arXiv subjects

Mikhail V. Bondarko

Publications and source records attributed to Mikhail V. Bondarko.

At least 19 recordsLinked to original sources

arXiv2025

On t-structures adjacent and orthogonal to weight structures

We study $t$-structures (on triangulated categories) that are closely related to weight structures. A $t$-structure couple $t=(C_{t\le 0},C_{t\ge 0})$ is said to be adjacent to a weight structure $w=(C_{w\le 0}, C_{w\ge 0})$ if $C_{t\ge 0}=C_{w\ge 0}$. For a category $C$ that satisfies the Brown representability property we prove that $t$ that is adjacent to $w$ exists if and only if $w$ is smashing (that is, "respects C-coproducts"). The heart $Ht$ of this $t$ is the category of those functors $Hw^{op}\to Ab$ that respect products (here $Hw$ is the heart of $w$); the result has important applications. We prove several more statements on constructing $t$-structures starting from weight structures; we look for a strictly orthogonal $t$-structure $t$ on some $C'$ (where $C,C'$ are triangulated subcategories of a common $D$) such that $C'_{t\le 0}$ (resp. $C'_{t\ge 0}$) is characterized by the vanishing of morphisms from $C_{w\ge 1}$ (resp. $C_{w\le -1}$). Some of these results generalize properties of semi-orthogonal decompositions proved in the previous paper, and can be applied to various derived categories of (quasi)coherent sheaves on a scheme $X$ that is projective over an affine noetherian one. We also study hearts of orthogonal $t$-structures and their restrictions, and prove some statements on "reconstructing" weight structures from orthogonal $t$-structures.

math.KT↗

arXiv2025

Bases of associated Galois modules in general wildly ramified extensions and in elementary abelian extensions of degree $p^2$

For a wildly ramified extension $K/k$ of complete discrete valuation fields we study collections of elements of $k[G]$ (where $G=Gal(K/k)$) that fit well for constructing bases of various associated Galois modules and orders. In the case $G=(Z/pZ)^2$ (where $p$ is the characteristic of residue fields) we are able to compute the action of the elements $(σ_1-1)^i(σ_2-1)^j,\ 0\le i,j\le p-1,$ on the valuation filtration; here $σ_1,σ_2$ are generators of $G$. If the ramification jumps of $K/k$ are distinct modulo $p^2$ then these elements do yield "good enough" bases in question.

math.AG↗

arXiv2025

Some general étale Weak Lefschetz-type theorems

We establish new general etale versions of theorems of Barth and Sommese. Respectively, we compute the lower etale cohomology of closed subvarieties of $P^N$ of small codimensions and of their preimages with respect to proper morphisms (that are not necessarily finite; this statement is completely new), and also of the zero loci of sections of ample vector bundles; all these statements are valid over fields of arbitrary characteristics. To obtain these results, we use a new 'fat hyperplane section' Weak Lefschetz-type theorem for etale cohomology of non-projective varieties that is related to a result of Goresky and MacPherson (over complex numbers).

math.AG↗

arXiv2024

Completing hearts of triangulated categories via weight-exact localizations

We study a weight-exact localization pi of a well generated triangulated category C along with the embedding of the hearts of adjacent t-structures coming from the functor right adjoint to pi. We prove that the functors relating the corresponding four hearts are completely determined by the heart Hw of the weight structure on C along with the set of Hw-morphisms that we invert via pi; it also suffices to know the corresponding embedding of the hearts of t-structures. Our results generalize the description of non-commutative localizations of rings in terms of weight-exact localizations given in an earlier paper of the first author. That paper was essentially devoted to weight-exact localizations by compactly generated subcategories, whereas in the current text we focus on "more complicated" localizations. We recall that two types of localizations of the sort we are interested in were studied by several authors. They took C=D(R-mod); the heart of the first t-structure was equivalent to R-mod, and the second heart was equivalent to the exact abelian category $U_{contra}\subset R-mod$ of U-contramodules (corresponding to a set of Proj R-mod-morphisms U related either to a homological ring epimorphism $u:R\to UU$ or to an ideal of I of R). The functor $R-mod\to U_{contra}$ induced by pi is a certain completion one. Consequently, the hearts of the corresponding weight structures are equivalent to Proj R-mod and to the subcategory of projective objects of U_{contra}, respectively. Moreover, the connecting functors between these categories are isomorphic to ones coming from any weight structure class-generated by a single compact object whose endomorphism ring is R^{op}; in particular, one can take R=Z and C=SH and re-prove some important statements due to Bousfield.

