Grigory Garkusha

• Professor
• Swansea University

We introduce and study matrix transfers to achieve elementary models for bivariant $K$-theory. They share lots of common properties with Voevodsky's framed correspondences and lead to symmetric matrix motives of algebraic varieties introduced in this paper. Symmetric matrix motives recover $K$-motives and fit in a closed symmetric monoidal triangul...

Using methods of stable homotopy theory, the category of symmetric quasi-coherent sheaves associated with non-commutative graded algebras with extra symmetries is introduced and studied in this paper. It is shown to be a closed symmetric monoidal Grothendieck category with invertible generators. It is proven that the category of quasi-coherent shea...

ECM Satellite Conference "Functor and Tensor Categories, Models, and Systems", Santiago de Compostela, Spain, June 2024.

The Ziegler spectrum for categories enriched in closed symmetric monoidal Grothendieck categories is defined and studied in this paper. It recovers the classical Ziegler spectrum of a ring. As an application, the Ziegler spectrum as well as the category of generalised quasi-coherent sheaves of a reasonable scheme is introduced and studied. It is sh...

An alternative approach to the classical Morel–Voevodsky stable motivic homotopy theory S H ( k ) SH(k) is suggested. The triangulated category of framed bispectra S H nis fr ( k ) SH_{\operatorname {nis}}^{\operatorname {fr}}(k) and effective framed bispectra S H nis fr , eff ( k ) SH_{\operatorname {nis}}^{\operatorname {fr},\operatorname {eff}}(...

We combine several mini miracles to achieve an elementary understanding of infinite loop spaces and very effective spectra in the algebro-geometric setting of motivic homotopy theory. Our approach combines $\Gamma$-spaces and Voevodsky's framed correspondences into the concept of framed motivic $\Gamma$-spaces; these are continuous or enriched func...

Собрав воедино несколько вычислительных значимых примеров, мы даем элементарное описание бесконечно-кратных пространств петель и вполне эффективных спектров в контексте мотивной гомотопической теории. Наш подход состоит в том, чтобы включить $\Gamma$-пространства Сигала и оснащенные соответствия Воеводского в единое понятие оснащенного мотивного $\...

In this paper, semilocal Milnor K-theory of fields is introduced and studied. A strongly convergent spectral sequence relating semilocal Milnor K-theory to semilocal motivic cohomology is constructed. In weight 2, the motivic cohomology groups $H^p_{zar} (k,\mathbb Z(2))$, $p\leq 1$, are computed as semilocal Milnor K-theory groups $\widehat{K}^M_{...

A general method of producing correspondences and spectral categories out of symmetric ring objects in general categories is given. As an application, stable homotopy theory of spectra $SH$ is recovered from modules over a commutative symmetric ring spectrum defined in terms of framed correspondences over an algebraically closed field. Another appl...

Homological Algebra for Enriched Grothendieck Categories

Complex cobordism spectrum via algebraic varieties

By a theorem of Mandell-May-Schwede-Shipley the stable homotopy theory of classical spectra is recovered from orthogonal spectra. In this paper general linear, special linear, symplectic, orthogonal and special orthogonal motivic spectra are introduced and studied. It is shown that the stable homotopy theory of motivic spectra is recovered from eac...

We investigate certain categorical aspects of Voevodsky's triangulated categories of motives. For this, various recollements for Grothendieck categories of enriched functors and their derived categories are established. In order to extend these recollements further with respect to Serre's localization, the concept of the (strict) Voevodsky property...

The machinery of framed (pre)sheaves was developed by Voevodsky. Based on the theory, framed motives of algebraic varieties are introduced and studied in [GP1]. An analog of Voevodsky's Cancellation Theorem is proved in this paper for framed motives stating that a natural map of framed $S^1$-spectra $M_{fr}(X)(n)\to\underline{Hom}(\mathbb{G},M_{fr}...

The category of framed correspondences $Fr_*(k)$, framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [20]. Based on the notes [20] a new approach to the classical Morel–Voevodsky motivic stable homotopy theory was developed in [8]. This approach converts the classical motivic stable homotopy theory into an equi...

We investigate certain categorical aspects of Voevodsky's triangulated categories of motives. For this, various recollements for Grothendieck categories of enriched functors and their derived categories are established. In order to extend these recollements further with respect to Serre's localization, the concept of the (strict) Voevodsky property...

We combine several mini miracles to achieve an elementary understanding of infinite loop spaces and very effective spectra in the algebro-geometric setting of motivic homotopy theory. Our approach combines $\Gamma$-spaces and framed correspondences into the concept of framed motivic $\Gamma$-spaces; these are continuous or enriched functors of two...

Using a recent computation of the rational minus part of SH(k) by Ananyevskiy--Levine--Panin, a theorem of Cisinski--Deglise and a version of the Roendigs--Ostvaer theorem, rational stable motivic homotopy theory over an infinite field of characteristic different from 2 is recovered in this paper from finite Milnor--Witt correspondences in the sens...

