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Zariski filtered sheaves. (English) Zbl 0961.16030

Let \(R\) be an associative ring with unit and with a filtration \(FR\). Let \(\widetilde R=\sum_{n\in\mathbb{Z}}F_nRX^n\subset R[X,X^{-1}]\) be the Rees ring, \(X\) being homogeneous of degree 1. The filtration \(FR\) is called a Zariskian filtration if \(\widetilde R\) is Noetherian and \(X\) belongs to the Jacobson graded radical of \(\widetilde R\). Assume that the associated graded ring \(G(R)=F_nR/F_{n-1}R\) is a commutative Noetherian domain. Let \(\sigma\colon R\to G(R)\) be the principal symbol map. For filtered \(R\)-modules \(M\) define microlocalisation \(Q^\mu_S(M)\) of \(M\) with respect to a multiplicative set \(S\) in \(R\) by \(Q^\mu_S(M)=Q^\mu_{{\widetilde S}}(\widetilde M)/(1-X)Q^\mu_S(\widetilde M)\) where \((\widetilde M)=\sum_nF_nMX^n\), \(Q^\mu_{{\widetilde S}}(\widetilde M)=\varprojlim_n(\widetilde M/X^n\widetilde M)_{{\widetilde S(n)}}\), \(\widetilde S=\{\widetilde r=sX^m\mid s\in S\) with filtration degree \(m\}\), \(\widetilde S(n)=\{\widetilde r\text{ mod }X^n\widetilde R,\;\widetilde r\in S\}\). Let \(Y=\text{Spec}^gG(R)\) be the graded prime spectrum. By associating to the basic open set \(Y_f\), \(f\in G(R)\), the microlocalisation \(Q^\mu_f(R)\) (resp. \(Q^\mu_f(M)\)) we get the sheaf \({\mathcal O}_Y^\mu\) (resp. \(M^\mu_Y\)) on \(Y\).
The main results of the paper are the following. Assume that \(R\) is a Zariski filtered ring with \(G(R)\) a commutative Noetherian domain. Then one has (1) The stalk \({\mathcal O}^\mu_{Y,p}=\bigcup_{f\notin p}Q^\mu_f(R)\) at \(p\) is a Zariski filtered ring. \(M^\mu_Y\) is coherently filtered if and only if \(M^\mu_{Y,p}\) has a good filtration for all \(p\in Y\). The same results hold for quantum sheaves obtained by looking at degree \(0\) part \({\mathcal F}_0{\mathcal O}^\mu_Y\), \({\mathcal F}_0M^\mu_Y\). (2) For a filtered \(R\)-module \(M\) with a good filtration, one has \(M^\mu_Y={\mathcal O}^\mu_Y+M\), \(\widetilde M^\mu_Y={\mathcal O}^\mu_Y+\widetilde M\). The same result holds for \(Y\) replaced by the formal scheme \(\widehat Y=\) the closed subscheme \(V(G(I))\) with \({\mathcal O}^\mu_{\widehat Y}=\varprojlim_n{\mathcal O}^\mu_Y/(I^\mu_Y)^n\) and \(M^\mu_{\widehat Y}=\varprojlim_n M^\mu_Y/(I^nM)^\mu_Y\).

MSC:

16W70 Filtered associative rings; filtrational and graded techniques
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
16S32 Rings of differential operators (associative algebraic aspects)
16S60 Associative rings of functions, subdirect products, sheaves of rings
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