Cohomological and projective dimensions (Q2841771)

Let \(\alpha\) be a homogeneous ideal of a polynomial ring \(R = k[x_1, \ldots, x_n]\) over a field \(k,\) and \(t =\text{depth}(R/\alpha),\) \(0\leq t\leq n.\) The author proves that if \(t = 3,\) then the local cohomology modules \(H_\alpha^i(M)\) vanish for all \(i > n-3\) and all \(R\)-modules \(M.\) In other terms, the cohomological dimension \(\text{cd}(R,\alpha)\leq n-3.\) In particular, if \(\alpha\) is perfect of height \(n-3\) then \(\text{cd}(R,\alpha)\leq n-3.\) As an application, in characteristic \(0\) the author constructs several examples of prime ideals \(\alpha\subseteq R\) such that \(R/\alpha\) is not set-theoretically Cohen-Macaulay; it is a generalization of a result in [\textit{A. K. Singh} and \textit{U. Walther}, Contemp. Math. 390, 147--155 (2005; Zbl 1191.14059)]. He also remarks that in positive characteristic the case of any \(\text{depth}(R/\alpha)\) was analyzed completely by \textit{C. Peskine} and \textit{L. Szpiro} [Publ. Math., Inst. Hautes Étud. Sci. 42, 47--119 (1972; Zbl 0268.13008)]. On the other hand, when \(\text{char}(k)=0\) and \(t < 3\) then the statement \(H_\alpha^i(M)=0\) for all \(i>n-t,\) follows directly from classical results of A. Grothendieck and many others, while for all \(t>3\) there are counterexamples.