User sasha - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:48:32Z http://mathoverflow.net/feeds/user/2095 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/125273/about-de-rham-and-l-adic-local-systems-comparison About "de-Rham" and "l-adic" local systems - comparison Sasha 2013-03-22T13:43:45Z 2013-03-23T05:12:37Z <p>Hello,</p> <p>Suppose that $k$ is an algebraically closed field of char. 0.</p> <p>Let $X$ be a smooth connected variety over $k$.</p> <p>Then I have the category $A$ of Regular Singular smooth $D$-modules on $X$ (i.e. algebraic vector bundles equipped with regular singular algebraic flat connections).</p> <p>For a category $B$, I am less sure. I would like to say "local systems of $k$-vector spaces for the etale topology on $X$", but maybe this is not good as $k$ is not finite or of $l$-adic nature in general.</p> <p>So:</p> <p>1) Can one make sense from category $B$ and wish $A$ and $B$ to be equivalent?</p> <p>2) Anyway, there seems to be a functor from the category of local systems (of finite sets) on the etale topology to $A$, by "tensoring" with the constant $D$-module (as $D$-modules are etale local). What can one say about this functor?</p> <p>3) On a "decategorified" level, what can one say about the etale fundamental group, versus the group which we get by Tannakian formalism from $A$?</p> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/123535/a-submodule-of-a-constant-d-module-is-constant A submodule of a constant D-module is constant Sasha 2013-03-04T14:50:47Z 2013-03-04T20:11:29Z <p>Hello,</p> <p>Let $X$ be a smooth variety in char. 0. Let us call a $D$-module on $X$ constant, if it is isomorphic to a finite direct sum of the $D$-modules $O$ (the sheaf of regular functions with the usual $D$-module structure). Then a subquotient of a constant $D$-module should be constant. But how to show it? The reason why it should be true is that Riemann-Hilbert correspondence (when over the complex numbers) translates as to representations of a the fundamental group, and there such a statement is trivial.</p> <p>Thanks, Sasha</p> http://mathoverflow.net/questions/121785/covering-of-verma-modules-by-translation-of-a-dominant-verma-module Covering of Verma modules by translation of a dominant Verma module Sasha 2013-02-14T10:19:19Z 2013-02-16T17:33:57Z <p>Hello,</p> <p>Could anyone give a reference or proof for the following fact (which is, probably, not very difficult):</p> <p>We work in category O for a semisimple complex Lie algebra. $M_{\chi}$ denotes the Verma module with shifted highest weight $\chi$.</p> <p>Fact: Let $\chi$ be dominant, $\lambda$ be such that $\chi - \lambda$ is integral. Then there exists a finite dimensional module $E$ and a surjection $E \otimes M_{\chi} \to M_{\lambda}$.</p> <p>It seems that one should first "move" from $\chi$ to a weight in the same Weyl orbit as $\lambda$, and then...</p> <p>Thanks, Sasha</p> http://mathoverflow.net/questions/121787/kostants-theorem-about-ug-being-free-over-zg-and-a-corollary-of-it Kostant's theorem about U(g) being free over Z(g) and a corollary of it Sasha 2013-02-14T10:52:19Z 2013-02-14T14:43:34Z <p>Hello,</p> <p>$g$ is a complex semisimple Lie algebra.</p> <p>There is the result that $U(g)$ is free over $Z(g)$.</p> <p>There is another result: If $E$ is a finite dimensional representation of $g$, then $Hom(E,U(g)^{ad})$ is a free $Z(g)$-module of rank equals the multiplicity of the zero weight in $E$ (here $U(g)^{ad}$ denotes $U(g)$ as a $g$-module for the adjoint action $v\cdot u = vu-uv$).</p> <p>My question is: how can one deduce the second result from the first one?</p> <p>Thanks, Sasha</p> http://mathoverflow.net/questions/115880/duality-in-category-o-vs-duality-of-d-modules Duality in category O vs. Duality of D-modules Sasha 2012-12-09T11:12:14Z 2012-12-09T11:12:14Z <p>Hello,</p> <p>I omit in the following all the words "derived, twisted, holonomic, finitely-generated...".</p> <p>We have the Bernstein-Beilinson equivalence between the category of $N$-equivariant $D$-modules on the flag variety and $\mathfrak{g}$-modules which are $\mathfrak{n}$ locally finite. Also, we have the equivalence between $K$-equivariant $D$-modules and Harish-Chandra modules.</p> <p>My question is about interaction of duality with this equivalences. We have duality for $D$-modules (like Verdier duality). In the categories of Lie modules, we also have dualities (where we take some finite vectors in the abstract dual). Is there a reference for the relation of these dualities? I think one should be careful with the twistings and such (duality will take a $D$-modules to a $D$-module with opposite twisting, a module in category $O$ to a module in the category $O$ for the opposite Borel, and so on).