math.CT↗

arXiv2023

Producing "new" semi-orthogonal decompositions in arithmetic geometry

This paper is devoted to constructing "new" admissible subcategories and semi-orthogonal decompositions of triangulated categories out of "old" ones. For two triangulated subcategories $T$ and $T'$ of a certain $D$ and a decomposition $(L,R)$ of $T$ we look either for a decomposition $(L',R')$ of $T'$ such that there are no non-zero $D$-morphisms from $L$ into $L'$ and from $R$ into $R'$, or for a decomposition $(L_D,R_D)$ of $D$ such that $L_D\cap T=L$ and $R_D\cap T=R$. We prove some general existence statements (that also extend to semi-orthogonal decompositions with any number of components) and apply them to various derived categories of coherent sheaves over a scheme $X$ that is proper over a noetherian ring $R$. This gives a one-to-one correspondence between semi-orthogonal decompositions of $D_{perf}(X)$ and $D^b_{coh}(X)$; the latter extend to $D^-_{coh}(X)$, $D^+_{coh}({Qcoh}(X))$, $D_{coh}({Qcoh}(X))$, and $D({Qcoh}(X))$ under very mild conditions. In particular, we obtain a vast generalization of a theorem of J. Karmazyn, A. Kuznetsov, and E. Shinder. These applications rely on recent results of Neeman that express $D^b_{coh}(X)$ and $D^-_{coh}(X)$ in terms of $D_{perf}(X)$ along with its new variations corresponding to $D^+_{coh}({Qcoh}(X))$ and $D_{coh}({Qcoh}(X))$. We also discuss an application of this theorem to the construction of certain adjoint functors.

math.AG↗

arXiv2022

Killing weights from the perspective of t-structures

This paper is devoted to morphisms killing weights in a range (as defined by the first author) and to objects without these weights (as essentially defined by J. Wildeshaus) in a triangulated category endowed with a weight structure w. We describe several new criteria for morphisms and objects to satisfy these conditions. In some of them we use virtual t-truncations and a t-structure adjacent to w. In the case where the latter exists we prove that a morphism kills weights $m,...,n$ if and only if it factors through an object without these weights; we also construct new families of torsion theories and projective and injective classes. As a consequence, we obtain some "weakly functorial decompositions" of spectra (in the stable homotopy category SH) and a new description of those morphisms that act trivially on degree zero singular cohomology with coefficients in every abelian group.

math.KT↗

arXiv2021

On morphisms killing weights and Hurewicz-type theorems

We study "canonical weight decompositions" slightly generalizing that defined by J. Wildeshaus. For an triangulated category $C$, any integer $n$, and a weight structure $w$ on $C$ a triangle $LM\to M\to RM\to LM[1]$, where $LM$ is of weights at most $m-1$ and $RM$ is of weights at least $n+1$ for some $m\le n$, is determined by $M$ if exists. This happens if and only if the weight complex $t(M)\in Obj K(Hw)$ ($Hw$ is the heart of $w$) is homotopy equivalent to a complex with zero terms in degrees $-n,\dots, -m$; hence this condition can be "detected" via pure functors. One can also take $m=-\infty$ or $n=+\infty$ to obtain that the weight complex functor is "conservative and detects weights up to objects of infinitely small and infinitely large weights"; this is a significant improvement over previously known bounded conservativity results. Applying this statement we "calculate intersections of purely generated subcategories" and prove that certain weight-exact functors are conservative up to weight-degenerate objects. The main tool is the new interesting notion of morphisms killing weights $m,\dots, n$ that we study in detail as well. We apply general results to equivariant stable homotopy categories and spherical weight structures for them (as introduced in the previous paper) and obtain a certain converse to the (equivariant) stable Hurewicz theorem. In particular, the singular homology of a spectrum $E$ vanishes in negative degrees if and only if $E$ is an extension of a connective spectrum by an acyclic one.

