Related papers: Adjointness properties for relative extensions of …

Building on previous work, we study the splitting of idempotents in the category of extensions $\mathbb{E}\operatorname{-Ext}(\mathcal{C})$ associated to a pair $(\mathcal{C},\mathbb{E})$ of an additive category and a biadditive functor to…

This paper is concerned with the characterizations of quasi self-adjoint extensions of a class of formally non-self-adjoint discrete Hamiltonian systems. Some properties of the solutions and the characterization of the minimal linear…

Let $F\colon \mathcal{C} \to \mathcal{E}$ be a functor from a category $\mathcal{C}$ to a homological (Borceux-Bourn) or semi-abelian (Janelidze-M\'arki-Tholen) category $\mathcal{E}$. We investigate conditions under which the homology of…

The functor that takes a ring to its category of modules has an adjoint if one remembers the forgetful functor to abelian groups: the endomorphism ring of linear natural transformations. This uses the self-enrichment of the category of…

Let $G$ be a connected complex algebraic group and $A$ a connected abelian algebraic group endowed with an algebraic action of $G$ by group automorphisms. In the present note we describe the abelian group $\Ext_{alg}(G,A)$ of algebraic…

We prove extension theorems for several geometric properties such as asymptotic property C (APC), finite decomposition complexity (FDC), strict finite decomposition complexity (sFDC) which are weakenings of Gromov's finite asymptotic…

We survey the cohomology jumping loci and the Alexander-type invariants associated to a space, or to its fundamental group. Though most of the material is expository, we provide new examples and applications, which in turn raise several…

For a certain full additive subcategory X of an additive category A, one defines the lower extension groups in relative homological algebra. We show that these groups are isomorphic to the suspended Hom groups in the Verdier quotient…

We study topological properties of the correspondence of prime spectra associated to a noncommutative ring homomorphism R -> S. Our main result provides criteria for the adjointness of certain functors between the categories of Zariski…

We develop a theory of $C_p$-Green functors of Lie type, unifying the axiomatic framework of Green functors with the structure of Lie algebras under the action of a cyclic group $C_p$ of prime order. Extending classical notions from…

Let $\mathscr{A}$ be a small abelian category. For a closed subbifunctor $F$ of $\Ext_{\mathscr{A}}^{1}(-,-)$, Buan has generalized the construction of the Verdier's quotient category to get a relative derived category, where he localized…

Complex Chern-Simons bundles are line bundles with connection, originating in the study of quantization of moduli spaces of flat connections with complex gauge groups. In this paper we introduce and study these bundles in the families…

We define the extension group between an atom and an object in a locally noetherian Grothendieck category as a module over a skew field. We show that the dimension of the i-th extension group between an atom and an object coincides with the…

We construct a correspondence between the cohomology groups of a group $G$ relative to a family of subgroups $\famS$ and the classes of `relative extensions' of $G$ by abelian groups, modulo a certain equivalence relation. We establish this…

For a compact space X we consider extending endomorphisms of the algebra C(X) to be endomorphisms of Arens-Hoffman and Cole extensions of C(X). Given a non-linear, monic polynomial p in C(X)[t], with C(X)[t]/pC(X)[t] semi-simple, we show…

Given a Fourier-Mukai functor $\Phi$ in the general setting of singular schemes, under various hypotheses we provide both left and a right adjoints to $\Phi$, and also give explicit formulas for them. These formulas are simple and natural,…

For each pair of lax-idempotent pseudomonads $R$ and $I$, for which $I$ is locally fully faithful and $R$ distributes over $I$, we establish an adjoint functor theorem, relating $R$-cocontinuity to adjointness relative to $I$. This provides…

For a given family $\{(\mathrm{q}_i, \mathrm{t}_i, \mathrm{p_i} )\}_{i \in I}$ of adjoint triples between exact categories $\mathcal{C}$ or $\mathcal{D}$, we show that any cotorsion pair in $\mathcal{C}$ and $\mathcal{D}$ yield two…

The aim of the present paper is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category C over a finite field FF_q and any sequence ii of simple objects in C the element X_{V,ii}…

Let $R$ be a commutative local ring. We study the subcategory of the homotopy category of $R$-complexes consisting of the totally acyclic $R$-complexes. In particular, in the context where $Q\to R$ is a surjective local ring homomorphism…