</p> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/110101/compact-objects-in-triangulated-and-infinity-categories Compact objects in triangulated and infinity categories Sasha 2012-10-19T16:22:19Z 2012-10-20T10:16:42Z <p>Hello,</p> <p>I think of a compact object in a category as an object, $Hom$ from which commutes with filtered colimits.</p> <p>I guess that in an infinity category, one also defines a compact object as an object, $Hom$ from which commutes with homotopy filtered colimits.</p> <p>1) In a derived category (in the classical sense), one defines a compact object as an object, $Hom$ from which commutes with direct sums. Is this equivalent to this object being compact in the above sense, when one thinks of it now in the derived category, as an infinity category?</p> <p>2) In the paper of Thomason-Trobaugh, formula 2.4.1.2, it is written that $Hom$ from a perfect complex, commutes with filtered colimits; Here $Hom$ is in the derived category, but the colimit is in the category of complexes. How should I relate it to the notion of compactness? I mean, the colimit is not a homotopy colimit in this formulation.</p> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/109903/embedding-of-a-scheme-into-a-regular-scheme Embedding of a scheme into a regular scheme Sasha 2012-10-17T12:39:47Z 2012-10-18T07:40:44Z <p>Hello,</p> <p>Is it true that if I have a scheme $X$ which is, say, Noetherian, of finite Krull dimension, and semi-separated (intersection of two open affines is again open affine), then I can find a locally closed embedding of it into a scheme of the same type, which is, in addition, regular?</p> <p>I ask just of curiosity, or minor desire not to say "quasi-projective" when I want to have enough locally free objects.</p> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/81707/localizability-of-differential-operators-a-la-grothendieck Localizability of differential operators a la Grothendieck Sasha 2011-11-23T14:00:24Z 2012-10-16T10:41:08Z <p>Hello,</p> <p>Maybe this question is trivial, so sorry</p> <p>Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1).</p> <p>Then we can define the module of differential operators $D^{\leq n} (A)$, a submodule of $End_k (A,A)$ (endomorphisms of vector spaces). $D^{\leq -1} = 0$, and then inductively $D^{\leq n} = { d | [d,a]\in D^{\leq n-1}}$.</p> <p>We have a lemma:</p> <p>Lemma. Let $f \in A$. Then for every $d \in D^{\leq n}(A)$ we can find unique $e \in D^{\leq n}(A_f )$, such that $l\circ d = e \circ l$, where $l: A \to A_f$ is the localization map.</p> <p>I think that I know how to prove the lemma, by induction on the order of diff. op. (just need to see how to apply operators to fractions). It gives us a map $D^{\leq n}(A) \to D^{\leq n}(A_f)$.</p> <p>Question 1. Under which assumptions on $A/k$ this map $D^{\leq n}(A) \to D^{\leq n}(A_f)$ is a localization map (i.e. becomes an isomorphism after tensoring (say on the left, it does not matter) with $A_f$)?</p> <p>Question 2 (my real question). If $A/k$ is finitely generated, or finitely presented, is this a localization map?</p> <p>Somehow, I am having trouble with the "surjectivness" part. Maybe there is some reference?</p> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/106370/gandhis-quote-formalized Gandhi's quote formalized Sasha 2012-09-04T17:56:45Z 2012-09-04T19:09:42Z <p>Hello,</p> <p>I hope this question is appropriate for Mathoverflow. Gandhi said, "Be the change that you wish to see in the world". I don't understand anything in Game/optimization theory (I don't know exactly what theory), but I am curious whether there are mathematical models, in which if the individual makes his local contribution the best, without thinking about the interaction with the decisions of the other, this makes the whole system act in the best possible way.</p> <p>Thank you, Sasha </p> http://mathoverflow.net/questions/105827/quiver-description-of-blocks-in-category-o-for-sl-2 Quiver description of blocks in category O for sl_2 Sasha 2012-08-29T12:37:26Z 2012-08-29T12:37:26Z <p>Hello,</p> <p>Let us consider the category $\mathcal{O}$ for $sl_2$. For every central character $\theta$, we have the corrseponding block $\mathcal{O}(\theta)$. If we consider the block $\mathcal{O}(O)$ with trivial central character (the one in which highest weights can be $0$ or $-2$), it has a functor to the category of things of the form $(V_{-2} , V_0 , X: V_{-2} \to V_0 , Y: V_0 \to V_{-2})$ such that $XY = 0$. This functor is clearly exact and faithful.