A connected space is called a C0-space if its rational cup product is trivial. A characterizing property of C0-spaces is obtained. This property is used to calculate the algebraic K0-group K0(C𝔽(X)) of the ring of continuous functions for infinite-dimensional complexes X.

Using that integrable derivations of symmetric algebras can be interpreted in terms of Bockstein homomorphisms in Hochschild cohomology, we show that integrable derivations are invariant under the transfer maps in Hochschild cohomology of symmetric algebras induced by stable equivalences of Morita type. With applications in block theory in mind, we allow complete discrete valuation rings of unequal characteristic.

It is shown that various definitions of $\unicode[STIX]{x1D711}$ -Connes amenability and $\unicode[STIX]{x1D711}$ -contractibility are equivalent to older and simpler concepts.

We introduce the notion of exact tilting objects, which are partial tilting objects $T$ inducing an equivalence between the abelian category generated by $T$ and the category of modules over the endomorphism algebra of $T$. Given a chain of sufficiently negative rational curves on a rational surface, we construct an exceptional sequence whose universal extension is an exact tilting object. For a chain of $(-2)$-curves, we obtain an equivalence with modules over a well-known algebra.

Inthis paper, for each $n\geqslant g\geqslant 0$ we consider the moduli stack $\widetilde{{\mathcal{U}}}_{g,n}^{ns}$ of curves $(C,p_{1},\ldots ,p_{n},v_{1},\ldots ,v_{n})$ of arithmetic genus $g$ with $n$ smooth marked points $p_{i}$ and nonzero tangent vectors $v_{i}$ at them, such that the divisor $p_{1}+\cdots +p_{n}$ is nonspecial (has $h^{1}=0$) and ample. With some mild restrictions on the characteristic we show that it is a scheme, affine over the Grassmannian $G(n-g,n)$. We also construct an isomorphism of $\widetilde{{\mathcal{U}}}_{g,n}^{ns}$ with a certain relative moduli of $A_{\infty }$-structures (up to an equivalence) over a family of graded associative algebras parametrized by $G(n-g,n)$.

We present a calculus that is well-adapted to homogeneous quadratic algebras. We define this calculus on Koszul cohomology – resp. homology – by cup products – resp. cap products. The Koszul homology and cohomology are interpreted in terms of derived categories. If the algebra is not Koszul, then Koszul (co)homology provides different information than Hochschild (co)homology. As an application of our calculus, the Koszul duality for Koszul cohomology algebras is proved for any quadratic algebra, and this duality is extended in some sense to Koszul homology. So, the true nature of the Koszul duality theorem is independent of any assumption on the quadratic algebra. We compute explicitly this calculus on a non-Koszul example.

In this paper we study categories ${\mathcal{O}}$ over quantizations of symplectic resolutions admitting Hamiltonian tori actions with finitely many fixed points. In this generality, these categories were introduced by Braden et al. We establish a family of standardly stratified structures (in the sense of the author and Webster) on these categories ${\mathcal{O}}$ . We use these structures to study shuffling functors of Braden et al. (called cross-walling functors in this paper). Most importantly, we prove that all cross-walling functors are derived equivalences that define an action of the Deligne groupoid of a suitable real hyperplane arrangement.

A quantum Frobenius map a la Lusztig for $\mathfrak{s}\mathfrak{l}_{2}$ is categorified at a prime root of unity.

We study cluster categories arising from marked surfaces (with punctures and non-empty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of $\operatorname{Ext}^{1}$ as intersection numbers of tagged curves and Auslander–Reiten translation as tagged rotation. An important consequence is that the cluster(-tilting) exchange graphs of such cluster categories are connected.

Khovanov–Lauda–Rouquier (KLR) algebras of finite Lie type come with families of standard modules, which under the Khovanov–Lauda–Rouquier categorification correspond to PBW bases of the positive part of the corresponding quantized enveloping algebra. We show that there are no non-zero homomorphisms between distinct standard modules and that all non-zero endomorphisms of a standard module are injective. We present applications to the extensions between standard modules and modular representation theory of KLR algebras.

In 2014, the first two authors proved an extension to modules of a theorem of Camillo and Yu that an exchange ring has stable range 1 if and only if every regular element is unit-regular. Here, we give a Morita context version of a stronger theorem. The definition of regular elements in a module goes back to Zelmanowitz in 1972, but the notion of a unit-regular element in a module is new. In this paper, we study unit-regular elements and give several characterizations of them in terms of “stable” elements and “lifting” elements. Along the way, we give natural extensions to the module case of many results about unit-regular rings. The paper concludes with a discussion of when the endomorphism ring of a unit-regular module is a unit-regular ring.

We consider the notion of a free resolution. In general, a free resolution can be of any length depending on the group ring under investigation. The metacyclic groups $G(pq)$ however admit periodic resolutions. In the particular case of $G(21)$ it is possible to achieve a fully diagonalized resolution. In order to achieve a diagonal resolution, we obtain a complete list of indecomposable modules over $\unicode[STIX]{x1D6EC}$ . Such a list aids the decomposition of the augmentation ideal (the first syzygy) into a direct sum of indecomposable modules. Therefore, we are able to achieve a diagonalized map here. From this point it is possible to decompose all of the remaining syzygies in terms of indecomposable modules, leaving a diagonal resolution.

