User - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:07:20Z http://mathoverflow.net/feeds/user/2330 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18593/what-are-the-worst-notations-in-your-opinion What are the worst notations, in your opinion ? nicojo 2010-03-18T14:49:17Z 2013-02-25T21:45:17Z <p>With which notation do you feel uncomfortable ?</p> http://mathoverflow.net/questions/20750/properties-stable-under-base-change-in-algebraic-geometry Properties stable under base change in algebraic geometry nicojo 2010-04-08T15:56:23Z 2012-04-24T16:28:04Z <p>I remember to have seen a big list in the EGA of properties $(P)$ such that: if $f : X \to Y$ has $(P)$ then, $f_{(S')} : X_{(S')}\to Y_{(S')}$ has $(P)$, where $f_{(S')}$ is the morphism $f$ after a base change $S'\to S$., etc. but I can't find it now...</p> <p>Does anyone know where I can find such a list ?</p> <p>(I am interested by $(P)$ = "to be a closed map")</p> http://mathoverflow.net/questions/42276/noethers-normalization-lemma-over-a-ring-a Noether's normalization lemma over a ring A nicojo 2010-10-15T11:41:38Z 2011-04-05T17:08:05Z <p>Given a field $k$ and a finitely generated $k$-algebra $R$ without zero divisors, one knows that there exist $x_1, \ldots, x_n$ algebraically independent such that $R$ is integral over $k[x_1, \ldots, x_n]$. </p> <p>Does one have a similar statement, under good assumptions, if $k$ is not a field but a ring ? In this discussion, I am also interested by geometric explanations.</p> http://mathoverflow.net/questions/54603/what-should-i-read-before-reading-about-arakelov-theory/54610#54610 Answer by nicojo for What should I read before reading about Arakelov theory? nicojo 2011-02-07T07:26:06Z 2011-02-07T07:26:06Z <p>You can try to read this wonderful <a href="http://www.ams.org/mathscinet/search/publdoc.html?arg3=&amp;co4=AND&amp;co5=AND&amp;co6=AND&amp;co7=AND&amp;dr=all&amp;pg4=AUCN&amp;pg5=TI&amp;pg6=PC&amp;pg7=ALLF&amp;pg8=ET&amp;review_format=html&amp;s4=&amp;s5=&amp;s6=&amp;s7=%2520Butl.%2520Soc.%2520Catalana%2520Mat.%2520&amp;s8=All&amp;vfpref=html&amp;yearRangeFirst=&amp;yearRangeSecond=&amp;yrop=eq&amp;r=72&amp;mx-pid=1872772" rel="nofollow">paper</a> by Burgos. It is in Catalan, though. If you are familiar with Spanish or French, you can manage to read it.</p> http://mathoverflow.net/questions/48798/non-finitely-generated-subalgebra-of-a-finitely-generated-algebra Non finitely-generated subalgebra of a finitely-generated algebra nicojo 2010-12-09T18:05:26Z 2010-12-09T21:28:16Z <p>Ok, I feel a little bit ashamed by my question.</p> <p>This afternoon in the train, I looked for a counter-example:<br/> — k a field <br/> — A a finitely generated k-algebra <br/> — B a sub-k-algebra of A that is not finitely generated <br/></p> <p>Finally, I have found this: <br/> — k any field<br/> — A=k[x,y]<br/> — B=k[xy, x.y^2, x.y^3, ...]<br/></p> <p>(proof : exercise)</p> <p>My questions are : </p> <p>1) What is your usual counter-example ?<br/> 2) Under which conditions can we conclude that B is f.g. ?<br/> 3) How would you interpret geometrically this counter-example ?</p> http://mathoverflow.net/questions/42275/choosing-the-algebraic-independent-elements-in-noethers-normalization-lemma Choosing the algebraic independent elements in Noether's normalization lemma nicojo 2010-10-15T11:39:55Z 2010-10-16T15:26:41Z <p>Given a field $k$ and a finitely generated $k$-algebra $R$ without zero divisors, one knows that there exist $x_1, \ldots, x_n$ algebraically independent such that $R$ is integral over $k[x_1, \ldots, x_n]$. How can one choose actually the $x_i$'s ? </p> <p>More precisely, if $a\in R$ is transcendent over $k$, can one find $x_2, \ldots x_n$ such that $R$ is integral over $k[a,x_2 \ldots, x_n]$ ? If it is false, can one get the conclusion under stronger assumptions ? In this discussion, I am also interested by geometric explanations.</p> http://mathoverflow.net/questions/28119/why-the-roots-of-unity-are-the-analogs-of-constants Why the roots of unity are the analogs of constants ? nicojo 2010-06-14T13:55:23Z 2010-06-23T08:15:24Z <p>Hello,</p> <p>Joel Dogde, in a comment on his question "Roots of unity in different completions of a number field", says the following, about the analogy between number fields and function fields : </p> <p><em>Number of roots of unity in number fields is something like the size of the constant field for function fields.