We observe that standard derived equivalences give a counterexample to a conjecture of Kalck on global dimension for weighted projective lines. For the root stack $X=\mathbb{P}^1\langle \infty,0,1;2,3,3\rangle$ we exhibit a $13$-dimensional radical-square-zero algebra $A$ such that $$ D^b(\mathrm{coh}X)\simeq D^b(\mathrm{mod}A), \qquad \mathrm{gldim}A=4>3. $$
Let $G$ be a finite group and let $V$ be a finite-dimensional $G$-module over a field $k$. We construct explicit counterexamples in characteristic $2$ to conjectures of Wehlau concerning norms. We give faithful four-dimensional representations of $C_2^3$ and $C_2^4$ over $\mathbb F_8$ for which every nonlinear orbit no...
Let $k$ be a field of characteristic $2$. We exhibit an explicit polynomial endomorphism $F: \mathbb{A}_k^3\to\mathbb{A}_k^3$ whose Jacobian determinant is identically $1$, whose induced extension of rational function fields has degree $3$, and which is nevertheless noninjective. Since $2\neq 3$, this gives a counterex...
We construct a complete local Gorenstein ring $R$ of dimension $1$ with a prime ideal $\mathfrak{p}\in \operatorname{Spec}(R)$ such that $R$ is an AB ring, but the localization $R_{\mathfrak{p}}$ is not an AB ring. This settles the localization problem for AB rings posed by Huneke and Jorgensen in the negative.
We consider infinite walks in $\mathbb{N}^k$ with standard unit basis vector steps that avoid $t$ collinear points, and show that these walks exist for $(k,t) \in \{(6,3), (4,4), (3,7)\}$. In particular, our construction for $k = 3$ improves the previous bound $189$, obtained by Lidbetter, to $7$. Our results also impl...
Stijn Cambie, Erik Kalviainen, J. Shallit· 0 citations
Let $-f<0$ be a fundamental discriminant, $w$ the number of roots of unity in $\mathbb{Q}(\sqrt{-f})$, and $\chi_{-f}$ the associated quadratic character. We prove that there is a polynomial $R_f\in\mathbb{Q}[x,y]$ such that $m(R_f)=4wL'(\chi_{-f},-1)$. Hence the weak two-variable Chinburg conjecture holds.
We construct a projective log-smooth surface pair $(X,\Delta)$ and a morphism $f:X\to E$ to an elliptic curve such that $f_*\OO_X(K_X+\Delta)$ is Atiyah's indecomposable rank-two bundle $F_2$ of degree zero. This gives a negative answer to Question 3.4 of Jiang \cite{Jiang}. We also construct a log canonical pair $(X,\...
Houari Benammar Ammar· 0 citations
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