Definable Subcategories Research Articles

A monoidal analogue of the 2-category anti-equivalence between ABEX and DEF

We prove that the 2-category of skeletally small abelian categories with exact monoidal structures is anti-equivalent to the 2-category of fp-hom-closed definable additive categories satisfying an exactness criterion. For a fixed finitely accessible category C with products and a monoidal structure satisfying the appropriate assumptions, we provide bijections between the fp-hom-closed definable subcategories of C , the Serre tensor-ideals of C fp - mod and the closed subsets of a Ziegler-type topology. For a skeletally small preadditive category A with an additive, symmetric, rigid monoidal structure we show that elementary duality induces a bijection between the fp-hom-closed definable subcategories of Mod - A and the definable tensor-ideals of A - Mod .

Pure Projective Tilting Modules

Let $T$ be a $1$-tilting module whose tilting torsion pair $({\mathcal T}, {\mathcal F})$ has the property that the heart ${\mathcal H}_t$ of the induced $t$-structure (in the derived category ${\mathcal D}({\rm Mod} \mbox{-} R)$ is Grothendieck. It is proved that such tilting torsion pairs are characterized in several ways: (1) the $1$-tilting module $T$ is pure projective; (2) ${\mathcal T}$ is a definable subcategory of ${\rm Mod} \mbox{-} R$ with enough pure projectives, and (3) both classes ${\mathcal T}$ and ${\mathcal F}$ are finitely axiomatizable. This study addresses the question of Saorin that asks whether the heart is equivalent to a module category, i.e., whether the pure projective $1$-tilting module is tilting equivalent to a finitely presented module. The answer is positive for a Krull-Schmidt ring and for a commutative ring, every pure projective $1$-tilting module is projective. A criterion is found that yields a negative answer to Saorin's Question for a left and right noetherian ring. A negative answer is also obtained for a Dubrovin-Puninski ring, whose theory is covered in the Appendix. Dubrovin-Puninski rings also provide examples of (1) a pure projective $2$-tilting module that is not classical; (2) a finendo quasi-tilting module that is not silting; and (3) a noninjective module $A$ for which there exists a left almost split morphism $m: A \to B,$ but no almost split sequence beginning with $A.$

Purity in compactly generated derivators and t-structures with Grothendieck hearts

We study t-structures with Grothendieck hearts on compactly generated triangulated categories $${\mathcal {T}}$$ that are underlying categories of strong and stable derivators. This setting includes all algebraic compactly generated triangulated categories. We give an intrinsic characterisation of pure triangles and the definable subcategories of $${\mathcal {T}}$$ in terms of directed homotopy colimits. For a left nondegenerate t-structure $$\mathbf{t}=({\mathcal {U}},{\mathcal {V}})$$ on $${\mathcal {T}}$$ , we show that $${\mathcal {V}}$$ is definable if and only if $$\mathbf{t}$$ is smashing and has a Grothendieck heart. Moreover, these conditions are equivalent to $$\mathbf{t}$$ being homotopically smashing and to $$\mathbf{t}$$ being cogenerated by a pure-injective partial cosilting object. Finally, we show that finiteness conditions on the heart of $$\mathbf{t}$$ are determined by purity conditions on the associated partial cosilting object.

Two definable subcategories of maximal Cohen-Macaulay modules

Over a Cohen-Macaulay local ring we consider two extensions of the maximal Cohen-Macaulay modules from the viewpoint of definable subcategories, which are closed under direct limits, direct products and pure submodules. After presenting these categories, we compare them and consider which properties they inherit from the maximal Cohen-Macaulay modules. We then consider some further properties of these classes and how they interact with the entire module category.

Duality and contravariant functors in the representation theory of artin algebras

We know that the model theory of modules leads to a way of obtaining definable categories of modules over a ring [Formula: see text] as the kernels of certain functors [Formula: see text] rather than of functors [Formula: see text] which are given by a pp-pair. This paper will give various algebraic characterizations of these functors in the case that [Formula: see text] is an artin algebra. Suppose that [Formula: see text] is an artin algebra. An additive functor [Formula: see text] preserves inverse limits and [Formula: see text] is finitely presented if and only if there is a sequence of natural transformations [Formula: see text] for some [Formula: see text] which is exact when evaluated at any left [Formula: see text]-module. Any additive functor [Formula: see text] with one of these equivalent properties has a definable kernel, and every definable subcategory of [Formula: see text] can be obtained as the kernel of a family of such functors. In the final section, a generalized setting is introduced, so that our results apply to more categories than those of the form [Formula: see text] for an artin algebra [Formula: see text]. That is, our results are extended to those locally finitely presented [Formula: see text]-linear categories whose finitely presented objects form a dualizing variety, where [Formula: see text] is a commutative artinian ring.

