We extend Gromov's conjecture on the sharp width estimate for Riemannian bands with positive scalar curvature to the family case and prove that it holds for fiber bundles with infinite family Acirc;-area. The method we employ is based on Dirac operators and the family index theory. Our proof relies on a product formula for index bundles established in this paper.
We study admissible subcategories of the derived categories of smooth noncommutative (nc) curves as classified by Reiten-van den Bergh. We prove that any admissible subcategory of the derived category of a smooth nc curve is again the derived category of a smooth nc curve. We use this result to classify semiorthogonal decompositions in derived categories of nc curves. The results obtained imply that phantom categories do not exist in these cases. As a further application, we prove an extension of the Bondal-Orlov reconstruction theorem to the case of orbifold curves.
In this paper, we investigate noncommutative resolutions of AS-Gorenstein isolated singularities. Noncommutative resolutions in graded case are achieved as the graded endomorphism rings of some finitely generated graded modules, which are seldom N-graded algebras but bounded-below Z-graded algebras. So the paper works on locally finite bounded-below Z-graded algebras. We first define and study noncommutative projective schemes after Artin-Zhang and define noncommutative quasi-projective spaces as the base spaces of noncommutative projective schemes. The equivalences between noncommutative quasi-projective spaces are proved to be induced by so-called modulo-torsion-invertible bimodules, which are in fact a Morita-like theory at the quotient category level. Based on the equivalences, we propose a definition of noncommutative resolutions of AS-Gorenstein isolated singularities and prove that such noncommutative resolutions are generalized AS-regular algebras. The center of any noncommutative resolution is isomorphic to the center of the original AS-Gorenstein isolated singularity. In the final part, we prove that a noncommutative resolution of an AS-Gorenstein isolated singularity of dimension d is given by an MCM generator M if and only if M is a (d-1)-cluster tilting module. A noncommutative version of the Bondal-Orlov conjecture is also proved to be true in dimensions 2 and 3.
We introduce an algebraic extension of the Turaev cobracket as a family of maps parametrised by a positive integer. It is defined by constructing the generalised version of the divergence maps for an arbitrary associative algebra with the help of a connection in non-commutative geometry, and we recover the above extension by applying the construction to the case of the group algebra of the fundamental group of a compact connected oriented surface with boundary. If the connection is flat, the generalised divergence maps define classes in the Lie algebra cohomology of the space of derivations, and in the case of the free associative algebra, we show that they are canonically identified with the standard generators of the cohomology ring of the matrix Lie algebra g(n)(l).
In this paper, we study p-adic Hodge theory for non-commutative algebraic varieties. Firstly, we propose a conjecture that for K a complete discretely valued nonarchimedean extension K of \mathbb{Q}_{p} with perfect residue field k and \mathcal{T} an \mathcal{O}_{K}-linear idempotent-complete, small smooth proper stable \infty-category, there exists a B_{\mathrm{crys}}-coefficient isomorphism preserving additional structures between the K(1)-local K-theory on the generic fiber of \mathcal{T} and the topological periodic cyclic homology on the special fiber. This conjecture can be regarded as a non-commutative analog of the crystalline comparison theorem. We then proceed to prove the following results: the topological negative cyclic homology \pi_{i}\mathrm{TC}<^>{-}(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_{p}) admits a Breuil-Kisin module structure, and the non-commutative analog of Bhatt-Morrow-Scholze's comparison theorems holds. Additionally, we demonstrate that the \mathbb{Z}_{p}[G_{K}]-module obtained from the topological negative cyclic homology is a \mathbb{Z}_{p}-lattice of a crystalline representation. Finally, we show that when the generic fiber of \mathcal{T} admits a geometric realization in the sense of Orlov, the non-commutative analog of the crystalline comparison theorem proposed by the author holds.
