On the relationship between depth and cohomological dimension

<jats:p>Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X15007678_inline1" /><jats:tex-math>$(S,\mathfrak{m})$</jats:tex-math></jats:alternatives></jats:inline-formula> be an <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X15007678_inline2" /><jats:tex-math>$n$</jats:tex-math></jats:alternatives></jats:inline-formula>-dimensional regular local ring essentially of finite type over a field and let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X15007678_inline3" /><jats:tex-math>$\mathfrak{a}$</jats:tex-math></jats:alternatives></jats:inline-formula> be an ideal of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X15007678_inline4" /><jats:tex-math>$S$</jats:tex-math></jats:alternatives></jats:inline-formula>. We prove that if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X15007678_inline5" /><jats:tex-math>$\text{depth}\,S/\mathfrak{a}\geqslant 3$</jats:tex-math></jats:alternatives></jats:inline-formula>, then the cohomological dimension <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X15007678_inline6" /><jats:tex-math>$\text{cd}(S,\mathfrak{a})$</jats:tex-math></jats:alternatives></jats:inline-formula> of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X15007678_inline7" /><jats:tex-math>$\mathfrak{a}$</jats:tex-math></jats:alternatives></jats:inline-formula> is less than or equal to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X15007678_inline8" /><jats:tex-math>$n-3$</jats:tex-math></jats:alternatives></jats:inline-formula>. This settles a conjecture of Varbaro for such an <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X15007678_inline9" /><jats:tex-math>$S$</jats:tex-math></jats:alternatives></jats:inline-formula>. We also show, under the assumption that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X15007678_inline10" /><jats:tex-math>$S$</jats:tex-math></jats:alternatives></jats:inline-formula> has an algebraically closed residue field of characteristic zero, that if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X15007678_inline11" /><jats:tex-math>$\text{depth}\,S/\mathfrak{a}\geqslant 4$</jats:tex-math></jats:alternatives></jats:inline-formula>, then <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X15007678_inline12" /><jats:tex-math>$\text{cd}(S,\mathfrak{a})\leqslant n-4$</jats:tex-math></jats:alternatives></jats:inline-formula> if and only if the local Picard group of the completion <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X15007678_inline13" /><jats:tex-math>$\widehat{S/\mathfrak{a}}$</jats:tex-math></jats:alternatives></jats:inline-formula> is torsion. We give a number of applications, including a vanishing result on Lyubeznik’s numbers, and sharp bounds on the cohomological dimension of ideals whose quotients satisfy good depth conditions such as Serre’s conditions <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0010437X15007678_inline14" /><jats:tex-math>$(S_{i})$</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>