Silting and cosilting classes in derived categories

An important result in tilting theory states that a class of modules over a ring is a tilting class if and only if it is the Ext-orthogonal class to a set of compact modules of bounded projective dimension. Moreover, cotilting classes are precisely the resolving and definable subcategories of the module category whose Ext-orthogonal class has bounded injective dimension.In this article, we prove a derived counterpart of the statements above in the context of silting theory. Silting and cosilting complexes in the derived category of a ring generalise tilting and cotilting modules. They give rise to subcategories of the derived category, called silting and cosilting classes, which are part of both a t-structure and a co-t-structure. We characterise these subcategories: silting classes are precisely those which are intermediate and Ext-orthogonal classes to a set of compact objects, and cosilting classes are precisely the cosuspended, definable and co-intermediate subcategories of the derived category.

Torsion pairs in silting theory

In the setting of compactly generated triangulated categories, we show that the heart of a (co)silting t-structure is a Grothendieck category if and only if the (co)silting object satisfies a purity assumption. Moreover, in the cosilting case the previous conditions are related to the coaisle of the t-structure being a definable subcategory. If we further assume our triangulated category to be algebraic, it follows that the heart of any nondegenerate compactly generated t-structure is a Grothendieck category.

Definable categories

We introduce the notion of a definable category–a category equivalent to a full subcategory of a locally finitely presentable category that is closed under products, directed colimits and pure subobjects. Definable subcategories are precisely the finite-injectivity classes. We prove a 2-duality between the 2-category of small exact categories and the 2-category of definable categories, and provide a new proof of its additive version. We further introduce a third vertex of the 2-category of regular toposes and show that the diagram of 2-(anti-)equivalences between three 2-categories commutes; the corresponding additive triangle is well-known.

Representation embeddings, interpretation functors and controlled wild algebras

We establish a number of results which say, roughly, that interpretation functors preserve algebraic complexity. First we show that representation embeddings between categories of modules of finite-dimensional algebras induce embeddings of lattices of pp formulas and hence are non-decreasing on Krull-Gabriel dimension and uniserial dimension. A consequence is that the category of modules of any wild finite-dimensional algebra has width $\infty$ and hence, if the algebra is countable, there is a superdecomposable pure-injective representation. It is conjectured that a stronger result is true: that a representation embedding from ${\rm Mod}\mbox{-}S$ to ${\rm Mod}\mbox{-}R$ admits an inverse interpretation functor from its image and hence that, in this case, ${\rm Mod}\mbox{-}R$ interprets ${\rm Mod}\mbox{-}S$. This would imply, for instance, that every wild category of modules interprets the (undecidable) word problem for (semi)groups. We show that the conjecture holds for finitely controlled representation embeddings. Finally we prove that if $R,S$ are finite dimensional algebras over an algebraically closed field and $I:{\rm Mod}\mbox{-}R\rightarrow{\rm Mod}\mbox{-}S$ is an interpretation functor such that the smallest definable subcategory containing the image of $I$ is the whole of ${\rm Mod}\mbox{-}S$ then, if $R$ is tame, so is $S$ and similarly, if $R$ is domestic, then $S$ also is domestic.

Torsion-free, Divisible, and Mittag-Leffler Modules

We study (relative) 𝒦-Mittag–Leffler modules, with emphasis on the class 𝒦 of absolutely pure modules. A final goal is to describe the 𝒦-Mittag–Leffler abelian groups as those that are, modulo their torsion part, ℵ1-free. Several more general results of independent interest are derived on the way. In particular, every flat 𝒦-Mittag–Leffler module (for 𝒦 as before) is Mittag–Leffler. A question about the definable subcategories generated by the divisible modules and the torsion-free modules, resp., has been left open.

Modules with irrational slope over tubular algebras

Let A be a tubular algebra and let r be a positive irrational. Let D r be the definable subcategory of A-modules of slope r. Then the width of the lattice of pp formulas for D r is ∞. It follows that if A is countable, then there is a superdecomposable pure-injective module of slope r.

Almost Dual Pairs and Definable Classes of Modules

In [7] Holm considers categories of right modules dual to those with support in a set of finitely presented modules. We extend some of his results by placing them in the context of elementary duality on definable subcategories. In doing so we also prove that dual modules have enough indecomposable direct summands.