We define canonical and $n$ -canonical modules of a module-finite algebra over a Noether commutative ring and study their basic properties. Using $n$ -canonical modules, we generalize a theorem on $(n,C)$ -syzygy by Araya and Iima which generalize a well-known theorem on syzygies by Evans and Griffith. Among others, we prove a noncommutative version of Aoyama’s theorem which states that a canonical module descends with respect to a flat local homomorphism.

We introduce a new method for expanding an abelian category and study it using recollements. In particular, we give a criterion for the existence of cotilting objects. We show, using techniques from noncommutative algebraic geometry, that our construction encompasses the category of coherent sheaves on Geigle–Lenzing weighted projective lines. We apply our construction to some concrete examples and obtain new weighted projective varieties, and analyze the endomorphism algebras of their tilting bundles.

Assume that $A$ is a finite-dimensional algebra over some field, and assume that $A$ is weakly symmetric and indecomposable, with radical cube zero and radical square nonzero. We show that such an algebra of wild representation type does not have a nonprojective module $M$ whose ext-algebra is finite dimensional. This gives a complete classification of weakly symmetric indecomposable algebras which have a nonprojective module whose ext-algebra is finite dimensional. This shows in particular that existence of ext-finite nonprojective modules is not equivalent with the failure of the finite generation condition (Fg), which ensures that modules have support varieties.

We study certain integer valued length functions on triangulated categories, and establish a correspondence between such functions and cohomological functors taking values in the category of finite length modules over some ring. The irreducible cohomological functions form a topological space. We discuss its basic properties, and include explicit calculations for the category of perfect complexes over some specific rings.

In this work we prove the so-called dimension property for the cohomological field theory associated with a homogeneous polynomial $W$ with an isolated singularity, in the algebraic framework of [A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1–122]. This amounts to showing that some cohomology classes on the Deligne–Mumford moduli spaces of stable curves, constructed using Fourier–Mukai-type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result established in [A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class, in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (American Mathematical Society, Providence, RI, 2001), 229–249] for certain characteristic classes of Koszul matrix factorizations of $0$ . To reduce to this result, we use the theory of Fourier–Mukai-type functors involving matrix factorizations and the natural rational lattices in the relevant Hochschild homology spaces, as well as a version of Hodge–Riemann bilinear relations for Hochschild homology of matrix factorizations. Our approach also gives a proof of the dimension property for the cohomological field theories associated with some quasihomogeneous polynomials with an isolated singularity.

In this paper, we consider several homological dimensions of crossed products A α σ G, where A is a left Noetherian ring and G is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of A σ αG are classified: global dimension of A σ αG is either infinity or equal to that of A, and finitistic dimension of Aσ αG coincides with that of A. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that A is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow p-subgroup S ≤ G, we show that A and Aα σ G share the same homological dimensions under extra assumptions, extending the main results in (Li, Representations of modular skew group algebras, Trans. Amer. Math. Soc.367(9) (2015), 6293–6314, Li, Finitistic dimensions and picewise hereditary property of skew group algebras, to Glasgow Math. J.57(3) (2015), 509–517).

In this paper we demonstrate that non-commutative localizations of arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category $\text{}\underline{C}$ by a set $S$ of morphisms in the heart $\text{}\underline{Hw}$ of a weight structure $w$ on it one obtains a triangulated category endowed with a weight structure $w^{\prime }$. The heart of $w^{\prime }$ is a certain version of the Karoubi envelope of the non-commutative localization $\text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$ (of $\text{}\underline{Hw}$ by $S$). The functor $\text{}\underline{Hw}\rightarrow \text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$ is the natural categorical version of Cohn’s localization of a ring, i.e., it is universal among additive functors that make all elements of $S$ invertible. For any additive category $\text{}\underline{A}$, taking $\text{}\underline{C}=K^{b}(\text{}\underline{A})$ we obtain a very efficient tool for computing $\text{}\underline{A}[S^{-1}]_{\mathit{add}}$; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that $\text{}\underline{A}[S^{-1}]_{\mathit{add}}$ coincides with the ‘abstract’ localization $\text{}\underline{A}[S^{-1}]$ (as constructed by Gabriel and Zisman) if $S$ contains all identity morphisms of $\text{}\underline{A}$ and is closed with respect to direct sums. We apply our results to certain categories of birational motives $DM_{gm}^{o}(U)$ (generalizing those defined by Kahn and Sujatha). We define $DM_{gm}^{o}(U)$ for an arbitrary $U$ as a certain localization of $K^{b}(Cor(U))$ and obtain a weight structure for it. When $U$ is the spectrum of a perfect field, the weight structure obtained this way is compatible with the corresponding Chow and Gersten weight structures defined by the first author in previous papers. For a general $U$ the result is completely new. The existence of the corresponding adjacent$t$-structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over $U$.

We show that Artin–Schelter regularity of a $\mathbb{Z}$-graded algebra can be examined by its associated $\mathbb{Z}$r-graded algebra. We prove that there is exactly one class of four-dimensional Artin–Schelter regular algebras with two generators of degree one in the Jordan type. This class is strongly noetherian, Auslander regular, and Cohen–Macaulay. Their automorphisms and point modules are described.