</em></p> <p>Could anyone explain that ?</p> <p>Thanks.</p> http://mathoverflow.net/questions/26947/is-there-a-reason-for-defining-the-differential-forms-before-the-vector-fields Is there a reason for defining the differential forms before the vector fields ? nicojo 2010-06-03T18:34:22Z 2010-06-03T23:41:37Z <p>Hi, my question is the following : </p> <p>In EGA IV chapter 16, given $X$ a scheme over $S$, Grothendieck defines first $\Omega^1_{X/S}$, the $O_{X}$-module of the 1-differentials. He then defines the tangent sheaf : $T_{X/S}:= Hom_{O_X} (\Omega^1_{X/S}, O_X)$, which is equal to $Der(O_X, O_X)$. Why, one does not do the opposite and first define $T_{X/S}:=Der(O_X, O_X)$ and then $\Omega^1_{X/S}:=Hom_{O_X} (T_{X/S}, O_X)$ ?</p> <p>I suspect there are several reasons for this : </p> <p>1) This new object, whose definition seems to be easier, is maybe less handy to work with.</p> <p>2) In some important cases, it gives the wrong object</p> <p>3) Some other philosophical reason</p> <p>I would like to have the opinion on this question of mathematicians who know more than me geometry and differential forms.</p> <p>Thanks.</p> http://mathoverflow.net/questions/25054/different-ways-of-thinking-about-the-derivative/25075#25075 Answer by nicojo for Different ways of thinking about the derivative nicojo 2010-05-18T00:27:09Z 2010-05-18T00:27:09Z <p>If $R$ is a ring, then a derivation $\partial : R \to R$ is a vector field over the scheme $\mathrm{Spec} R$.</p> http://mathoverflow.net/questions/24361/are-two-sheaves-that-are-locally-isomorphic-globally-isomorphic Are two sheaves that are locally isomorphic globally isomorphic ? nicojo 2010-05-12T10:21:48Z 2010-05-12T18:31:09Z <p>Let $X$ be a topological space and let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves over $X$.</p> <p>Of course, if one has a morphism $f : \mathcal{F} \to \mathcal{G}$ such that for all $x\in X$, $f_x : \mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, then it is known that $f$ itself is an isomorphism.</p> <p>My question is the following: if we don't have such a morphism $f$, but if we know that for all $x\in X$, $\mathcal{F}_x$ and $\mathcal{G}_x$ are isomorphic, is it true that $\mathcal{F}$ and $\mathcal{G}$ are isomorphic ?</p> http://mathoverflow.net/questions/7911/is-a-functor-which-has-a-left-adjoint-which-is-also-its-right-adjoint-an-equivale Is a functor which has a left adjoint which is also its right adjoint an equivalence ? nicojo 2009-12-05T22:27:50Z 2010-04-28T13:12:39Z <p>I am looking for a counter-example of two functors F : C -> D and G : D->C such that</p> <p>1) F is left adjoint to G</p> <p>2) F is right adjoint to G</p> <p>3) F is not an equivalence (ie F is not a quasi-inverse of G)</p> http://mathoverflow.net/questions/21577/is-there-a-relation-between-the-tangent-bundle-of-a-scheme-and-the-tangent-bundle Is there a relation between the tangent bundle of a scheme and the tangent bundle of the associated reduced scheme ? nicojo 2010-04-16T14:02:20Z 2010-04-17T19:17:20Z <p>My question is vague and general, and, if you want, naive.</p> <p>$\bullet$ Given $X/S$ a scheme. We denote by $X_{red}$ the reduced scheme associated to $X$.</p> <p>Grothendieck defines in EGA 4 (16.5.12.1) the tangent bundle $T_{X/S}$ of $X$ relatively to $S$. Is there any relation between $T_{X_{red}/S}$ and $T_{X/S}$ ? More precisely, we know that there is a morphism $$f : T_{X_{red}/S} \to T_{X/S}\times_{X} X_{red} ;$$ what can be said of $f$ ? I am interested, in particular, in the case when $S=Spec \ k$, with $k$ a field.</p> <p>$\bullet$ A related question (but in which I am less interested) is the comparison of the Zariski tangent space at one point (as defined in Hartshorne p. 80) of $X$ and $X_{red}$. In general, if $\pi : X_{red} \to X$ denotes the canonical morphism, and if $x\in X_{red}$, the linear map $$T_x \ \pi : T_x \ X_{red} \to T_{\pi(x)} \ X$$ is injective, since $\pi$ is a closed immersion. Is it an isomorphism ?