DEFINABLE SUBCATEGORIES OVER PURE SEMISIMPLE RINGS

Let R be a right pure semisimple ring, and [Formula: see text] be a family of sources of left almost split morphisms in mod-R. We study the definable subcategory [Formula: see text] in Mod-R determined by the family [Formula: see text], and show that [Formula: see text] has several nice properties similar to those of the category Mod-R. For example, its functor category [Formula: see text] is a module category, and preinjective objects of [Formula: see text] are sources of left almost split morphisms in [Formula: see text] and in mod-R. As an application, it is shown that if R is a right pure semisimple ring with no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in mod-R (in particular, if R is right pure semisimple hereditary), then any definable subcategory of Mod-R determined by a finite set of indecomposable right R-modules contains only finitely many non-isomorphic indecomposable modules.

WHEN COTORSION MODULES ARE PURE INJECTIVE

We characterize rings over which every cotorsion module is pure injective (Xu rings) in terms of certain descending chain conditions and the Ziegler spectrum, which renders the classes of von Neumann regular rings and of pure semisimple rings as two possible extremes. As preparation, descriptions of pure projective and Mittag–Leffler preenvelopes with respect to so-called definable subcategories and of pure generation for such are derived, which may be of interest on their own. Infinitary axiomatizations lead to coherence results previously known for the special case of flat modules. Along with pseudoflat modules we introduce quasiflat modules, which arise naturally in the model-theoretic and the category-theoretic contexts.

Pure Projective Approximations

AbstractA notion of good behavior is introduced for a definable subcategory of left R-modules. It is proved that every finitely presented left R-module has a pure projective left -approximation if and only if the associated torsion class of finite type in the functor category (mod-R, Ab) is coherent, i.e., the torsion subobject of every finitely presented object is finitely presented. This yields a bijective correspondence between such well-behaved definable subcategories and preenveloping subcategories of the category Add(R-mod) of pure projective left R-modules. An example is given of a preenveloping subcategory ⊆ Add(R-mod) that does not arise from a covariantly finite subcategory of finitely presented left R-modules. As a general example of this good behavior, it is shown that if R is a ring over which every left cotorsion R-module is pure injective, then the smallest definable subcategory (R-proj) containing every finitely generated projective module is well-behaved.

Identification of Common Transcriptional Regulatory Elements in Interleukin-17 Target Genes

Interleukin (IL)-17 is the founding member of a novel family of inflammatory cytokines. Although produced by T cells, IL-17 activates genes and signals typical of innate immune mediators such as tumor necrosis factor (TNF)-alpha and IL-1beta. Most IL-17 target genes characterized to date are cytokines or neutrophil-attractive chemokines. Our recent microarray studies identified an acute phase response gene, 24p3/lipocalin 2, as a novel IL-17-induced gene. Here we describe a detailed analysis of the 24p3 promoter. We find that, unlike cytokine or chemokine gene target genes, 24p3 is regulated primarily at the level of transcription rather than mRNA stability and that synergy between IL-17 and TNFalpha occurs at the level of the 24p3 promoter. Two key transcription factor binding sites (TFBS) were identified, corresponding to NF-kappaB and CCAAT/enhancer-binding protein (C/EBP). Deletion of either site eliminated 24p3 promoter activity in response to IL-17. These findings were strikingly similar to the IL-6 promoter, where IL-17-mediated regulation of both NF-kappaB and C/EBP is essential. To determine whether joint use of NF-kappaB and C/EBP is common to all IL-17 target genes, we performed a computational analysis on 18 well documented IL-17 target promoters to assess statistical enrichment of specific TFBSs. Indeed, NF-kappaB and C/EBP sites were over-represented in these genes, as were AP1 and OCT1 sites. Moreover, these promoters fell into three definable subcategories based on TFBS location and usage. Analysis of IL-17 target gene regulation is key for understanding this important host-defense molecule and also contributes to an understanding of upstream signaling mechanisms used by IL-17, either alone or in concert with TNFalpha.

Coherent Functors and Covariantly Finite Subcategories

Given a locally presentable additive category A, we study a class of covariantly finite subcategories which we call definable. A definable subcategory arises from a set of coherent functors F i on A by taking all objects X in A such that F i X=0 for all i. We give various characterizations of definable subcategories, demonstrating that all covariantly finite subcategories which arise in practice are of this form. This is based on a filtration of the category of all coherent functors on A.

The spectrum of a module category

Introduction The functor category Definable subcategories Left approximations duality Ideals in the category of finitely presented modules Endofinite modules Krull-Gabriel dimension The infinite radical Functors between module categories Tame algebras Rings of definable scalars Reflective definable subcategories Sheaves Tame hereditary algebras Coherent rings Appendix A. Locally coherent Grothendieck categories Appendix B. Dimensions Appendix C. Finitely presented functors and ideals Bibliography.