</p> <p>My question (I would like an answer, or a reference...) are:</p> <p>1) How to show that this functor is an equivalence (maybe first, how to define an adjoint).</p> <p>2) What will be the corresponding quiver data for other blocks (probably $(V_n , V_{-n-2} , X^{n+1} , Y^{n+1}$) with some condition (?)).</p> <p>3) What will be the corresponding assertions for a general semisimple group.</p> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/100509/decomposition-of-a-proper-morphism Decomposition of a proper morphism Sasha 2012-06-24T07:57:35Z 2012-06-24T10:02:27Z <p>Hello,</p> <p>Does any proper morphism $X \to Y$ of, say, algebraic varieties, can be factored as $X \to Z \times Y \to Y$, where $X \to Z \times Y$ is a closed embedding, and $Z \times Y \to Y$ is the projection (and $Z$ is complete).</p> <p>I ask just out of curiosity.</p> <p>Sasha</p> http://mathoverflow.net/questions/99361/the-anticanonical-bundle-on-a-flag-variety-is-ample The anticanonical bundle on a flag variety is ample Sasha 2012-06-12T12:49:22Z 2012-06-14T12:54:51Z <p>Hello,</p> <p>I would like to get references or answers, for the following. How do I show that the anti-canonical line bundle (i.e. dual to top wedge power of cotangent bundle) on a flag variety (of a semisimple algebraic group in char. 0) is ample?</p> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/95026/connective-spectra-versus-simplicial-abelian-groups-very-basic-question Connective spectra versus simplicial abelian groups - very basic question Sasha 2012-04-24T14:49:45Z 2012-04-26T16:55:02Z <p>Hello,</p> <p>I have very general , "introductory" questions (It is quite hard for me to seek for specific things in the algebraic topology literature).</p> <p>I guess that connective spectra have a model structure. So do simplicial abelian groups. Are these Quillen equivalent?</p> <p>Secondly, I think of a simplicial abelian group as a space with strictly associative and commutative operation, while I think of a connective spectrum as a space with an operation which is associative and commutative up to all higher coherences (i.e. some words like $E_{\infty}$). So these are similar. How do I see what extra richness is encoded in a spectrum? For example, what mental pictures do I lose when I think of a connective spectrum as a right-bounded chain complex?</p> <p>I think that the last is the most important for me, to have some small mental picture which I should have for spectra but not for simplicial abelian groups / chain complexes.</p> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/94846/reference-request-cdga-vs-salg-in-char-0 Reference request - CDGA vs. sAlg in char. 0 Sasha 2012-04-22T13:32:27Z 2012-04-23T15:32:33Z <p>Hello,</p> <p>Are the model categories of simplicial commutative algebras over $k$ and that of commutative differential graded algebras (in negative cohmological dimension) Quillen equivalent in char. 0 (or maybe if $k$ is a $Q$-algebra)? What would be a reference?</p> <p>Thank you</p> http://mathoverflow.net/questions/94431/technical-question-about-cell-complexes Technical question about cell complexes Sasha 2012-04-18T17:21:27Z 2012-04-18T19:41:48Z <p>Hello,</p> <p>I have a technical question. My terminology:</p> <p>I - set of standard inclusions $\partial I^n \to I^n$.</p> <p>I-cell (Relative Cell Complexes) - transfinite compositions of pushouts of maps in $I$.</p> <p>CW (CW complexes) - the usual definition (like I-cell but with "cells attached by order of dimension).</p> <p>I-cof - retracts of maps in I-cell, the same as maps having Left Lifting Property w.r.t. maps having Right lifting property w.r.t. I (by Quillen small object argument).</p> <p>My first question is to confirm that CW in I-cell in I-cof are all different</p> <p>My second question is: In this page: <a href="http://ncatlab.org/nlab/show/model+structure+on+topological+spaces" rel="nofollow">http://ncatlab.org/nlab/show/model+structure+on+topological+spaces</a> it is written under "Mixed model structure" (the model structure on $Top$ for which equivalences are weak equivalences and fibrations are Hurewicz fibrations) that cofibrant objects are spaces homotopy equivalent to CW complexes. Is this exactly true (or is it for example spaces homotopy equivalent to cell complexes?). I also read somewhere around nlab that Milnor advocated that spaces homotopy equivalent to CW complexes are nice, so I wanted to know if the cofibrant objects in this mixed model structure are exactly such, or a bit more general.