On the essential dimension of infinitesimal group schemes

We discuss essential dimension of group schemes, with particular attention to infinitesimal group schemes. We prove that the essential dimension of a group scheme of finite type over a field k is at least equal to the difference between the dimension of its Lie algebra and its dimension. Furthermore, we show that the essential dimension of a trigonalizable group scheme of length p^{n} over a field of characteristic p>0 is at most n. We give several examples.Comment: 11 pages; proof of Theorem 1.2 slightly changed; improved the exposition in section 4 and added Proposition 4.3. Accepted for publication in The American Journal of Mathematic

Unexpected subgroup schemes of PGL_{2,k} in characteristic 2

If the characteristic of a field $k$ is odd any infinitesimal group scheme of $PGL_{2,k}$ lifts to $SL_{2,k}$. In this paper, we prove that this is not true in characteristic $2$ and we give a complete description, up to isomorphism, of infinitesimal unipotent subgroup schemes of $PGL_{2,k}$. Also, the infinitesimal trigonalizable case is considered.Comment: 9 page

Models of the group schemes of roots of unity

Let O_K be a discrete valuation ring of mixed characteristics (0,p), with residue field k. Using work of Sekiguchi and Suwa, we construct some finite flat O_K-models of the group scheme \mu_{p^n,K} of p^n-th roots of unity, which we call Kummer group schemes. We set carefully the general framework and algebraic properties of this construction. When k is perfect and O_K is a complete totally ramified extension of the ring of Witt vectors W(k), we provide a parallel study of the Breuil-Kisin modules of finite flat models of \mu_{p^n,K}, in such a way that the construction of Kummer groups and Breuil-Kisin modules can be compared. We compute these objects for n < 4. This leads us to conjecture that all finite flat models of \mu_{p^n,K} are Kummer group schemes.Comment: 63 pages, final version, minor changes. To appear in Annales de l'Institut Fourie

Nonisotrivial families over curves with fixed point free automorphisms

We construct for any smooth projective curve of genus q ≥ 2 with a fixed point free automorphism a nonisotrivial family of curves. Moreover we study the space of the modular curves and one of the parameters

A lithium-ion battery based on LiFePO4 and silicon/reduced graphene oxide nanocomposite

In this paper, the preparation and chemical–physical characterization of a composite material made of silicon nanoparticles (nSi) and reduced graphene oxide (RGO) for using as an anode for lithium-ion batteries are report- ed. The nSi/RGO composite was synthesized by microwave irradiation followed by a thermal treatment under reducing atmosphere of a mixture of nSi and graphene oxide, and characterized by XRD, SEM, and TGA. The nano- structured material was used to prepare an electrode, and its electrochemical performance was evaluated in a lithium cell by galvanostatic cycles at various charge rates. The electrode was then coupled with a LiFePO4 cathode to fabricate a full lithium-ion battery cell and the cell performance evaluated as a function of the discharge rate and cycle number

Sustainable Anodes for Lithium- and Sodium-Ion Batteries Based on Coffee Ground-Derived Hard Carbon and Green Binders

The reuse and recycling of products, leading to the utilization of wastes as key resources in a closed loop, is a great opportunity for the market in terms of added value and reduced environmental impact. In this context, producing carbonaceous anode materials starting from raw materials derived from food waste appears to be a possible approach to enhance the overall sustainability of the energy storage value chain, including Li-ion (LIBs) and Na-ion batteries (NIBs). In this framework, we show the behavior of anodes for LIBs and NIBs prepared with coffee ground-derived hard carbon as active material, combined with green binders such as Na-carboxymethyl cellulose (CMC), alginate (Alg), or polyacrylic acid (PAA). In order to evaluate the effect of the various binders on the charge/discharge performance, structural and electrochemical investigations are carried out. The electrochemical characterization reveals that the alginate-based anode, used for NIBs, delivers much enhanced charge/discharge performance and capacity retention. On the other hand, the use of the CMC-based electrode as LIBs anode delivers the best performance in terms of discharge capacity, while the PAA-based electrode shows enhanced cycling stability. As a result, the utilization of anode materials derived from an abundant food waste, in synergy with the use of green binders and formulations, appears to be a viable opportunity for the development of efficient and sustainable Li-ion and Na-ion batteries