Related papers: Ideals in Triangulated Categories: Phantoms, Ghost…
In this paper we give a construction of phantom categories, i.e. admissible triangulated subcategories in bounded derived categories of coherent sheaves on smooth projective varieties that have trivial Hochschild homology and trivial…
We define the spectrum of a tensor triangulated category $K$ as the set of so-called prime ideals, endowed with a suitable topology. In this very generality, the spectrum is the universal space in which one can define supports for objects…
Motivated by Freyd's famous unsolved problem in stable homotopy theory, the generating hypothesis for the stable module category of a finite group is the statement that if a map in the thick subcategory generated by the trivial…
Suppose that $G$ is a finite group and $k$ is a field of characteristic $p>0$. A ghost map is a map in the stable category of finitely generated $kG$-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper…
We construct explicit examples of globally regular static, spherically symmetric solutions in general relativity with scalar and electromagnetic fields, describing traversable wormholes with flat and AdS asymptotics and regular black holes,…
We use homological ideals in triangulated categories to get a sufficient criterion for a pair of subcategories in a triangulated category to be complementary. We apply this criterion to construct the Baum-Connes assembly map for locally…
We study phantom maps and homology theories in a stable homotopy category S via a certain Abelian category A. We express the group P(X,Y) of phantom maps X -> Y as an Ext group in A, and give conditions on X or Y which guarantee that it…
We study ideal cotorsion pairs associated to weak proper classes of triangles in extension closed subcategories of triangulated categories. This approach allows us to extend the recent ideal approximations theory developed by Fu, Herzog et…
We extend the existence of ghost measures beyond nonnegative primitive regular sequences to a large class of nonnegative real-valued regular sequences. In the general case, where the ghost measure is not unique, we show that they can be…
Approximation sequences and derived equivalences occur frequently in the research of mutation of tilting objects in representation theory, algebraic geometry and noncommutative geometry. In this paper, we introduce symmetric approximation…
We study the spectrum of prime ideals in the tensor-triangulated category of compact equivariant spectra over a finite group. We completely describe this spectrum as a set for all finite groups. We also make significant progress in…
We study several closely related invariants of the group algebra $kG$ of a finite group. The basic invariant is the ghost number, which measures the failure of the generating hypothesis and involves finding non-trivial composites of maps…
We give sufficient conditions to ensure that the ideal $\Phi(\mathcal E)$ of $\mathcal E$-phantom maps in a locally $\lambda$-presentable exact category $(\mathcal{A}, \mathcal{E})$ is (special) (pre)covering ideal, where $\mathcal E$ is an…
For a tensor triangulated category and any regular cardinal $\alpha$ we study the frame of $\alpha$-localizing tensor ideals and its associated space of points. For a well-generated category and its frame of localizing tensor ideals we…
Ghost modules were introduced in [I3] without definitions or proofs. We also introduced stability diagrams or "relative pictures" for torsion classes and torsion-free classes for representations of Dynkin quivers. Modules which were not in…
Given a symbolic power of a homogeneous ideal in a polynomial ring, we study the problem of determining which powers of the ideal contain it. For ideals defining 0-dimensional subschemes of projective space, as an immediate corollary of our…
Bivariant (equivariant) K-theory is the standard setting for non-commutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from…
In this paper, we introduce and study relative phantom morphisms in extriangulated categories defined by Nakaoka and Palu. Then using their properties, we show that if $(\C,\E,\s)$ is an extriangulated category with enough injective objects…
It is proved that if G is a finite group, then the order of G is a proper upper bound for the phantom number of G. More specifically, if k is a field whose characteristic divides the order of G, and $\Phi$ is the ideal of phantom morphisms…
We construct an abelian category A(G) of sheaves over a category of closed subgroups of the r-torus G and show it is of finite injective dimension. It can be used as a model for rational $G$-spectra in the sense that there is a homology…