arXiv subjects
Publications and source records attributed to Takeshi Saito.
For a constructible sheaf on a smooth scheme over a field of positive characteristic, the singular support and the characteristic cycle are defined on the cotangent bundle. We study if a construction using the microlocalization similar to that in a transcendental setting due to Kashiwara--Schapira works in the algebraic setting and prove partial positive results.
Building on Beilinson's work, ``constructible sheaves are holonomic,'' we introduce the notion of holonomicity for étale sheaves, without assuming a priori constructibility. Over a perfect base field, we establish the converse of Beilinson's result, showing that holonomic sheaves are indeed constructible. This can be seen as an étale analogue of Kashiwara's theorem on holonomic ${\mathcal D}_X$-modules.
We fix an excellent regular noetherian scheme $S$ over ${\mathbf Z}_{(p)}$ satisfying a certain finiteness condition. For a constructible étale sheaf ${\cal F}$ on a regular scheme $X$ of finite type over $S$, we introduce a variant of the singular support relatively to $S$ and prove the existence of a saturated relative variant of the singular support by adopting the method of Beilinson using the Radon transform. We may deduce the existence of the singular support itself, if we admit an expected property on the micro support of tensor product and if the scheme $X$ is sufficiently ramified over the base $S$.
Armchair graphene nanoribbons (AGNRs) with sub-nanometer width are potential materials for fabrication of novel nanodevices thanks to their moderate direct band gaps. AGNRs are usually synthesized by polymerizing precursor molecules on substrate surface. However, it is time-consuming and not suitable for large-scale production. AGNRs can also be grown by transforming precursor molecules inside single-walled carbon nanotubes via furnace annealing, but the obtained AGNRs are normally twisted. In this work, microwave heating is applied for transforming precursor molecules into AGNRs. The fast heating process allows synthesizing the AGNRs in seconds. Several different molecules were successfully transformed into AGNRs, suggesting that it is a universal method. More importantly, as demonstrated by Raman spectroscopy, aberration-corrected high-resolution transmission electron microscopy and theoretical calculations, less twisted AGNRs are synthesized by the microwave heating than the furnace annealing. Our results reveal a route for rapid production of AGNRs in large scale, which would benefit future applications in novel AGNRs-based semiconductor devices.
T. Dupuy, E. Katz, J. Rabinoff, D. Zureick-Brown introduced the module of total $p$-differentials for a ring over $Z/p^2Z$. We study the same construction for a ring over $Z_{(p)}$ and prove a regularity criterion. For a local ring, the tensor product with the residue field is constructed in a different way by O. Gabber, L. Ramero. In another article arXiv:2006.00448, we use the sheaf of FW-differentials to define the cotangent bundle and the micro-support of an etale sheaf.
We give a characterization of ramification groups of local fields with imperfect residue fields, using those for local fields with perfect residue fields. As an application, we reprove an equality of ramification groups for abelian extensions defined in different ways.
Three typical one-dimensional (1D)/quasi-1D nanocarbons, linear carbon chains, carbon nanotubes, and graphene nanoribbons have been proven to grow inside single-wall carbon nanotubes. This gives rise to three types of hybrid materials whose behaviour and properties compared among each other are far from understood. After proving successful the synthesis of these nanostructured materials in recently published work, we have now been able to study their oxidation stability systematically by using resonance Raman spectroscopy. Surprisingly, the linear carbon chains, which have been theoretically predicted to be very unstable, are actually thermally stable up to 500 °C assisted by the protection of the carbon nanotube hosts. Besides, longer linear carbon chains inside narrower CNTs are more stable than the shorter ones inside larger tubes, suggesting that the thermal stability not only depends on the length of linear carbon chains alone, but it is correlated with the confinement of the host tubes in a more complicated manner. In addition, graphene nanoribbons come overall in view as the most stable confined structures. On the other hand, peculiarities like the higher stability of the (6,5) CNT over its (6,4) counterpart allow this study to provide a solid platform for further studies on the application of these 1D nanocarbons (including true 1D linear carbon chains) at ambient conditions.
For a regular scheme and a prime number $p$, we define the FW-cotangent bundle as a vector bundle on the closed subscheme defined by $p=0$, under a certain finiteness condition. For a constructible complex on the etale site of the scheme, we introduce the condition to be micro-supported on a closed conical subset in the FW-cotangent bundle. At the end of the article, we compute the singular supports in some cases.
Graphene nanoribbons with sub-nanometer widths are extremely interesting for nanoscale electronics and devices as they combine the unusual transport properties of graphene with the opening of a band gap due to quantum confinement in the lateral dimension. Strong research efforts are presently paid to grow such nanoribbons. Here we show the synthesis of 6- and 7-armchair graphene nanoribbons, with widths of 0.61 and 0.74 nm, and excitonic gaps of 1.83 and 2.18 eV, by high-temperature vacuum annealing of ferrocene molecules inside single-walled carbon nanotubes. The encapsulation of the so-obtained graphene nanoribbons is proved by atomic resolution electron microscopy, while their assignment is provided by a combination of an extensive wavelength-dependent Raman scattering characterization and quantum-chemical calculations. These findings enable a facile and scalable approach leading to the controlled growth and detailed analysis of well-defined sub-nanometer graphene nanoribbons.
