arXiv subjects

Gleb Terentiuk

Publications and source records attributed to Gleb Terentiuk.

3 recordsLinked to original sources

arXiv2026

Ogus-Vologodsky equivalence via stacks

Using the relative de Rham stack for a family $X \to S$ in characteristic $p,$ we reprove the (local and global) Ogus-Vologodsky equivalence. Moreover, we observe that a lift of $S$ is not necessary. Instead, we use a lift of $X$ to the second Witt vectors of $S.$ The main ingredient is that, for a quasi-syntomic family $X/S,$ the relative de Rham stack admits a structure of a torsor over $X'$ which is the analogue of the Azumaya property of the algebra of differential operators. This can be applied to families of (reasonable) algebraic stacks, which gives rise to a logarithmic version of the Cartier equivalence. Along the way, we also obtain a decompleted version of the global Cartier equivalence.

math.AG↗

arXiv2024

Prismatic $F$-Gauges and Fontaine--Laffaille modules

Let $k$ be a perfect field of characteristic $p$ and $W(k)$ its ring of Witt vectors. We construct an equivalence of categories between the full subcategory of the derived category of quasi-coherent sheaves on the syntomification of $W(k)$ spanned by objects whose Hodge-Tate weights are between $[0,p-2]$ and an appropriate derived category of Fontaine-Laffaille modules.

math.NT↗

arXiv2019

Chern Classes via Derived Determinant

Motivated by the Chern-Weil theory, we prove that for a given vector bundle $E$ on a smooth scheme $X$ over a field $k$ of any characteristic, the Chern classes of $E$ in the Hodge cohomology can be recovered from the Atiyah class. Although this problem was solved by Illusie in \cite{i}, we present another proof by means of derived algebraic geometry. Also, for a scheme $X$ over a field $k$ of characteristic $p$ with a vector bundle $E$ we construct elements $c^{cris}_n (E, α(E)) \in H_{dR}^{2n} (X) $ using an obstruction $α(E)$ to a lifting of $F^* E$ to a crystal modulo $p^2$ and prove that $c^{cris}_n (E, α(E)) = n! \cdot c_{n}^{dR} (E)$, where $c_{n}^{dR} (E)$ are the Chern classes of $E$ in the de Rham cohomology and $F$ is the Frobenius map.

math.AG↗