arXiv subjects
Publications and source records attributed to Georgy Belousov.
For an algebraic variety $X$ we introduce generalized first Chern classes, which are defined for coherent sheaves on $X$ with support in codimension $p$ and take values in $CH^p(X)$. We use them to provide an explicit formula for the differentials ${d_2^p: E_2^{p,-p-1} \to E_2^{p+2, -p-2} \cong CH^{p+2}(X)}$ in the Quillen spectral sequence.
Let $φ$ be a rational map $\mathbb{P}^2 \dashrightarrow\mathbb{P}^2$ that preserves the rational volume form $\frac{\mathrm{d}x}{x}\wedge\frac{\mathrm{d}y}{y}$. Sergey Galkin conjectured that in this case $φ$ is necessarily birational. We show that such a map preserves the element $\{x,y\}$ of the second K-group $K_2(\mathbf{k}(x,y))$ up to multiplication by a constant, and restate this condition explicitly in terms of mutual intersections of the divisors of coordinates of $φ$ in a way suitable for computations.