ideals of $\[{\cal A}\]$ are described. The simple unfaithful $\[{\cal A}\]$-modules are classified and an explicit description of all prime factor algebras of $\[{\cal A}\]$ is given. The following classes of simple U(𝔟⋉V2)-modules are classified: the Whittaker modules, the 𝕂[X]-torsion modules and the 𝕂[E]-torsion modules.
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.
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.
A ring is called right annelidan if the right annihilator of any subset of the ring is comparable with every other right ideal. In this paper we develop the connections between this class of rings and the classes of right Bézout rings and rings whose right ideals form a distributive lattice. We obtain results on localization of right annelidan rings at prime ideals, chain conditions that entail left-right symmetry of the annelidan condition, and construction of completely prime ideals.
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 provide a complete classification of all algebras of generalized dihedral type, which are natural generalizations of algebras which occurred in the study of blocks of group algebras with dihedral defect groups. This gives a description by quivers and relations coming from surface triangulations.
A theorem of Burgess and Stephenson asserts that in an exchange ring with central idempotents, every maximal left ideal is also a right ideal. The proof uses sheaf-theoretic techniques. In this paper, we give a short elementary proof of this important theorem.