</p> <p>$\bullet$ Finally, a related question is the comparison of $\Omega^1_{X/S}$ and $\Omega^1_{X_{red}/S}$.</p> <p>$\bullet$ Ideally, I would like to transport sections of $T_{X_{red}/S}$ to sections of $T_{X/S}$.</p> http://mathoverflow.net/questions/20065/topological-limits-as-categorical-limits Topological limits as categorical limits nicojo 2010-04-01T12:28:57Z 2010-04-01T12:28:57Z <blockquote> <p><strong>Possible Duplicate:</strong><br> <a href="http://mathoverflow.net/questions/9951/limits-in-category-theory-and-analysis" rel="nofollow">limits in category theory and analysis</a> </p> </blockquote> <p>Let $X$ be topological space, $(u_n)$ a sequence converging to $l$.</p> <p>Can one find a categorical setting such that $l$ appears to be the limit a of certain functor ? </p> <p>Ie, can one find a category $C$, a functor $F : \mathbb{N}\to C$ that represents the sequence $(u_n)$ and such that $\lim F=l$ (or something like that) ?</p> http://mathoverflow.net/questions/18593/what-are-the-worst-notations-in-your-opinion/18599#18599 Answer by nicojo for What are the worst notations, in your opinion ? nicojo 2010-03-18T15:07:40Z 2010-03-18T15:07:40Z <p>I don't like (but maybe for a bad reason) the notation $F\vdash G$ for $F$ is left adjoint to $G$. </p> <p>Any comment ?</p> http://mathoverflow.net/questions/16760/cohomology-of-sheaves-for-sites-and-galois-cohomology Cohomology of sheaves for sites and Galois cohomology nicojo 2010-03-01T15:18:04Z 2010-03-01T23:50:54Z <p>Hello,</p> <p>I am looking for a reference (if it exists) that makes the link between cohomology of sheaves for sites and Galois cohomology : </p> <p>quickly said, I would like to see Galois cohomology (at least in the commutative case) as the cohomology of a sheaf over the étale site of extensions of k.</p> <p>By the way, what is a reference for cohomology of sites ?</p> <p>Thanks</p> http://mathoverflow.net/questions/14151/a-good-place-where-to-learn-about-derived-functors A good place where to learn about derived functors nicojo 2010-02-04T14:34:01Z 2010-02-04T23:36:38Z <p>I would like to learn about derived functors. Which reference do you advise ?</p> http://mathoverflow.net/questions/8879/are-coproducts-and-patchings-filtered-colimits Are coproducts and patchings filtered colimits ? nicojo 2009-12-14T15:25:01Z 2009-12-16T06:41:44Z <p>1) Is the coproduct $\coprod_{i\in I} X_i$ a filtered colimit ?</p> <p>2) Is the colimit colim($X => Y$) (two arrows from X to Y) a filtered colimit ?</p> http://mathoverflow.net/questions/8451/definition-of-etale-for-rings Definition of étale for rings nicojo 2009-12-10T13:05:03Z 2009-12-14T15:01:29Z <p>Let $A \to B$ be a ring extension.</p> <p>What is the definition of $B/A$ étale ?</p> <p>When $A$ is a field, do we get a nice characterization ?</p> http://mathoverflow.net/questions/8376/a-technical-question-about-derivations-of-sheaves-on-group-schemes A technical question about derivations of sheaves on group schemes nicojo 2009-12-09T17:51:40Z 2009-12-09T19:12:14Z <p>Let $G$ be a group scheme (for instance, over $k$ a field of characteristic 0). Let $e$ be its unit. I denote by $O_G$ the structural sheaf of $G$.</p> <p>Let $D_e : O_{G,e} \to k$ a derivation.</p> <p>I would like to get directly (ie, without any consideration about the cotangent bundle, or some canonical isomorphisms...) a derivation $D : O_G\to O_G$ that extends $D_e$, and which is compatible with the action of $G$. That is, I would like to get this derivation by the mean of the multiplication map : $m : G \times G \to G$, etc., etc.</p> <p>I have guessed this question would not be difficult, and would only be a matter of technics, but I can't manage to do it.</p> http://mathoverflow.net/questions/7911/is-a-functor-which-has-a-left-adjoint-which-is-also-its-right-adjoint-an-equivale/7922#7922 Answer by nicojo for Is a functor which has a left adjoint which is also its right adjoint an equivalence ? nicojo 2009-12-05T23:36:25Z 2009-12-06T00:18:17Z <p>The answer of Ben Webster, can be made easier. Consider the functor F : (A-mod) -> (A-mod) which maps any A-module on (0). Then, F is a left adjoint to F ; and so, is a also a right adjoint to F. This is clear because for all A-modules N, M, one has Hom_A(0,N)=Hom_A(M,0). But, F is not an equivalence.