Related papers: Unbounded derived categories and the finitistic di…

For a locally presentable abelian category $\mathsf B$ with a projective generator, we construct the projective derived and contraderived model structures on the category of complexes, proving in particular the existence of enough homotopy…

We classify all tilting and cotilting classes over commutative noetherian rings in terms of descending sequences of specialization closed subsets of the Zariski spectrum. Consequently, all resolving subcategories of finitely generated…

We prove that over a commutative noetherian ring the three approaches to introducing depth for complexes: via Koszul homology, via Ext modules, and via local cohomology, all yield the same invariant. Using this result, we establish a far…

Let R be a commutative Noetherian local ring, and denote by mod R the category of finitely generated R-modules. In this paper, we consider when mod R has a nontrivial extension-closed subcategory. We prove that this is the case if there are…

A noncommutative projective variety is defined, after Artin and Zhang, by a graded coherent algebra A, where the category of coherent sheaves is the quotient qgr(A) of the category of finitely presented graded modules by the subcategory of…

We show that the category of finitely generated free modules over certain local rings is n-angulated for every n at least 3. In fact, we construct several classes of n-angles, parametrized by equivalence classes of units in the local rings.…

We develop a technique to construct finitely injective modules which are non trivial, in the sense that they are not direct sums of injective modules. As a consequence, we prove that a ring $R$ is left noetherian if and only if each…

We introduce the notion of composition series of triangulated categories, which generalizes full exceptional sequences. The lengths of composition series yield invariants for triangulated categories. We study composition series of derived…

Let $\mathscr{A}$ be an abelian category having enough projective and injective objects, and let $\mathscr{T}$ be an additive subcategory of $\mathscr{A}$ closed under direct summands. A known assertion is that in a short exact sequence in…

In our previous papers we introduced categorical invariants, which are, roughly speaking, sets of triangulated subcategories in a given triangulated category and their quotients. Here is extended the list of examples, where these sets are…

We give a characterisation of those local not necessary commutative rings, for which the category of projective modules admits a triangulation with the identity as translation functor. By "admits a triangulation" we mean that the category…

We study cyclic adjoint modules arising from the relative locally finite part of the adjoint action of a quantum Levi subalgebra on a quantized enveloping algebra. We classify embeddings of finite-dimensional irreducible modules inside of…

Silting modules are abundant. Indeed, they parametrise the definable torsion classes over a noetherian ring, and the hereditary torsion pairs of finite type over a commutative ring. Also the universal localisations of a hereditary ring, or…

An extension $B\subset A$ of finite dimensional algebras is bounded if the $B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is finite and $\mathrm{Tor}_i^B(A/B, (A/B)^{\otimes_B j})=0$ for all $i, j\geq 1$. We show…

We compute the number of finite dimensional irreducible modules for the algebras quantizing Nakajima quiver varieties. We get a lower bound for all quivers and vectors of framing and provide an exact count in the case when the quiver is of…

We show that the cosupport of a commutative noetherian ring is precisely the set of primes appearing in a minimal pure-injective resolution of the ring. As an application of this, we prove that every countable commutative noetherian ring…

Let $S$ be a commutative noetherian ring. The extensions of matrix factorizations of non-zerodivisors $x_1,\dots,x_n$ of $S$ form a full subcategory of finitely generated modules over the quotient ring $S/(x_1\cdots x_n)$. In this paper, we…

For any graded commutative noetherian ring, where the grading group is abelian and where commutativity is allowed to hold in a quite general sense, we establish an inclusion-preserving bijection between, on the one hand, the twist-closed…

Let $R$ be a noetherian normal domain. We investigate when $R$ admits a faithful module whose endomorphism ring has finite global dimension. This can be viewed as a non-commutative desingularization of $\Spec(R)$. We show that the existence…

For a finite dimensional algebra $A$, we prove that the bounded homotopy category of projective $A$-modules and the bounded derived category of $A$-modules are dual to each other via certain categories of locally-finite cohomological…