We prove that the graded quotients of the filtration by ramification groups of any henselian discrete valuation field of residue characteristic $p>0$ are $F_p$-vector spaces. We define an injection of the character group of each graded quotient to a twisted cotangent space defined using a cotangent complex.
We prove the functoriality for proper push-forward of the characteristic cycles of constructible complexes by morphisms of smooth projective schemes over a perfect field, under the assumption that the direct image of the singular support has the dimension at most that of the target of the morphism. The functoriality is deduced from a conductor formula which is a special case for morphisms to curves. The conductor formula in the constant coefficient case gives the geometric case of a formula conjectured by Bloch.
There are two ways to define the Swan conductor of an abelian character of the absolute Galois group of a complete discrete valuation field. We prove that these two Swan conductors coincide.
Double-walled carbon nanotubes (DWCNTs) combined the advantages of multi-walled (MW-) and single-walled (SW-) CNTs can be obtained by transforming the precursors (e.g. fullerene, ferrocene) into thin inner CNTs inside SWCNTs as templates. However, this method is limited since the DWCNT yield is strongly influenced by the filling efficiency (depending on the type of the filled molecules), opening and cutting the SWCNTs, and the diameter of the host SWCNTs. Therefore, it cannot be applied to all types of SWCNT templates. Here we show a universal route to synthesize ultra-thin DWCNTs via making SWCNTs stable at high temperature in vacuum. This method applies to different types of SWCNTs including metallicity-sorted ones without using any precursors since the carbon sources were from the reconstructed SWCNTs and the residue carbons. The resulting DWCNTs are with high quality and the yield of inner tubes is comparable to/higher than that of the DWCNTs made from the transformation of ferrocene/fullerene peapods.
The Raman response of new structures grown after filling SWCNTs with ferrocene and transformation at moderate high temperatures is demonstrated to be very strong, even stronger than the response from the tubes. Transmission electron microscopy demonstrates that the new objects are flat and exhibit a structure similar to short fragments of nanoribbons. The growth process is controlled by two different activation energies for low and high transformation temperatures, respectively. Immediately after filling Raman pattern from a precursor molecule are detected. Two different types of nanoribbons were identified by selecting special laser energies for the Raman excitation. These ribbons have the signature of quaterrylene and terrylene, respectively.
We used scattering-type scanning near-field optical microscopy (s-SNOM) to characterize nickel nanoclusters grown inside single-walled carbon nanotubes (SWCNT). The nanotubes were filled with Ni(II) acetylacetonate and the molecules were transformed into nickel clusters via annealing. The metal clusters give high local contrast enhancement in near-field phase maps caused by the excitation of free charge carriers. The near-field contrast was simulated using the finite dipole model, approximating the clusters with elliptical nanoparticles. Compared to magnetic force microscopy, s-SNOM appears much more sensitive to localize metal clusters inside carbon nanotubes. We estimate the detection threshold to be ~600 Ni atoms.
For a character of the absolute Galois group of a complete discrete valuation field, we define a lifting of the refined Swan conductor, using higher dimensional class field theory.
We give a purely scheme theoretic construction of the filtration by ramification groups of the Galois group of a covering. The valuation need not be discrete but the normalizations are required to be locally of complete intersection.
The electronic states in isolated single-wall carbon nanotubes (SWCNTs) have been considered as an ideal realization of a Tomonaga-Luttinger liquid (TLL). However, it remains unclear whether one-dimensional correlated states are realized under local environmental effects such as the formation of a bundle structure. Intertube effects originating from other adjacent SWCNTs within a bundle may drastically alter the one-dimensional correlated state. In order to test the validity of the TLL model in bundled SWCNTs, low-energy spin excitation is investigated by nuclear magnetic resonance (NMR). The NMR relaxation rate in bundled mixtures of metallic and semiconducting SWCNTs shows a power-law temperature dependence with a theoretically predicted exponent. This demonstrates that a TLL state with the same strength as that for effective Coulomb interactions is realized in a bundled sample, as in isolated SWCNTs. In bundled metallic SWCNTs, we found a power-law temperature dependence of the relaxation rate, but the magnitude of the relaxation rate is one order of magnitude smaller than that predicted by theory. Furthermore, we found an almost doubled magnitude of the Luttinger parameter. These results indicate suppressed spin excitations with reduced Coulomb interactions in bundled metallic SWCNTs, which are attributable to intertube interactions originating from adjacent metallic SWCNTs within a bundle. Our findings give direct evidence that bundling reduces the effective Coulomb interactions via intertube interactions within bundled metallic SWCNTs.