The exact Mayer-Vietoris sequence in Čech cohomology (Q402441)

Let \(A\) denote a commutative noetherian ring and \(\mathfrak{a} = (a_1,\ldots,a_n)R\) an ideal. Then let \(H^i_ {\mathfrak{a}}(M), i \in \mathbb{N},\) denote the local cohomology functors of an \(A\)-module \(M\). They can be realized as the cohomology of the Čech complex \(\check{C}_{\underline{a}}\) or as the left derived functors of the section functor \(\Gamma_{\mathfrak{a}}(\cdot)\). For two ideals \(\mathfrak{a}, \mathfrak{b} = (b_1,\ldots,b_m)\) there is the Mayer-Vietoris sequence for local cohomology NEWLINE\[NEWLINE \cdots \to H^i_{(\mathfrak{a},\mathfrak{b})}(M) \to H^i_{\mathfrak{a}}(M) \oplus H^i_{\mathfrak{b}}(M) \to H^i_{\mathfrak{a}\mathfrak{b}}(M) \to H^{i+1}_{(\mathfrak{a},\mathfrak{b})}(M) \to \cdots. NEWLINE\]NEWLINE The proof requires the ring \(A\) to be noetherian. The aim of the paper is a generalization of the Mayer-Vietoris sequence to the general situation of an arbitrary commutative ring (not necessarily noetherian) and finitely generated ideals. This is done by the aid of variations of the Čech complex, namely there is an exact sequence NEWLINE\[NEWLINE \cdots \to \check{H}^i_{(\underline{a},\underline{b})}(M) \to \check{H}^i_{\underline{a}}(M) \oplus \check{H}^i_{\underline{b}}(M) \to \check{H}^i_{\underline{a}\underline{b}}(M) \to \check{H}^{i+1}_{(\underline{a},\underline{b})}(M) \to \cdots, NEWLINE\]NEWLINE where \(\underline{a} = a_1,\ldots,a_n, \underline{b} = b_1,\ldots,b_m\) and \(\underline{a}\underline{b}\) is the sequence formed by the \(mn\) products \(a_ib_j\) in a certain order. To be more precise: the author introduces \(\Delta\) a simplicial complex associated to the indices of \(\underline{a},\underline{b},\underline{ab}\) and a subcomplex \(\check{C}^{\Delta}\) of \(\check{C}_{\underline{a},\underline{b},\underline{ab}}\) such that there is a short exact sequence of complexes NEWLINE\[NEWLINE 0 \to \check{C}_{\underline{a},\underline{b},\underline{ab}}/\check{C}^{\Delta} \to \check{C}_{\underline{a},\underline{ab}} \oplus \check{C}_{\underline{b},\underline{ab}} \to \check{C}_{\underline{ab}} \to 0. NEWLINE\]NEWLINE By showing the acyclicity of \(\check{C}^{\Delta}\) this provides the main result. For the proofs the author used some interesting combinatorial constructions for complexes.