User lalit jain - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T12:13:57Z http://mathoverflow.net/feeds/user/12693 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104344/representable-presheaf Representable Presheaf Lalit Jain 2012-08-09T11:30:12Z 2012-08-09T23:26:17Z <p>I have a very quick question. Is there an easy example of a representable presheaf on a site that is not a sheaf? This certainly can't happen on a small FPPF site so I would expect a counterexample to arise on the big Zariski site. </p> <p>Thanks!</p> http://mathoverflow.net/questions/95514/incidence-correspondence Incidence Correspondence Lalit Jain 2012-04-29T17:46:17Z 2012-04-30T07:05:42Z <p>A useful tool in Algebraic Geometry is the incidence correspondence. Loosely speaking, it is a set of the form $$\{(p,X): p \text{ a fixed dimension subscheme of } Y \text{ and } X \text{ a specific type of subscheme} \}.$$ For example one could consider the incidence correspondence of lines in $\mathbb{P}^2$ with a point on them, or cubics in $\mathbb{P}^3$ with a line on them. </p> <p>It is not too hard to see that in each of the previous cases, the resulting scheme is a variety by writing down explicit equations in coordinates. In the first case the variety lives in $\mathbb{P}^2\times \mathbb{P}^2$ and with a bit of work it is not too hard to show it is a projective bundle over $\mathbb{P}^2.$ The second case is a subvariety of $\mathbb{P}^{19}\times G(1,3)$.</p> <p>My question is the following. Is there a way to get both of the previous examples in a more natural way then explicitly writing down equations? Should I even expect there to be one? Generally when working with incidence correspondences one is interested in properties such as smoothness and irreducibility and most authors I have seen conclude these from the equations. Since the first case above does end up being a projective bundle, I would really hope for there to be a natural way to construct it.</p> <p>Thanks.</p> http://mathoverflow.net/questions/23297/simple-examples-for-the-use-of-spectral-sequences/86976#86976 Answer by Lalit Jain for Simple examples for the use of spectral sequences Lalit Jain 2012-01-29T19:29:48Z 2012-01-29T19:29:48Z <p>I can't recommend the following document by Tom Weston enough. It introduces spectral sequences rapidly and at a comfortable level of generality. It then applies the Hochschild-Serre sequence to group cohomology.</p> <p>www.math.mcgill.ca/goren/SeminarOnCohomology/infres.pdf</p> http://mathoverflow.net/questions/82292/action-of-non-split-torus-in-deligne-lustzig-induction Action of Non-Split Torus in Deligne-Lustzig induction Lalit Jain 2011-11-30T17:19:38Z 2011-11-30T23:34:32Z <p>Recently I have been trying to understand Deigne-Lustzig induction in the case of $G = \text{Sl}(2,\mathbb{F}_p).$</p> <p>In this case the appropriate Deligne Lustzig variety is given by $X:xy^q-y^qx = 1,$ aka the Drinfeld curve. The action of $G$ on affine space fixes $X$ and in addition commutes with the action of the $q+1$ roots of unity, $\mu_{q+1},$ by scaling. In almost all expositions I have read on Deligne Lustzig induction, $\mu_{q+1}$ is noncanonically identified with a nonsplit torus of $G.$ However for the action of $T$ we choose does not seem to be the action of $G$ restricted to $T.$</p> <p>My question is rather vague. If we are only interested in the action of $T$ after identifying it with the roots of unity, why is it important to mention it at all? Perhaps this generalizes in some way that explains this but it is not clear in the case of $\text{Sl}(2,\mathbb{F}_p).$ Thanks.</p> http://mathoverflow.net/questions/60101/density-of-irreducible-polynomials-in-mathbbzx/60114#60114 Answer by Lalit Jain for Density of Irreducible Polynomials in $\mathbb{Z}[x]$ Lalit Jain 2011-03-30T19:47:04Z 2011-03-30T19:47:04Z <p>Take a look at the book: <em>An introduction to sieve methods and their applications</em> By Alina Cojocaru, Maruti Ram Murty</p> <p>In section 4.3 the Turan Sieve is used to prove that the probability of a random polynomial with integer coefficients is irreducible is 1. It is available online at <a href="http://books.google.com/books?id=1swo9Yf3d2YC&amp;lpg=PA135&amp;ots=gzJrlmjXuu&amp;dq=murty%2520sieves&amp;pg=PA51#v=onepage&amp;q=murty%2520sieves&amp;f=false" rel="nofollow">Google books</a>.</p> http://mathoverflow.net/questions/54612/minimal-prerequisite-to-reading-wiles-proof-of-fermats-last-theorem/54845#54845 Answer by Lalit Jain for Minimal prerequisite to reading Wiles' proof of Fermat's Last Theorem Lalit Jain 2011-02-09T04:05:05Z 2011-02-09T04:05:05Z <p>Here is a good set of notes by Nigel Boston. I find them very readable and fairly self contained.</p> <p><a href="http://www.math.wisc.edu/~boston/869.pdf" rel="nofollow">http://www.math.wisc.edu/~boston/869.pdf</a></p> http://mathoverflow.net/questions/104609/algebraic-curve-approximation Comment by Lalit Jain Lalit Jain 2012-08-13T13:09:01Z 2012-08-13T13:09:01Z See <a href="http://en.wikipedia.org/wiki/Polynomial_interpolation" rel="nofollow">en.wikipedia.org/wiki/Polynomial_interpolation</a> or <a href="http://en.wikipedia.org/wiki/Bernstein_polynomial" rel="nofollow">en.wikipedia.org/wiki/Bernstein_polynomial</a>. http://mathoverflow.net/questions/104344/representable-presheaf Comment by Lalit Jain Lalit Jain 2012-08-12T14:38:00Z 2012-08-12T14:38:00Z @Damian - Thanks, I didn't know the fpqc fact. http://mathoverflow.net/questions/104344/representable-presheaf/104360#104360 Comment by Lalit Jain Lalit Jain 2012-08-12T14:35:23Z 2012-08-12T14:35:23Z That's a great example! Thanks. http://mathoverflow.net/questions/95514/incidence-correspondence/95559#95559 Comment by Lalit Jain Lalit Jain 2012-05-02T18:25:00Z 2012-05-02T18:25:00Z Thanks. That was an helpful explanation. http://mathoverflow.net/questions/90087/errata-of-the-treatise-of-analysis-of-dieudonne-another-example Comment by Lalit Jain Lalit Jain 2012-03-02T23:28:24Z 2012-03-02T23:28:24Z You may want to add this to your previous post. http://mathoverflow.net/questions/82292/action-of-non-split-torus-in-deligne-lustzig-induction/82316#82316 Comment by Lalit Jain Lalit Jain 2011-12-01T19:15:40Z 2011-12-01T19:15:40Z Thanks for the insight! I'll take a closer look at Bonnafe's book.