</p> http://mathoverflow.net/questions/56677/what-notions-are-used-but-not-clearly-defined-in-modern-mathematics Comment by 2011-03-03T08:38:02Z 2011-03-03T08:38:02Z Wow ! It is crazy how things turned about this question (see the above comments) !! which is, in my opinion, a very interesting question !! http://mathoverflow.net/questions/52728/good-points-for-the-job-of-being-a-mathematician Comment by 2011-01-24T06:01:45Z 2011-01-24T06:01:45Z @Mariano : i don't agree with you. There is a point in having two different lists, one with the good points, another one with the bad ones. http://mathoverflow.net/questions/52728/good-points-for-the-job-of-being-a-mathematician Comment by 2011-01-21T05:53:05Z 2011-01-21T05:53:05Z @Mariano ; what is the point of saving a question ? http://mathoverflow.net/questions/48798/non-finitely-generated-subalgebra-of-a-finitely-generated-algebra/48805#48805 Comment by 2010-12-09T19:53:13Z 2010-12-09T19:53:13Z I think that your description corresponds actually to B=k[x, xy, x.y^2, x.y^3, ...]. http://mathoverflow.net/questions/28119/why-the-roots-of-unity-are-the-analogs-of-constants Comment by 2010-06-14T14:59:23Z 2010-06-14T14:59:23Z What is Malle conjecture ? http://mathoverflow.net/questions/26947/is-there-a-reason-for-defining-the-differential-forms-before-the-vector-fields/26953#26953 Comment by 2010-06-03T20:07:14Z 2010-06-03T20:07:14Z Yes, but, in the case $X/S$ not smooth, is $\Omega^1_{X/S}$ useful ? http://mathoverflow.net/questions/25054/different-ways-of-thinking-about-the-derivative/25075#25075 Comment by 2010-05-19T09:06:42Z 2010-05-19T09:06:42Z This is true !! More precisely, if X=Spec A and if T is the tangent sheaf of X (over the base Z), as defined in EGA 4, then a global section of T is the same as a derivation of A. http://mathoverflow.net/questions/23564/is-all-categorical-reasoning-formally-contradictory Comment by 2010-05-05T10:53:50Z 2010-05-05T10:53:50Z The interview goes on like that : &quot;Grothendieck provided a partial solution in terms of universes but a revolution of the foundations similar to what Cauchy and Weierstrass did for analysis is still to arrive.&quot; http://mathoverflow.net/questions/21577/is-there-a-relation-between-the-tangent-bundle-of-a-scheme-and-the-tangent-bundle Comment by 2010-04-16T14:47:34Z 2010-04-16T14:47:34Z Yes, I meant injective ! I've changed it in the question. Hence, with your example, we see that $T_x \ \pi$ is not surjective in general. http://mathoverflow.net/questions/20750/properties-stable-under-base-change-in-algebraic-geometry/20751#20751 Comment by 2010-04-08T16:54:38Z 2010-04-08T16:54:38Z OK, sorry... Indeed, Poonen's notes contains references to EGA. http://mathoverflow.net/questions/20750/properties-stable-under-base-change-in-algebraic-geometry Comment by 2010-04-08T16:12:12Z 2010-04-08T16:12:12Z Thanks for your comment Emerton. Fortunately, in my case, $f$ is a closed immersion ! http://mathoverflow.net/questions/20750/properties-stable-under-base-change-in-algebraic-geometry/20751#20751 Comment by 2010-04-08T16:11:08Z 2010-04-08T16:11:08Z Thanks for this reference ! However, a reference to EGA should be better for me. http://mathoverflow.net/questions/20065/topological-limits-as-categorical-limits Comment by 2010-04-01T12:53:39Z 2010-04-01T12:53:39Z Yes, thank you for pointing it to me ! http://mathoverflow.net/questions/14151/a-good-place-where-to-learn-about-derived-functors Comment by 2010-02-04T15:42:57Z 2010-02-04T15:42:57Z I don't really need them. But I feel like I need to know more about cohomology. I know a little bit of singular/simplicial/de Rham/Cech cohomologies. If often hear (in seminars) of the R^i f, and I would like to know who they are. Thanks for all for the answers. I was thinking of the Hartshorne, I know the Schapira-Kashiwara and I like it for it preciseness. I was more looking for some typed lectures than for a book. Thanks again. http://mathoverflow.net/questions/9611/eigenvalue-characters Comment by 2009-12-23T15:01:53Z 2009-12-23T15:01:53Z You should add the reference to the previous question. Thanks.