K-theory of n-coherent rings

Let [Formula: see text] be a strong [Formula: see text]-coherent ring such that each finitely [Formula: see text]-presented [Formula: see text]-module has finite projective dimension. We consider [Formula: see text] the full subcategory of [Formula: see text]-Mod of finitely [Formula: see text]-presented modules. We prove that [Formula: see text] is an exact category, [Formula: see text] for every [Formula: see text] and we obtain an expression of [Formula: see text].

On the Frobenius functor for symmetric tensor categories in positive characteristic

We develop a theory of Frobenius functors for symmetric tensor categories (STC) {\mathcal{C}} over a field {\boldsymbol{k}} of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor {F:\mathcal{C}\to\mathcal{C}\boxtimes{\rm Ver}_{p}}, where {{\rm Ver}_{p}} is the Verlinde category (the semisimplification of {\mathop{\mathrm{Rep}}\nolimits_{\mathbf{k}}(\mathbb{Z}/p)}); a similar construction of the underlying additive functor appeared independently in [K. Coulembier, Tannakian categories in positive characteristic, preprint 2019]. This generalizes the usual Frobenius twist functor in modular representation theory and also the one defined in [V. Ostrik, On symmetric fusion categories in positive characteristic, Selecta Math. (N.S.) 26 2020, 3, Paper No. 36], where it is used to show that if {\mathcal{C}} is finite and semisimple, then it admits a fiber functor to {{\rm Ver}_{p}}. The main new feature is that when {\mathcal{C}} is not semisimple, F need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor {\mathcal{C}\to{\rm Ver}_{p}}. We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of F, and use it to show that for categories with finitely many simple objects F does not increase the Frobenius–Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which F is exact, and define the canonical maximal Frobenius exact subcategory {\mathcal{C}_{\rm ex}} inside any STC {\mathcal{C}} with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius–Perron dimension is preserved by F. One of our main results is that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to {{\rm Ver}_{p}}. This is the strongest currently available characteristic p version of Deligne’s theorem (stating that a STC of moderate growth in characteristic zero is the representation category of a supergroup). We also show that a sufficiently large power of F lands in {\mathcal{C}_{\rm ex}}. Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category (namely, one having a fiber functor into the category of representations of the triangular Hopf algebra {\boldsymbol{k}[d]/d^{2}} with d primitive and R-matrix {R=1\otimes 1+d\otimes d}), and show that a STC with Chevalley property is (almost) Frobenius exact. Finally, as a by-product, we resolve Question 2.15 of [P. Etingof and S. Gelaki, Exact sequences of tensor categories with respect to a module category, Adv. Math. 308 2017, 1187–1208].

$$d\mathbb {Z}$$-Cluster tilting subcategories of singularity categories

For an exact category $${{\mathcal {E}}}$$ E with enough projectives and with a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory, we show that the singularity category of $${{\mathcal {E}}}$$ E admits a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory. To do this we introduce cluster tilting subcategories of left triangulated categories, and we show that there is a correspondence between cluster tilting subcategories of $${{\mathcal {E}}}$$ E and $${\underline{{{\mathcal {E}}}}}$$ E ̲ . We also deduce that the Gorenstein projectives of $${{\mathcal {E}}}$$ E admit a $$d\mathbb {Z}$$ d Z -cluster tilting subcategory under some assumptions. Finally, we compute the $$d\mathbb {Z}$$ d Z -cluster tilting subcategory of the singularity category for a finite-dimensional algebra which is not Iwanaga–Gorenstein.

Relatively divisible and relatively flat objects in exact categories

We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair [Formula: see text] in an exact category [Formula: see text], [Formula: see text] coincides with the class of relatively flat objects of [Formula: see text] for some relative projectively generated exact structure, while [Formula: see text] coincides with the class of relatively divisible objects of [Formula: see text] for some relative injectively cogenerated exact structure. We exhibit Galois connections between relative cotorsion pairs in exact categories, relative projectively generated exact structures and relative injectively cogenerated exact structures in additive categories. We establish closure properties and characterizations in terms of the approximation theory.

Purity and flatness in symmetric monoidal closed exact categories

Let [Formula: see text] be a symmetric monoidal closed exact category. This category is a natural framework to define the notions of purity and flatness. When [Formula: see text] is endowed with an injective cogenerator with respect to the exact structure, we show that an object [Formula: see text] in [Formula: see text] is flat if and only if any conflation ending in [Formula: see text] is pure. Furthermore, we prove a generalization of the Lambek Theorem (J. Lambek, A module is flat if and only if its character module is injective, Canad. Math. Bull. 7 (1964) 237–243) in [Formula: see text]. In the case [Formula: see text] is a quasi-abelian category, we prove that [Formula: see text] has enough pure injective objects.

The generalized Auslander–Reiten duality on an exact category

We introduce a notion of generalized Auslander–Reiten duality on a Hom-finite Krull–Schmidt exact category [Formula: see text]. This duality induces the generalized Auslander–Reiten translation functors [Formula: see text] and [Formula: see text]. They are mutually quasi-inverse equivalences between the stable categories of two full subcategories [Formula: see text] and [Formula: see text] of [Formula: see text]. A non-projective indecomposable object lies in the domain of [Formula: see text] if and only if it appears as the third term of an almost split conflation; dually, a non-injective indecomposable object lies in the domain of [Formula: see text] if and only if it appears as the first term of an almost split conflation. We study the generalized Auslander–Reiten duality on the category of finitely presented representations of locally finite interval-finite quivers.