Auslander’s formula shows that any abelian category C is equivalent to the category of coherent functors on C modulo the Serre subcategory of all effaceable functors. We establish a derived version of this equivalence. This amounts to showing that the homotopy category of injective objects of some appropriate Grothendieck abelian category (the category of ind-objects of C) is compactly generate...

Abstract. Let R be a quotient ring of a commutative coherent regular ring by a finitely generated ideal. Hovey gave a bijection between the set of coherent subcategories of the category of finitely presented R-modules and the set of thick subcategories of the derived category of perfect R-complexes. Using this isomorphism, he proved that every coherent subcategory of finitely presented R-module...

Consider the abelian category Ck of commutative group schemes of finite type over a field k. By results of Serre and Oort, Ck has homological dimension 1 (resp. 2) if k is algebraically closed of characteristic 0 (resp. positive). In this article, we explore the abelian category of commutative algebraic groups up to isogeny, defined as the quotient of Ck by the full subcategory Fk of finite k-g...

Let R be an associative ring with non-zero identity. For a Serre subcategory C of the category R-mod of left R-modules, we consider the class AC of all modules that do not belong to C, but all of their proper submodules belong to C. Alongside of basic properties of such associated classes of modules, we will prove that every uniform module of AC has a local endomorphism ring. Moreover, if R is ...

Let R be a commutative ring. A full additive subcategory C of R-modules is triangulated if whenever two terms of a short exact sequence belong to C, then so does the third term. In this note we give a classification of triangulated subcategories of finitely generated modules over a principal ideal domain. As a corollary we show that in the category of finitely generated modules over a PID, thic...

Let R be a commutative noetherian ring and let $$\mathfrak {a}$$ an ideal of R. In this paper, we study certain condition, namely $$C_{\mathfrak {a}}$$ , introduced by Aghapournahr Melkersson, on the extension two subcategories R-modules. We develop some main results Yoshizawa (Proc Am Math Soc 140(7):2293–2305, 2012; J Commut Algebra 13:137–155, 2021). As example subcategories, weakly Laskeria...

We prove non-degeneracy of the cohomological bulk and boundary traces for general open-closed Landau-Ginzburg models associated to a pair $(X,W)$, where $X$ is non-compact complex manifold with trivial canonical line bundle $W$ complex-valued holomorphic function defined on $X$, assuming only that critical locus compact (but may not consist isolated points). These results can be viewed as certa...

Abstract Let $\mathscr{C}$ be a $(d+2)$ -angulated category with d -suspension functor $\Sigma^d$ . Our main results show that every Serre on is functor. We also has $\mathbb{S}$ if and only Auslander–Reiten -angles. Moreover, $\tau_d=\mathbb{S}\Sigma^{-d}$ where $\tau_d$ -Auslander–Reiten translation. These generalize work by Bondal–Kapranov Reiten–Van den Bergh. As an application, we prove fo...

For a suitable triangulated category $\mathcal{T}$ with Serre functor $S$ and full precovering subcategory $\mathcal{C}$ closed under summands extensions, an indecomposable object $C$ in is called Ext-projective if Ext$^1(C,\mathcal{C})=0$. Then there no Auslander-Reiten triangle end term $C$. In this paper, we show that if, for such $C$, minimal right almost split morphism $\beta:B\rightarrow ...

0 Introduction This is a reasonably faithful account of the ve lectures I delivered at the summer course \Geometria Algebraica no Commutativa y Espacios Cuanti-cos" for graduate students, in Spain, July 25{29, 1994. The material covered was, for the most part, an abridged version of Artin and Zhang's paper 2]. Fix a eld k. Given a Z-graded k-algebra, A say, which for simplicity is assumed to be...