The polyhedral product constructed from a collection of pairs of cones and their bases and a simplicial complex $K$ is studied by investigating its filtration called the fat-wedge filtration. We give a sufficient condition for decomposing the polyhedral product in terms of the fat-wedge filtration of the real moment-angle complex for $K$, which is a desuspension of the decomposition of the suspension of the polyhedral product due to Bahri, Bendersky, Cohen, and Gitler. We show that the condition also implies a strong connection with the Golodness of $K$, and it is satisfied when $K$ is dual sequentially Cohen–Macaulay over $\mathbb{Z}$ or $\lceil \frac{dimK}{2}\rceil$-neighborly so that the polyhedral product decomposes. Specializing to the moment-angle complex, we prove that the similar condition on its fat-wedge filtrations is necessary and sufficient for its decomposition.

We introduce a cohomology theory for spatial superproduct systems and compute the $2$-cocycles for some basic examples called Clifford superproduct systems, thereby distinguishing them up to isomorphism. This consequently proves that a family of $E_0$-semigroups on type III factors, which we call CAR flows, are noncocycle-conjugate for different ranks. Similar results follow for the even CAR flows as well. We also compute the automorphism group of the Clifford superproduct systems.

With an action $\alpha$ of ${\mathbb{R}}^n$ on a $C^{\ast }$-algebra $A$ and a skew-symmetric $n\times n$ matrix $\Theta$, one can consider the Rieffel deformation $A_{\Theta }$ of $A$, which is a $C^{\ast }$-algebra generated by the $\alpha$-smooth elements of $A$ with a new multiplication. The purpose of this article is to obtain explicit formulas for $K$-theoretical quantities defined by elements of $A_{\Theta }$. We give an explicit realization of the Thom class in $\mathit{KK}$ in any dimension $n$ and use it in the index pairings. For local index formulas we assume that there is a densely defined trace on $A$, invariant under the action. When $n$ is odd, for example, we give a formula for the index of operators of the form $P{\pi }^{\Theta }\left(u\right)P$, where ${\pi }^{\Theta }\left(u\right)$ is the operator of left Rieffel multiplication by an invertible element $u$ over the unitization of $A$ and $P$ is the projection onto the nonnegative eigenspace of a Dirac operator constructed from the action $\alpha$. The results are new also for the undeformed case $\Theta =0$. The construction relies on two approaches to Rieffel deformations in addition to Rieffel’s original one: Kasprzak deformation and warped convolution. We end by outlining potential applications in mathematical physics.

We study the Verlinde bundles of generalized theta functions constructed from moduli spaces of sheaves over abelian surfaces. In degree $0$, the splitting type of these bundles is expressed in terms of indecomposable semihomogeneous factors. Furthermore, Fourier–Mukai symmetries of the Verlinde bundles are found consistently with strange duality. Along the way, a transformation formula for the theta bundles is derived, extending a theorem of Drézet–Narasimhan from curves to abelian surfaces.

The balanced tensor product $M{\otimes }_{A}N$ of two modules over an algebra $A$ is the vector space corepresenting $A$-balanced bilinear maps out of the product $M\times N$. The balanced tensor product $\mathcal{M}{⊠}_{\mathcal{C}}\mathcal{N}$ of two module categories over a monoidal linear category $\mathcal{C}$ is the linear category corepresenting $\mathcal{C}$-balanced right-exact bilinear functors out of the product category $\mathcal{M}\times \mathcal{N}$. We show that the balanced tensor product can be realized as a category of bimodule objects in $\mathcal{C}$, provided the monoidal linear category is finite and rigid.

We show that all complements of cuspidal hyperplane sections of smooth projective cubic surfaces have isomorphic ${\mathbb{A}}^2$-cylinders. As a consequence, we derive that the ${\mathbb{A}}^2$-cancellation problem fails in every dimension greater than or equal to $2$.

In the context of irreducible holomorphic symplectic manifolds, we say that (anti)holomorphic (anti)symplectic involutions are brane involutions since their fixed point locus is a brane in the physicists’ language, that is, a submanifold which is either a complex or Lagrangian submanifold with respect to each of the three Kähler structures of the associated hyper-Kähler structure. Starting from a brane involution on a $\mathrm{K3}$ or Abelian surface, one can construct a natural brane involution on its moduli space of sheaves. We study these natural involutions and their relation with the Fourier–Mukai transform. Later, we recall the lattice-theoretical approach to mirror symmetry. We provide two ways of obtaining a brane involution on the mirror, and we study the behavior of the brane involutions under both mirror transformations, giving examples in the case of a $\mathrm{K3}$ surface and ${\mathrm{K3}}^{[2]}$-type manifolds.

Let $(R,\mathfrak{m})$ be a $d$-dimensional commutative Noetherian local ring. Let $\mathcal{M}$ denote the morphism category of finitely generated $R$-modules, and let $\mathcal{S}$ be the full subcategory of $\mathcal{M}$ consisting of monomorphisms, known as the submodule category. This article reveals that the Auslander transpose in the category $\mathcal{S}$ can be described explicitly within $modR$, the category of finitely generated $R$-modules. This result is exploited to study the linkage theory as well as the Auslander–Reiten theory in $\mathcal{S}$. In addition, motivated by a result of Ringel and Schmidmeier, we show that the Auslander–Reiten translations in the subcategories $\mathcal{H}$ and $\mathcal{G}$, consisting of all morphisms which are maximal Cohen–Macaulay $R$-modules and Gorenstein projective morphisms, respectively, may be computed within $modR$ via $\mathcal{G}$-covers. The corresponding result for the subcategory of epimorphisms in $\mathcal{H}$ is also obtained.