</p> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/59282/sums-compact-objects-f-g-objects-in-categories-of-modules "Sums-compact" objects = f.g. objects in categories of modules? Sasha 2011-03-23T10:39:29Z 2012-04-18T18:34:34Z <p>Hello,</p> <p>Let us call an object of an additive category sumpact (contraction of "sums" and "compact") if taking $Hom$ from it (considered as functor from the category to $Ab$) commutes with coproducts. Note that to be sumpact is weaker than to be compact (which means that $Hom$ from you commutes with filtered colimits).</p> <p>Let us take, for our additive category, the category of left modules over some ring. It is known that compact objects in this category are exactly the finitely presented objects. What about sumpact objects?</p> <p>It is clear that every finitely generated module is sumpact. When I try to prove the converse, I get into some pathological things.</p> <p>Say, if a module has an increasing $\mathbb{N}$-sequence of submodules whose union is the whole module, and such that the union of every finite subsequence is not the whole module, then it is clear that this module is not a sumpact object (by considering the morphism from it to the direct sum of the quotients by members of our sequence). But it seems not clear (perhaps not true) that every non finitely generated module has such a sequence.</p> <p>Also, when I check in the internet, it seems people put some condition: the ring is assumed to be perfect. Then indeed sumpact = f.g.</p> <p>So my question is: for a general ring it is not true that sumpact implies f.g.? Can you give an example? Can you give an example when the ring is commutative? Can you indicate what perfect means and why then everything is OK?</p> <p>Thank you</p> http://mathoverflow.net/questions/93688/distinguished-triangle-of-closed-open-partition-for-d-modules Distinguished triangle of closed - open partition, for D-modules Sasha 2012-04-10T17:16:37Z 2012-04-15T19:11:07Z <p>Hello,</p> <p>I am sorry if this question is not appropriate for MO.</p> <p>Suppose $X$ is the affine line, $i:Z\to X$ is the origin, and $j: U \to X$ is the complement to $Z$ in $X$.</p> <p>I then have a distinguished triangle: $i_! i^! O \to O \to j_* j^* O \to$, where $O$ is any $D$-module on $X$, but I take it to be for this example just the structure sheaf.</p> <p>I want to see "explicitly" what do I get. If I am not wrong, $j_* j^* O$ is Laurent polynomials, while $i_! i^! O$ is $\Delta = \oplus k \partial^i$, in degree $1$. Thus the distinguished triangle is equivalent to the data of an exact sequence: $0 \to O \to j_* O_U \to \Delta \to 0$.</p> <p>My question is:</p> <blockquote> <p>How in principle should I compute the arrow $j_* O_U \to \Delta$ in the s.e.s. above? It is some connecting arrow in the distinguished triangle, which seems abstract to me.</p> </blockquote> <p>The second question I have is how to "compute" $j_! O_U$. I have two strategies, about both of which I am not sure exactly. The worse one is to compute $D (j_* O_U)$, the dual of $j_* O_U$. I don't know how to do it, a resolution seems complicated. The other method would be applying duality $D$ to the s.e.s. above, getting $0 \to \Delta \to j_! O_U \to O \to 0$. Then: </p> <blockquote> <p>How can I compute explicitly how $j_! O_U$ is an extension of $\Delta$ and $O$?</p> </blockquote> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/88453/continued-fractions-and-projective-resolutions Continued fractions and projective resolutions Sasha 2012-02-14T19:44:20Z 2012-04-13T04:14:48Z <p>Hello,</p> <p>This question might be vague and not thought-through enough.</p> <p>If we have a real positive number $x$, we can start to write it as a continued fraction: $x = a_0 + \frac{1}{x_1} , \ldots , x_n=a_n + \frac{1}{x_{n+1}}$ where $a_i$ are non-negative integers, $x_i$ non-nogative real numbers less than $1$. So we can write $x=[a_0,a_1,\ldots]$. But we also may write $x=[a_0,\ldots,a_{n-1},x_n]$, i.e. we might decide that our continued fraction is finite, allowing the last term to be non-integer.</p> <p>If we have a module, we can start taking a projective resolution of it. Again, we can take an infinite projective resolution, or decide to truncate it at some finite level, but then the last term will not be maybe projective.