math.KT↗

arXiv2021

Smooth weight structures and birationality filtrations on motivic categories

We study various triangulated motivic categories and introduce a vast family of aisles (these are certain classes of objects) in them. These aisles are defined in terms of the corresponding "motives" (or motivic spectra) of smooth varieties in them; we relate them to the corresponding homotopy t-structures. We describe our aisles in terms of stalks at function fields and prove that they widely generalize the ones corresponding to slice filtrations. Further, the filtrations on the "homotopy hearts" $Ht_{hom}^{eff}$ of the corresponding effective subcategories that are induced by these aisles can be described in terms of (Nisnevich) sheaf cohomology as well as in terms of the Voevodsky contractions $-_{-1}$. Respectively, we express the condition for an object of $Ht_{hom}^{eff}$ to be weakly birational (i.e., that its $n+1$th contraction is trivial or, equivalently, the Nisnevich cohomology vanishes in degrees $>n$ for some $n\ge 0$) in terms of these aisles; this statement generalizes well-known results of Kahn and Sujatha. Next, these classes define weight structures $w_{Smooth}^{s}$ (where $s=(s_{j})$ are non-decreasing sequences parameterizing our aisles) that vastly generalize the Chow weight structures $w_{Chow}$ defined earlier. Using general abstract nonsense we also construct the corresponding adjacent $t-$structures $t_{Smooth}^{s}$ and prove that they give the birationality filtrations on $Ht^{eff}_{hom}$. Moreover, some of these weight structures induce weight structures on the corresponding $n-$birational motivic categories (these are the localizations by the levels of the slice filtrations). Our results also yield some new unramified cohomology calculations.

math.AG↗

arXiv2021

On perfectly generated weight structures and adjacent $t$-structures

This paper is dedicated to the study of smashing weight structures (one may say that these are weight structures "coherent with arbitrary coproducts"), and the application of their properties to $t$-structures. In particular, we prove that hearts of compactly generated $t$-structures are Grothendieck abelian; this statement strengthens earlier results of several other authors. The central theorem of the paper is as follows: any perfect set of objects of a triangulated category generates a weight structure; we say that weight structures obtained this way are perfectly generated. An important family of perfectly generated weight structures are the ones adjacent to compactly generated $t$-structures; they give injective cogenerators for the hearts of the latter. We also establish the following not so explicit result: any smashing weight structure on a well generated triangulated category (this class of categories contains compactly generated ones) is perfectly generated; actually, we prove more than that. Moreover, we give a classification of compactly generated torsion theories (these generalize both weight structures and $t$-structures) that extends the corresponding result of D. Pospisil D. and J. Šťoviček to arbitrary smashing triangulated categories.

math.KT↗

arXiv2020

On weight complexes, pure functors, and detecting weights

This paper is dedicated to the study of weight complexes (defined on triangulated categories endowed with weight structures) and their applications. We introduce pure (co)homological functors that "ignore all non-zero weights"; these have a nice description in terms of weight complexes. For the weight structure $w^G$ generated by the orbit category in the $G$-equivariant stable homotopy category $SH(G)$ the corresponding pure cohomological functors into abelian groups are the Bredon cohomology associated to Mackey functors ones; pure functors related to motivic weight structures are also quite useful. Our results also give some (more) new weight structures. Moreover, we prove that certain exact functors are conservative and "detect weights".

math.KT↗

arXiv2020

On Chow-weight homology of motivic complexes and its relation to motivic homology

We study in detail the so-called Chow-weight homology of Voevodsky motivic complexes and relate it to motivic homology. We generalize earlier results and prove that the vanishing of higher motivic homology groups of a motif $M$ implies similar vanishing for its Chow-weight homology along with effectivity properties of the higher terms of its weight complex $t(M)$ and of higher Deligne weight quotients of its cohomology. Applying this statement to motives with compact support we obtain a similar relation between the vanishing of Chow groups and the cohomology with compact support of varieties. Moreover, we prove that if higher motivic homology groups of a geometric motif or a variety over a universal domain are torsion (in a certain "range") then the exponents of these groups are uniformly bounded. To prove our main results we study Voevodsky slices of motives. Since the slice functors do not respect the compactness of motives, the results of the previous Chow-weight homology paper are not sufficient for our purposes; this is our main reason to extend them to ($w_{Chow}$-bounded below) motivic complexes.

