For a positive integer $n$, let $A_n=\mathbb{C}[t_1^{\pm1},\ldots,t_n^{\pm1},x_1,\ldots,x_n]$ and $\mathfrak{g}_n=\bigoplus_{i=1}^n A_nd_i$, where $d_i=t_i\frac{\partial}{\partial t_i} +\frac{\partial}{\partial x_i}$. We first determine when the tensor module $T(P,V)=P\otimes V$ is simple, where $P$ is a simple module over the Weyl type algebra $D_n$ and $V$ is a simple $\mathfrak{gl}_n$-module. We then prove a canonical algebra isomorphism $A_n\#U(\mathfrak{g}_n)\cong D_n\otimes U(\mathfrak{m}_{\mathbf{1},\mathbf{0}}\Delta)$, and use it to show that every simple cuspidal $\mathfrak{g}_n$-module is isomorphic to a simple quotient of some $T(A_n(\lambda),V)$, where $V$ is a finite-dimensional simple $\mathfrak{gl}_n$-module.
Let $X/\mathbb{Q}$ be a smooth, projective, geometrically integral curve of genus $g \geq 2$ with a point $ b \in X(\mathbb{Q})$ and geometrically CM Jacobian. Fix an odd prime $p$ of good reduction, and let $U$ be the \'etale pro-unipotent $\mathbb{Q}_p$-fundamental group of $X$. We prove, for all $n \gg 0,$ the inequ...
Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$, residue field $k$ and let $I$ be an $\mathfrak{m}$-primary ideal. Assume $k$ is infinite. Let $N$ be a perfect $A$-module of dimension $t \geq 1$. Let $M$ be a MCM $A$-module. The function $n \rightarrow \ell(\text{Tor}^A_j(M, N/I^{n+1}N))$ is of p...
Let $S=\kk[x_1,\ldots,x_n]$ with its standard $\mathbb N^n$-grading. For $0\le i<n$, we identify the finite-box local-cohomology category $\mathcal H_i(\mathbf t)$ with the representation category of the upper support poset $U_{n-i}(\mathbf t)$. The local-cohomology functor admits an exact functorial section selecting...
Let $X=\mathbb{P}^3$ and let \[ \alpha_n=(0,1,-\tfrac12,\tfrac16-n)\in H^{\mathrm{even}}(X,\mathbb{Q}) \] with respect to the basis $1,H,H^2,H^3$, where $H=c_1(\mathcal{O}_X(1))$. We prove that every Gieseker semistable sheaf on $X$ with Chern character $\alpha_n$ is stable and uniquely of the form $\iota_{P*}\mathcal{...
R. Anderson· Journal of Geometry and Phys...· 0 citations
Let $f: X\rightarrow S$ be a faithful affine map. In this article we construct a group homomorphism $\theta_{{fppf}}: \text{Br}(f) \longrightarrow H^1_{{fppf}}\!\left(S, (f_*\mathcal{O}_X^\times / \mathcal{O}_S^\times)_{{fppf}}\right)$, extending the formerly defined morphism over an \'etale site $\theta_{et}$. In doin...
The $n$-$(n+1)$-$(n+2)$ theorem, recently established by Cohen and Shusterman, says that if two groups of type $\mathrm{F}_{n+1}$ map onto a common quotient $Q$ of type $\mathrm{F}_{n+2}$ and one of the two kernels is of type $\mathrm{F}_n$, then the associated fibre product is of type $\mathrm{F}_{n+1}$. We prove a st...
Benno Kuckuck· 0 citations
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