Given a perfect field of exponential characteristic $e$ and a functor $f:\mathcal A\to\mathcal B$ between symmetric monoidal strict $V$-categories of correspondences satisfying the cancellation property such that the induced morphisms of complexes of Nisnevich sheaves $$f_*:\mathbb Z_{\mathcal A}(q)[1/e]\to\mathbb Z_{\mathcal B}(q)[1/e],\quad q\geq...

An alternative approach to the classical Morel-Voevodsky stable motivic homotopy theory $SH(k)$ is suggested. The triangulated category of framed bispectra $SH_{nis}^{fr}(k)$ and effective framed bispectra $SH_{nis}^{fr,eff}(k)$ are introduced in the paper. Both triangulated categories only use Nisnevich local equivalences and have nothing to do wi...

Using the theory of framed correspondences developed by Voevodsky [24] and the machinery of framed motives introduced and developed in [6], various explicit fibrant resolutions for a motivic Thom spectrum E are constructed in this paper. It is shown that the bispectrum M G E (X) = (M E (X), M E (X)(1), M E (X)(2),. . .), each term of which is a twi...

The derived category D[C , V ] of the Grothendieck category of enriched functors [C , V ], where V is a closed symmetric monoidal Grothendieck category and C is a small V-category, is studied. We prove that if the derived category D(V) of V is a compactly generated triangulated category with certain reasonable assumptions on compact generators or K...

It is shown that the Grayson tower for $K$-theory of smooth algebraic varieties is isomorphic to the slice tower of $S^1$-spectra. We also extend the Grayson tower to bispectra and show that the Grayson motivic spectral sequence is isomorphic to the motivic spectral sequence produced by the Voevodsky slice tower for the motivic $K$-theory spectrum...

Given a perfect field of exponential characteristic $e$ and a functor $f:\mathcal A\to\mathcal B$ between symmetric monoidal strict $V$-categories of correspondences satisfying the cancellation property such that the induced morphisms of complexes of Nisnevich sheaves $$f_*:\mathbb Z_{\mathcal A}(q)[1/e]\to\mathbb Z_{\mathcal B}(q)[1/e],\quad q\geq...

Using a recent computation of the rational minus part of $SH(k)$ by Ananyevskiy-Levine-Panin, a theorem of Cisinski-Deglise and a version of the Roendigs-Ostvaer theorem, rational stable motivic homotopy theory over an infinite perfect field of characteristic different from 2 is recovered in this paper from finite Milnor-Witt correspondences in the...

The category of framed correspondences $Fr_*(k)$, framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [V2]. Based on the theory, framed motives are introduced and studied in [GP1]. The aim of this paper is to prove the following results stated in [GP1]: for any $k$-smooth scheme $X$ and any $n\geq 1$ the map of...

The category of framed correspondences $Fr_*(k)$ and framed sheaves were invented by Voevodsky in his unpublished notes [V2]. Based on the theory, framed motives are introduced and studied in [GP1]. These are Nisnivich sheaves of $S^1$-spectra and the major computational tool of [GP1]. The aim of this paper is to show the following result which is...

It is shown that the category of enriched functors [C , V ] is Grothendieck whenever V is a closed symmetric monoidal Grothendieck category and C is a category enriched over V. Localizations in [C , V ] associated to collections of objects of C are studied. Also, the category of chain complexes of generalized modules Ch(C R) is shown to be identifi...

The machinery of framed (pre)sheaves was developed by Voevodsky [V1]. Based on the theory, framed motives of algebraic varieties are introduced and studied in [GP1]. An analog of Voevodsky's Cancellation Theorem [V1] is proved in this paper for framed motives stating that a natural map of framed $S^1$-spectra $$M_{fr}(X)(n)\to\underline{\textrm{Hom...

This paper is to construct unstable, Morita stable and stable bivariant algebraic Kasparov K-theory spectra of k-algebras. These are shown to be homotopy invariant, excisive in each variable K-theories. We prove that the spectra represent universal unstable, Morita stable and stable bivariant homology theories respectively.

Using the theory of framed correspondences developed by Voevodsky, we introduce and study framed motives of algebraic varieties. They are the major computational tool for constructing an explicit quasi-fibrant motivic replacement of the suspension P^1-spectrum of any smooth scheme $X\in Sm/k$. Moreover, it is shown that the bispectrum $$(M_{fr}(X),...

To any admissible category of algebras and a family of fibrations on it a universal bivariant excisive homotopy invariant algebraic K-theory is associated. Also, Morita invariant and stable universal bivariant K-theories are studied. We introduce an additive category of correspondencies for non-unital algebras and study the problem of when stable b...

For any perfect field k a triangulated category of K-motives DK_(k) is constructed in the style of Voevodsky's construction of the category DM_(k). To each smooth k-variety X the K-motive is associated in the category DK_(k). Also, it is shown that K_n(X)=DK_(k)(M_K(X)[n],M_K(pt)), where K(X) is Quillen's K-theory of X.

A kind of motivic stable homotopy theory of algebras is developed. Explicit fibrant replacements for the S^1-spectrum and (S^1,\G)-bispectrum of an algebra are constructed. As an application, unstable, Morita stable and stable universal bivariant theories are recovered. These are shown to be embedded by means of contravariant equivalences as full t...

