Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T21:13:21Z http://mathoverflow.net/feeds/question/80548 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80548/affine-scheme-on-speca-of-a-ring-a-as-the-sheafification-of-a-pre-sheave-on-spe Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)? urelement 2011-11-10T02:58:37Z 2011-11-13T00:05:07Z <p>It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$ versus the sheafification of a pre-sheaf.</p> <p>The definition of the sheaf $\mathscr F^+$ associated to pre-sheaf $\mathscr F$ is (Hartshorne p.64):</p> <blockquote> <p>For any open set $U$, let $\mathscr F^+ (U)$ be the set of functions $s$ from $U$ to the union of stalks $\mathscr F_P$ of $\mathscr F$ over points $P$ of $U$ such that:</p> <ol> <li><p>For each $P$ in $U$, $s(P)$ is in $\mathscr F_p$.</p></li> <li><p>For each $P$ in $U$, there is a neighborhood $V$ of $P$ , contained in $U$, and an element $t$ in $\mathscr F(V)$, such that for all $Q$ in $V$, the germ $t_Q$ of $t$ at $Q$ is equal to $s(Q)$. </p></li> </ol> </blockquote> <p>While, in (Hartshorne p.70), the definition of the sheaf of rings $\mathscr O$ on $\operatorname{Spec}(A)$ is:</p> <blockquote> <p>For any open set $U$ of $\operatorname{Spec}(A)$, let $\mathscr O(U)$ be the set of functions $s$ from $U$ to the union of localizations $A_\mathscr{p}$ of $A$ at $\mathscr{p}$ such that: </p> <p>For each $\mathscr{p}$ in $U$, there is a neighborhood $V$ of $\mathscr{p}$, contained in $U$, and elements $a,f$ of $A$, such that for each $\mathscr{q}$ in $V$, $f$ not in $\mathscr{q}$, and $s(\mathscr{q}) = a/f$ .</p> </blockquote> <p>So, is there a naturally occurring pre-sheaf on $\operatorname{Spec}(A)$ (which in general is not a sheaf) that exists for any ring $A$ such that its sheafification gives exactly the structure sheaf $\mathscr O$ of $\operatorname{Spec}(A)$?</p> http://mathoverflow.net/questions/80548/affine-scheme-on-speca-of-a-ring-a-as-the-sheafification-of-a-pre-sheave-on-spe/80560#80560 Answer by a-fortiori for Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)? a-fortiori 2011-11-10T06:17:44Z 2011-11-10T06:27:50Z <p>For any open subset $U\subseteq\mathrm{Spec}(A)$ let $S_U=A\setminus\bigcup_{\mathfrak p\in U}\mathfrak p$ and $\mathscr O'(U)=A[S_U^{-1}]$. It is obviously a presheaf.</p> <p>Claim: For open subsets of the form $U=\mathrm{Spec}(A_f)$ with $f\in A$ we have $\mathscr O'(U)=A_f$. (This shows that the associated sheaf of $\mathscr O'$ is indeed $\mathscr O_{\mathrm{Spec}(A)}$.)</p> <p>Proof: Assume there is an $s\in S_U$ which does not divide $f^n$ for any $n$. The ideal $(s)$ does not meet the multiplicative set $S_f=\{1,f,f^2,\dots\}$, so it is contained in an ideal $\mathfrak q$ which is maximal with respect to this property, but it is well-known that such an ideal $\mathfrak q$ is prime. By construction, $s\in\mathfrak q\in U$, contradicting $s\in S_U$.</p> <p>Applying the usual associated sheaf construction to $\mathscr O'$ seems to be what Hartshorne does when he defines $\mathscr O_{\mathrm{Spec}(A)}$.</p>