User anonymous - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:39:24Z http://mathoverflow.net/feeds/user/12129 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56865/the-etale-site-of-a-closed-subscheme-and-its-etale-grothendieck-subtopology The etale site of a closed subscheme and its etale Grothendieck subtopology Anonymous 2011-02-28T01:24:17Z 2011-02-28T05:08:30Z <p>There is a very basic theorem for the Zariski topology.</p> <p>Let X = Spec(R) and Y=Spec(R/I) for I some reduced ideal. Y obtains a topology two ways, one is the subspace topology as a subset of X and another as the spectrum of a ring. These topologies are the same by the correspondence between ideals of R containing I and ideals in R/I.</p> <p>Is there a close statement to this in the etale toplogy? There are two natural ways to understand open sets on Y, those which come from etale neighborhoods of X base changed to Y and those which are etale neighborhoods of Y.</p> <p>I did a computation today in a very special case and it seems that both of these topologies seem to be 'the same'.</p> <p>Does anyone know if this statement is true in a general context and where I might locate this resource?</p> <p>Thanks.</p> http://mathoverflow.net/questions/51598/sheaves-on-stacks-and-interesting-functors Sheaves on stacks and interesting functors Anonymous 2011-01-09T23:00:22Z 2011-01-11T17:10:20Z <p>Let $G$ be a finite group and $H \subset G$ a normal subgroup. Consider $G$, $H$, and $X=G/H$ as affine algebraic groups over some algebraically closed base field $k$.</p> <p>I hear that there is an isomorphism of stacks $[X/G] \cong [pt/H]$.</p> <p>I have the following question:</p> <p>To give a sheaf (of vector spaces) on the stack $[X/G]$ is the same as giving a $G$-equivariant sheaf on $X$. By the isomorphism above, it is the same as giving a vector space with an $H$ action.</p> <p>What is this functor taking $G$-equivariant modules over the ring $k[G/H]=k[G]^H$ to vector spaces with $H$ actions?</p> <p>For example, what happens to the $G$-equivariant $k[G/H]$-module $M=k[G]$?</p> <p><em>*</em> Edit to more general situation</p> <p>The answers are getting stuck in a very basic situation, I want to think of a more general situation.</p> <p>Suppose that $G$ is an affine algebraic group over an algebraically closed subfield, $H$ a normal subgroup, and $X$ an affine $G$-variety with action factoring through $G/H$. Suppose that $G/H$ acts properly and freely on $X$. The stack $[X/(G/H)]$ is representable by a scheme $X/(G/H)$.</p> <p>Question 1: Do we still have $[X/G] \cong [(X/(G/H))/H]$?</p> <p>If so, Question 2: For any $G$-equivariant sheaf $\mathcal{M}$ on the space $X$, by descent for Cartesian sheaves $\mathcal{M}(X/(G/H))$ is computed by the kernel of the diagram,</p> <p>$$\mathcal{M}(X \times_{[X/G]} X/(G/H)) \rightarrow \mathcal{M}(X \times_{[X/G]} X/(G/H) \times_{X/(G/H)} X \times_{[X/G]} X/(G/H))$$</p> <p>What is this functor?</p> <p>Example: When $H = e$, this equalizer takes the difference between the action and projection pull-backs yielding the functor of invariants.</p> http://mathoverflow.net/questions/56865/the-etale-site-of-a-closed-subscheme-and-its-etale-grothendieck-subtopology/56875#56875 Comment by Anonymous Anonymous 2011-02-28T05:08:37Z 2011-02-28T05:08:37Z Thanks for rephrasing it in this language. I'll browse SGA for statement like this tomorrow. http://mathoverflow.net/questions/56865/the-etale-site-of-a-closed-subscheme-and-its-etale-grothendieck-subtopology/56874#56874 Comment by Anonymous Anonymous 2011-02-28T05:06:16Z 2011-02-28T05:06:16Z Thanks. This proof was very close to my original proof. I felt that my extra assumptions had more to do with personal comfort than actual mathematics. http://mathoverflow.net/questions/51598/sheaves-on-stacks-and-interesting-functors/51622#51622 Comment by Anonymous Anonymous 2011-01-11T17:10:43Z 2011-01-11T17:10:43Z Thanks for the response, Please see the new edit. http://mathoverflow.net/questions/51598/sheaves-on-stacks-and-interesting-functors/51599#51599 Comment by Anonymous Anonymous 2011-01-10T03:33:49Z 2011-01-10T03:33:49Z Thanks for the comment. I am aware what the answer should be. As in my example, $k[G] \cong \oplus_{\overline{g} \in G/H} g k[H]$. We want to pick out one of any of these copies $k[H]$ (with $H$ action), all of which are isomorphic under the action of $G/H$. There should be descent data describing how to pass to a sheaf on the point. If $H=\{e\}$ then it is just taking $G$-invariants. It should be something like taking $G/H$-invariants -- but this doesn't make any sense on most $G$ modules (e.g. $k[G]$). I am curious how the descent data would encode this interesting example.