Let $R$ be a commutative Noetherian ring, let $\mathbf{x}=x_1,\ldots,x_n$ be an $R$-regular sequence, and let $\mathbf{y}=y_1,\ldots,y_m$ be a sequence of elements of $R$. Put $I=(\mathbf y)$. Let $\mathcal S$ be a Serre subcategory of the category of $R$-modules. We consider the double complex obtained from the Koszul co-complex with respect to $\mathbf x$ and the \v{C}ech complex with respect to $\mathbf y$. Using the two spectral sequences associated with this double complex, we prove that \[ Ext_R^i(R/(\mathbf x),H_I^j(R))\in\mathcal S \quad\text{for all }i,j\in \mathbb N_0 \] implies \[ H_I^j(R/(\mathbf x))\in\mathcal S \quad\text{for all }j\in\mathbb N_0. \]
Let $R$ be a Noetherian standard $\mathbb{N}$-graded ring, and let $I_1,\ldots,I_r$ be homogeneous ideals of $R$. Let $L,M$, and $N$ be finitely generated $\mathbb{Z}$-graded $R$-modules with $N\subseteq M$. For $\underline{n} :=(n_1,\dots,n_r)\in\mathbb N^r$, set ${\bf I}^{\underline{n}}:=I_1^{n_1}\cdots I_r^{n_r}$. W...
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 $\kk$ be a \emph{field of characteristic $p>0$, not assumed algebraically closed}, and let $V=\kk[x]/(x^p)$ be the Novikov algebra with product $a\circ b=ab'$. For $\lambda\in\kk$, let $M(\lambda)$ be Xu's module. We compute the second cohomology $\Ht(V,M(\lambda))$ for all $\lambda$ and $p$, and describe the assoc...
Let $G$ be a compact, connected, simply connected semisimple Lie group with Weyl group $W$ and simple reflections $S$. For a simplicial complex $\mathcal K$ on $S$, form the homotopy colimit $X_{\mathcal K}(G)=\operatorname*{hocolim}_{I\in\mathcal K}G/G_I$ of standard partial flag manifolds. We compute its integral hom...
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 $\ell$ and $p$ be distinct prime numbers, $K$ a local field with residue characteristic $p$, $C$ a complete algebraically closed extension of $K$, and $X$ a quasi-compact quasi-separated rigid space over $K$. When $\mathop{\mathrm{char}} K = 0$, we prove that the alternating sum of traces of elements in the positiv...
Xian Wu· 0 citations
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