A kind of motivic algebra of spectral categories and modules over them is developed to introduce K-motives of algebraic varieties. As an application, bivariant algebraic K-theory as well as bivariant motivic kohomology groups are defined and studied. We use Grayson's machinery to produce the Grayson motivic spectral sequence connecting bivariant K-...

Various authors classified the thick triangulated ⊗-subcategories of the category of compact objects for appropriate compactly generated tensor triangulated categories by using supports of objects. In this paper we introduce R-supports for ring objects, showing that these completely determine the thick triangulated ⊗-subcategories. R-supports give...

Given a commutative ring R (respectively a positively graded commutative ring $A=\ps_{j\geq 0}A_j$ which is finitely generated as an A_0-algebra), a bijection between the torsion classes of finite type in Mod R (respectively tensor torsion classes of finite type in QGr A) and the set of all subsets Y\subset Spec R (respectively Y\subset Proj A) of...

Given a positively graded commutative coherent ring A=⊕j⩾0Aj, finitely generated as an A0-algebra, a bijection between the tensor Serre subcategories of qgrA and the set of all subsets Y⊆ProjA of the form Y=⋃i∈ΩYi with quasi-compact open complement ProjA∖Yi for all i∈Ω is established. To construct this correspondence, properties of the Ziegler and...

Given a quasi-compact, quasi-separated scheme X, a bijection between the tensor localizing subcategories of finite type in Qcoh(X) and the set of all subsets $Y\subseteq X$ of the form $Y=\bigcup_{i\in\Omega}Y_i$, with $X\setminus Y_i$ quasi-compact and open for all $i\in\Omega$, is established. As an application, there is constructed an isomorphis...

A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a quasi-isomorphism (or weak equivalence) for rings and shows that—similar to spaces—the derived category obtain...

Given a commutative coherent ring R, a bijective correspondence between the thick subcategories of perfect complexes D_{per}(R) and the Serre subcategories of finitely presented modules is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of (iso-classes for) indecomposable injective modules...

To any left system of diagram categories or to any left pointed derivateur (in the sense of Grothendieck) a K-theory space is associated. This K-theory space is shown to be canonically an infinite loop space and to have a lot of common properties with Waldhausen's K-theory. A weaker version of additivity is shown. Also, Quillen's K-theory of a larg...

A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a quasi-isomorphism (or weak equivalence) for rings and shows that - similar to spaces - the derived category ob...

The relationship between the Ziegler spectrum of (the category of modules over) a ring and the Ziegler spectrum of its derived category is investigated. Over von Neumann regular rings and hereditary rings the spectrum of the derived category is a disjoint union of copies of the spectrum of the ring but in general there are further indecomposable pu...

The additivity theorem for drivateurs associated to complicial biWaldhausen categories is proved. Also, to any exact category in the sense of Quillen a K-theory space is associated. This K-theory is shown to satisfy the additivity, approximation and resolution theorems.

The classes of fp-injective and fp-flat modules have been introduced in [8] to study F P-injective and weakly quasi-Frobenius rings. Both classes are definable (=elementary) in the first order language of modules and naturally generalize the corresponding classes of F P-injective and flat modules. Homological algebra based on F P-injective and flat...

Given a compactly generated triangulated category $\mathcal{T}$ and a generating set ℛ of compact objects, the class of ℛ-injective objects $\mathsf{Inj}_\mathcal{R}\ \mathcal{T}$ is introduced. If $X\in\mathcal{T}$ is compact and ℛ={ΣⁿX}n∈ℤ it is shown that there is a functor \[ \Gamma:\mathsf{Inj}\ \mathcal{S} \to \mathsf{Inj}_{\mathcal R}\ \math...

It is proved that a group ring R = AG is almost regular if and only if (i) the ring A is almost regular; (ii) the group G is locally finite; (iii) the order |H| of every finite subgroup H of G is invertible in A. Bibliography: 7 titles.

The additivity theorem for derivateurs associated to complicial biWaldhausen categories is proved. Also, to any exact category in the sense of Quillen a K-theory space is associated. This K-theory is shown to satisfy the additivity, approximation and resolution theorems.

The natural map of K-spectra K(R) → G(R) fits into a homotopy cofibre sequence \[ K (R)\to G (R)\to K ({\mathcal A}, w) \] where $({\mathcal A}, w)$ is the Waldhausen category of bounded chain complexes over the category of finitely presented modules, with weak equivalences being stable quasi-isomorphisms.

The classes of FP-injective and weakly quasi-Frobenius rings are investigated. The properties for both classes of rings are closely linked with embedding of finitely presented modules in fp-flat and free modules respectively. Using these properties, we describe the classes of coherent CF and FGF-rings. Moreover, it is proved that the group ring R(G...

The general theory of Grothendieck categories is presented. We systemize the principle methods and results of the theory, showing how these results can be used for studying rings and modules.

Duality between categories of finitely presented modules mod-R and R-mod is investigated. Also, the classes of absolutely pure, almost regular rings, fp-injective and fp-flat modules as well as their relationships with the Ziegler spectrum are investigated by using localising subcategories and their torsion functors in the category of functors (mod...