math.AG↗

arXiv2020

The hearts of weight structures are the weakly idempotent complete categories

In this note we prove that additive categories that occur as hearts of weight structures are precisely the weakly idempotent completecategories, that is, the categories where all split monomorphisms give direct sum decompositions. We also give several other conditions equivalent to weak idempotent completeness (some of them are completely new), and discuss weak idempotent completions of additive categories.

math.CT↗

arXiv2020

On Chow-pure cohomology and Euler characteristics for motives and varieties, and their relation to unramified cohomology and Brauer groups

We study Grothedieck groups of triangulated categories using weight structures, weight complexes, and the corresponding pure (co)homological functors. We prove some general statements on $K_0$ of weighted categories and apply it to Voevodsky motives endowed with so-called Chow weight structures. We obtain certain "motivic substitutes" for smooth compactifications of smooth varieties over arbitrary perfect fields; this enables us to make certain unramified cohomology and Euler characteristic calculations that are closely related to results of T. Ekedahl and B. Kahn.

math.AG↗

arXiv2019

On infinite effectivity of motivic spectra and the vanishing of their motives

This paper is dedicated to the study of the kernel of the "compact motivization" functor $M_{k}^c:SH^c(k)\to DM^c(k)$ (i.e., we try to describe those compact objects of $SH(k)$ whose associated motives vanish. Moreover, we study the question when the $m$-connectivity of $M^c_{k}(E)$ ensures the $m$-connectivity of $E$ itself (with respect to the corresponding homotopy t-structures). We prove that the kernel of $M_{k}^c$ vanishes and the corresponding "connectivity detection" statement is also valid if and only if $k$ is a non-orderable field; this is an easy consequence of the corresponding results of T. Bachmann (who considered the case where the $2$-adic cohomological dimension of $k$ is finite). We also sketch a deduction of these statements from the "slice-convergence" results of M. Levine. Moreover, for a general $k$ we prove that this kernel does not contain any $2$-torsion; the author also suspects that all its elements are odd torsion. Besides we prove that the kernel in question consists exactly of "infinitely effective" (in the sense of Voevodsky's slice filtration) objects of $SH^c(k)$ (assuming that the exponential characteristic of $k$ is inverted in the coefficient ring). These result allow (following another idea of Bachmann) to carry over his results on the tensor invertibility of certain motives of affine quadrics to the corresponding motivic spectra whenever $k$ is non-orderable. We also generalize a theorem of A. Asok.

math.AG↗

arXiv2019

On torsion pairs, (well generated) weight structures, adjacent $t$-structures, and related (co)homological functors

The paper contains a collection of results related to weight structures, $t$-structures, and (more generally) to torsion pairs. For any weight structure $w$ we study (co)homological pure functors; these "ignore all weights except weight zero" and have already found several applications. We also study virtual $t$-truncations of cohomological functors coming from $w$. These are closely related to $t$-structures; so we prove in several cases (including certain categories of coherent sheaves) that $w$ "gives" a $t$-structure (that is adjacent or $Φ$-orthogonal to it). We also study in detail "well generated" weight structures (and prove that any perfect set of objects generates a weight structure). The existence of weight structures right adjacent to compactly generated $t$-structures (and constructed using Brown-Comenetz duality) implies that the hearts of the latter have injective cogenerators and satisfy the AB3* axiom; actually, "most of them" are Grothendieck abelian (due to the existence of "regularly orthogonal" weight structures). It is convenient for us to use the notion of torsion pairs; these essentially generalize both weight structures and $t$-structures. We prove several properties of torsion pairs (that are rather parallel to that of weight structures); we also generalize a theorem of D. Pospisil and J. Stovicek to obtain a classification of compactly generated torsion pairs.

math.KT↗

arXiv2019

Detecting effectivity of motives, their weights, connectivity, and dimension via Chow-weight (co)homology: a "mixed motivic decomposition of the diagonal"

