arXiv subjects

Mattia Ornaghi

Publications and source records attributed to Mattia Ornaghi.

11 recordsLinked to original sources

arXiv2026

New examples of non-Fourier-Mukai exact functors via non-isomorphic octahedra

We study a triangulated category $\mathscr S$ that admits a full and strong exceptional sequence of three objects with one-dimensional Hom spaces. We show that the isomorphism classes of exact functors from $\mathscr S$ to another triangulated category $\mathscr T$ are in bijection with the isomorphism classes of octahedra in $\mathscr T$ satisfying a natural condition. As an application, we construct an exact functor from $\mathscr S$ to $\mathbf D^b(\Bbbk[x]\text{-}\mathrm{mod})$ that does not admit a dg lift. This provides an explicit example of a non-Fourier-Mukai exact functor between $\mathbf D^b(\mathbb P^2)$ and $\mathbf D^b(\mathbb P^1)$.

math.AG↗

arXiv2026

Some properties of the A$_{\infty}$-nerve

The aim of this paper is to prove that the A$_{\infty}$-nerve of two quasi-equivalent A$_{\infty}$-categories (linear over a commutative ring) are weak-equivalent in the Joyal model structure. As a consequence we prove that the A$_{\infty}$-nerve of a pretriangulated A$_{\infty}$-category is a stable $\infty$-category.

math.AG↗

arXiv2025

A model structure on the category of A$_\infty$-categories with strict morphisms

We prove that the category of (strictly unital) A$_\infty$-categories, linear over a commutative ring $R$, with strict A$_\infty$-morphisms has a cofibrantly generated model structure. In this model structure every object is fibrant and the cofibrant objects have cofibrant morphisms. As a consequence we prove that the semi-free A$_\infty$-categories (resp. resolutions) are cofibrant objects (resp. resolution) in this model structure.

math.CT↗

arXiv2024

A remark on fibrancy of ($\mbox{A$_{\infty}$Cat}$,$W^{\tiny\mbox{A}_{\infty}}_{\tiny\mbox{qe}}$)

In this note we prove the existence, in the category of (strictly unital) A$_{\infty}$categories, of the pullback of a (strictly unital) A$_{\infty}$functor, satisfying a particular property (denoted by F1), along any A$_{\infty}$functor. As a consequence we provide a positive answer to Pascaleff's question whether (A$_{\infty}$Cat,$W^{\tiny\mbox{A}_{\infty}}_{\tiny\mbox{qe}}$) is a fibrant object in RelCat.

math.CT↗

arXiv2024

Tensor product of A$_{\infty}$-categories

In this paper we define the tensor product of two A$_{\infty}$-categories and two A$_{\infty}$-functors. This tensor product makes the category of A$_{\infty}$-categories symmetric monoidal (up to homotopy), and the category A$_{\infty}$Cat$^u$/$_{\approx}$ a closed symmetric monoidal category. Moreover, we define the derived tensor product making Ho(A$_{\infty}$Cat), the homotopy category of the A$_{\infty}$-categories, a closed symmetric monoidal category. We provide also an explicit description of the internal homs in terms of A$_{\infty}$- functors.

math.AG↗

arXiv2024

Localizations of the categories of $A_\infty$ categories and internal Homs over a ring

We show that, over an arbitrary commutative ring, the localizations of the categories of dg categories, of cohomologically unital, of unital and of strictly unital $A_\infty$ categories with respect to the corresponding classes of quasi-equivalences are all equivalent. The result is proven at the $\infty$-categorical level by considering the natural $\infty$-categorical models of the categories above. As an application of the techniques we develop to compare the localizations mentioned above, we provide a new proof of the existence of internal Homs for the homotopy category of dg categories in terms of the category of (strictly) unital $A_\infty$ functors. This yields a complete proof of a claim by Kontsevich and Keller.

math.CT↗

arXiv2024

The Homotopy Theory of $A_\infty$Categories

In this paper we describe the homotopy category of the $A_\infty$categories. To do that we introduce the notion of semi-free $A_\infty$category, which plays the role of standard cofibration. Moreover, we define the non unital $A_\infty$ (resp. DG)categories with cofibrant morphisms and we prove that any non unital $A_\infty$ (resp. DG)category has a resolution of this kind.

math.AG↗

arXiv2024

Rigid Dualizing Complexes over Commutative Rings and their Functorial Properties

In this paper we treat Grothendieck Duality for noetherian rings via rigid dualizing complexes. In particular, we prove that every ring, essentially finite type over a regular base ring, has a unique rigid dualizing complex. The rigid dualizing complexes have strong functorial properties, allowing us to construct the twisted induction pseudofunctor, which is our ring-theoretic version of the twisted inverse pseudofunctor $f^{!}$. This is the first article of a bigger project, whose final goal is establishing Grothendieck Duality, including global duality for proper maps, for Deligne-Mumford stacks.

math.AG↗

arXiv2020

Localizations of the category of $A_\infty$ categories and internal Homs

We prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital $A_\infty$ categories with respect to the corresponding classes of quasi-equivalences are all equivalent. Moreover we show that the last two localizations are equivalent to the corresponding quotients by the relation of being isomorphic in the cohomology of the $A_\infty$ category of $A_\infty$ functors. As an application we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly unital $A_\infty$ functors between them.

math.AG↗

arXiv2019

A descent criterion for equivalences between equivariant derived categories

We investigate equivalences between the categories of perfects complexes of the quotients of two smooth projective schemes by the action of a finite group. As a result we give a necessary and sufficient condition for an equivalence between the equivariant derived categories to descend to the categories of perfect complexes.

math.AG↗

arXiv2018

Voevodsky's conjecture for cubic fourfolds and Gushel-Mukai fourfolds via noncommutative K3 surfaces

In the first part of this paper we will prove the Voevodsky's nilpotence conjecture for smooth cubic fourfolds and ordinary generic Gushel-Mukai fourfolds. Then, making use of noncommutative motives, we will prove the Voevodsky's nilpotence conjecture for generic Gushel-Mukai fourfolds containing a $τ$-plane $\G(2,3)$ and for ordinary Gushel-Mukai fourfolds containing a quintic del Pezzo surface.

math.AG↗