HYPERSURFACE SUPPORT FOR NONCOMMUTATIVE COMPLETE INTERSECTIONS

We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category $\operatorname {Sing}(R)$ , and that the support of an object in $\operatorname {Sing}(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz’ support theory for (commutative) local complete intersections. In the companion piece [27], we employ hypersurface support for infinite-dimensional modules, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.

The existence of n-Auslander–Reiten sequences via determined morphisms

Let [Formula: see text] be a commutative artinian ring and [Formula: see text] a small Ext-finite Krull–Schmidt [Formula: see text]-abelian [Formula: see text]-category with enough projectives and injectives. We introduce two full subcategories [Formula: see text] and [Formula: see text] of [Formula: see text] in terms of the representable functors from the stable category of [Formula: see text] to category of finitely generated [Formula: see text]-modules. Moreover, we define two additive functors [Formula: see text] and [Formula: see text], which are mutually quasi-inverse equivalences between the stable categories of this two full subcategories. We give an equivalent characterization on the existence of [Formula: see text]-Auslander–Reiten sequences using determined morphisms.

Mass Media and Consumer Culture in 20th-Century Mexico

In 20th-century Mexico, as in many other places, consumer culture and mass media have shaped everyday experiences, helped give meaning to ordinary lives, and opened up spaces in which political ideologies could be created and contested. Cultural forms such as dance, song, cuisine, clothing, and sports have been deployed to distinguish regions from one another, while at the same time, print media, radio, television, recorded music, film, and other cultural forms have connected Mexicans across regional and international borders (and across lines of gender, class, ethnicity, language, religion, political affiliation, and more) from the 1880s to the present day. Consumer culture—meaning the distribution, sale, and use of mass-produced goods such as clothing, as well as agricultural commodities like sugar and coffee—linked Mexico to a wider world in the historical era in which Mexico joined in the global process of rapid-fire modernization. The study of mass media and consumer culture in Mexico has been, at its best, highly interdisciplinary: historians and art historians, literary critics and cinema studies specialists, sociologists, and ethnographers have worked with journalists, folklorists, ethnomusicologists, and others in developing a sophisticated scholarly literature. This literature has its roots in two interrelated schools of scholarship: one that interpreted the products of culture industries as well as the creativity of ordinary people in a search for clues to Mexican national identity, and another that interpreted both locally made and imported mass media to understand how they shaped and supported the political, social, and economic status quo, both locally and globally. Since the 1980s, however, scholarly attention has broadened its focus from the images, narratives, movements, sounds, and objects produced by Mexican and foreign culture industries, and recent scholarship has looked to processes of creation, distribution, criticism, and consumption as well. Identities—whether regional, national, local, ideological, sexual, or political—are no longer understood as stable categories, but rather as a highly contested set of ideas, stories, and pictures that have changed radically over time. Much scholarship on mass media and consumer culture now begins with the understanding that culture industries have provided the tools with which discourses of identity could be shaped and reshaped, and that audiences and consumers have sometimes picked up those tools and turned them to their own purposes. And they have moved beyond taking the nation as a central category of analysis to ask how Mexican consumers and culture industries have participated in international and transnational processes of modernization.

Lifting of recollements and gluing of partial silting sets

This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.

A Hundred Forms of Spirituality in the Least Religious Country in the World

This paper is mainly based on interviews and observations that the author made during the process of writing a book about a hundred forms of religious and spiritual movements, teachings, and techniques in Estonia, thus being a reflection of trends and transformations of spiritual thought and practice in a country that has been repeatedly called the least religious country in Europe or even the whole world. Bringing some topical case analyses from this empirical material, the article will offer an amended interpretative framework for discussing features that are relevant in the research of Western contemporary spiritualities, for example multiple, situational, and fluctuating spiritual identities incongruent with the use of stable categories in religiosity statistics; children as important spiritual agents; mediatized liquidity and hybridity of spiritual thought being part of the ‘all-inclusive’ and ‘open-ended’ spiritual environment; and public conflicts and private symbioses of scientific, spiritual, and religious worldviews.

Combinatorial study of stable categories of graded Cohen–Macaulay modules over skew quadric hypersurfaces

In this paper, we present a new connection between representation theory of noncommutative hypersurfaces and combinatorics. Let S be a graded ($$\pm 1$$ ± 1 )-skew polynomial algebra in n variables of degree 1 and $$f =x_1^2 + \cdots +x_n^2 \in S$$ f = x 1 2 + ⋯ + x n 2 ∈ S . We prove that the stable category $$\mathsf {\underline{CM}}^{\mathbb Z}(S/(f))$$ CM ̲ Z ( S / ( f ) ) of graded maximal Cohen–Macaulay module over S/(f) can be completely computed using the four graphical operations. As a consequence, $$\mathsf {\underline{CM}}^{\mathbb Z}(S/(f))$$ CM ̲ Z ( S / ( f ) ) is equivalent to the derived category $$\mathsf {D}^{\mathsf {b}}({\mathsf {mod}}\,k^{2^r})$$ D b ( mod k 2 r ) , and this r is obtained as the nullity of a certain matrix over $${\mathbb F}_2$$ F 2 . Using the properties of Stanley–Reisner ideals, we also show that the number of irreducible components of the point scheme of S that are isomorphic to $${\mathbb P}^1$$ P 1 is less than or equal to $$\left( {\begin{array}{c}r+1\\ 2\end{array}}\right) $$ r + 1 2 .

Recollements of stable categories of Gorenstein injective modules

Let [Formula: see text] and [Formula: see text] be rings, [Formula: see text] a [Formula: see text]-bimodule and [Formula: see text] be a triangular matrix ring. We first give an explicit description for Gorenstein injective [Formula: see text]-modules over the triangular matrix ring [Formula: see text], and then construct a (right) recollement of stable categories of Gorenstein injective modules.