We describe certain criteria for a motif $M$ to be $r$-effective, i.e., to belong to the $r$th Tate twist $Obj DM^{eff}_{gm,R}(r)=Obj DM^{eff}_{gm,R} \otimes L^{\otimes r}$ of effective Voevodsky motives (for $r\ge 1$; $R$ is the coefficient ring). In particular, $M$ is 1-effective if and only if a complex whose terms are certain Chow groups of zero-cycles is acyclic. The dual to this statement checks whether an effective motif $M$ belongs to the subcategory of $DM^{eff}_{gm,R}$ generated by motives of varieties of dimension $\le r$. These criteria are formulated in terms of the Chow-weight (co)homology of $M$. These (co)homology theories are introduced in the current paper and have several (other) remarkable properties: they yield a bound on the "weights" of $M$ (in the sense of the Chow weight structure defined by the first author) and detect the effectivity of "the lower weight pieces" of $M$. We also calculate the "connectivity" of $M$ (in the sense of Voevodsky's homotopy t-structure) and prove that the exponents of the higher motivic homology groups (of an "integral" motif) are bounded whenever these groups are torsion. These motivic properties of $M$ have important consequences for its cohomology. As a corollary, we prove that if Chow groups of an arbitrary variety $X$ vanish up to dimension $r-1$ then the highest Deligne weight factors of the (singular or étale) cohomology of $X$ with compact support are $r$-effective in the naturally defined sense. Our results yield a vast generalization of the so-called "decomposition of the diagonal" statements.

math.AG↗

arXiv2019

On weakly negative subcategories, weight structures, and (weakly) approximable triangulated categories

We prove that certain triangulated categories are (weakly) approximable in the sense of A. Neeman. We prove that a triangulated $C$ that is compactly generated by a single object $G$ is weakly approximable if $C(G,G[i])=0$ for $i>1$ (we say that $G$ is weakly negative if this assumption is fulfilled; the case where the equality $C(G,G[1])=0$ is fulfilled as well was mentioned by Neeman himself). Moreover, if $G\cong \bigoplus_{0\le i\le n}G_i$ and $C(G_i,G_j[1])=0$ whenever $i\le j$ then $C$ is also approximable. The latter result can be useful since (under a few more additional assumptions) it allows to characterize a certain explicit subcategory of $C$ as the category of finite cohomological functors from the subcategory $C^c$ of compact objects of $C$ into $R$-modules (for a noetherian commutative ring $R$ such that $C$ is $R$-linear). One may apply this statement to the construction of certain adjoint functors and $t$-structures. Our proof of (weak) approximability of $C$ under the aforementioned assumptions is closely related to (weight decompositions for) certain (weak) weight structures, and we discuss this relationship in detail.

math.KT↗

arXiv2019

From weight structures to (orthogonal) $t$-structures and back

A $t$-structure $t=(C_{t\le 0},C_{t\ge 0})$ on a triangulated category $C$ is right adjacent to a weight structure $w=(C_{w\le 0}, C_{w\ge 0})$ if $C_{t\ge 0}=C_{w\ge 0}$; then $t$ can be uniquely recovered from $w$ and vice versa. We prove that if $C$ satisfies the Brown representability property then $t$ that is adjacent to $w$ exists if and only if $w$ is smashing (i.e., coproducts respect weight decompositions); then the heart $Ht$ is the category of those functors $Hw^{op}\to Ab$ that respect products. The dual to this statement is related to results of B. Keller and P. Nicolas. We also prove that an adjacent $t$ exists whenever $w$ is a bounded weight structure on a saturated $R$-linear category $C$ (for a noetherian ring $R$); for $C=D^{perf}(X)$, where the scheme $X$ is regular and proper over $R$, this gives 1-to-1 correspondences between bounded weights structures on $C$ and the classes of those bounded $t$-structures on it such that $Ht$ has either enough projectives or injectives. We generalize this existence statement to construct (under certain assumptions) a $t$-structure $t$ on a triangulated category $C'$ such that $C$ and $C'$ are subcategories of a common triangulated category $D$ and $t$ is right orthogonal to $w$. In particular, if $X$ is proper over $R$ but not necessarily regular then one can take $C=D^{perf}(X)$, $C'=D^b_{coh}(X)$ or $C'=D^-_{coh}(X)$, and $D=D_{qc}(X)$. We also study hearts of orthogonal $t$-structures and their restrictions, and prove some statements on "reconstructing" weight structures from orthogonal $t$-structures. The main tool of this paper are virtual $t$-truncations of (cohomological) functors; these are defined in terms of weight structures and "behave as if they come from $t$-truncations" whether $t$ exists or not.

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