</p> <p>The last term will be projective if our module was "good", i.e. of small enough (in particular, finite) cohomological dimension. In the continued fraction setting, the last term will be integer if our number was "good", i.e. rational...</p> <p>Is there any (wild?) relation between rational numbers among real numbers, and modules of finite cohomological dimension among modules?</p> <p>Sasha</p> http://mathoverflow.net/questions/90638/can-one-prove-vanishing-of-higher-direct-images-fiber-wise/93711#93711 Answer by Sasha for Can one prove vanishing of higher direct images fiber-wise? Sasha 2012-04-10T22:08:10Z 2012-04-10T22:08:10Z <p>Hi Rami,</p> <p>What about using the Grothendieck complex? Thus, there is a bounded complex $K$ of coherent locally free sheaves on Y, such that $K_y$, the application of "fiber at y" functor elementwise to $K$, computes cohomologies of fibers.Then backwards induction seems to give that if (2) holds, the direct image of $O_X$ is concentrated in degree 0.</p> <p>Maybe I am wrong? I did not check carefully. Also, I did not check if we can extract information about image being exactly $O_Y$.</p> <p>Sasha</p> http://mathoverflow.net/questions/92826/about-a-statement-in-jardine-and-goerss-simplicial-homotopy-theory About a statement in Jardine and Goerss "Simplicial Homotopy Theory" Sasha 2012-04-01T17:25:12Z 2012-04-02T07:46:54Z <p>Hello,</p> <p>This probably just technical, but anyway:</p> <p>In "Simplicial Homotopy Theory" by Goerss and Jardine, chap. III, par. 2, after cor. 2.12, they describe a model structure on $Ch^{+}$, the category of chain complexes in non-negative degress. Equivalences are quasi-isom., fibrations are those maps which are surjective in degrees $n \ge 1$ (notice the $1$!).</p> <p>Then they note: "After the fact, it turns out that the cofibrations are those monomorphisms of chain complexes having degreewise projective cokernels".</p> <p>It seems this is not correct as stated. Maybe if it would say $n \ge 0$ upstairs it would be correct, but we do have $n \ge 1$ (corresponding to the picture of simplicial abelian groups).</p> <p>Is it a typo or I don't understand something? Is there a nice description of cofibrations in this case?</p> <p>Thank you, Sasha</p> <p>Update: I was just confused with homological/cohmological indexing conventions. One needs either chains in positive degrees, or cochains in negative degrees. I by mistake used positive degrees as they, but with cochains - by habit.</p> http://mathoverflow.net/questions/92538/transporting-model-structures-via-adjunctions Transporting model structures via adjunctions Sasha 2012-03-29T08:21:18Z 2012-03-29T10:07:37Z <p>Hello,</p> <p>If $F$ is a left adjoint between $C$ and $D$, and $D$ has a model structure; We can define cofibrations and equivalences in $C$ to be those that are so after applying $F$. What are criterions for this to define a model structure? Where can I find a discussion? Also, the dual question about a right adjoint.</p> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/92254/a-infty-structure-questions $A_{\infty}$ structure questions Sasha 2012-03-26T11:25:27Z 2012-03-28T01:06:20Z <p>Hello,</p> <p>I would like explanation or clear source for some things related to $A_{\infty}$-spaces, via Stasheff's polytopes.</p> <p>I tried not to think about them, because they seem too complicated for me; I thought that the small $1$-cubes operad, and abstract $A_{\infty}$-operads (each $A(n)$ is contractible), would be enough. But still, when I want to derive, at least for myself, at least heuristically, the axioms of $A_{\infty}$-algebra (in the algebraic sense), I see that I would like to understand those polytopes a bit.</p> <blockquote> <p>There are different descriptions of $K_n$, Stasheffs polytopes. What would be a clear description, which shows all of the following three features: 1) $K_n$ embed into the small $1$-cubes (non-symmetric) operad; 2) This embedding makes $K_n$ a suboperad. 3) The boundary of $K_n$ breaks to different $K_s \times K_t$, and moreover, I can read the orientations from this, i.e. the signs which I will need to put in the dg-version.</p> </blockquote> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/91618/local-model-structure-on-simplicial-presheaves local model structure on simplicial presheaves Sasha 2012-03-19T13:30:20Z 2012-03-19T16:29:32Z <p>Hello,</p> <p>Let $\mathcal{C}$ be a (small) category equipped with a Grothendieck pretopology.</p> <p>Let $sPSh(\mathcal{C})$ be the category of simplicial presheaves on $\mathcal{C}$, together with its projective model structure (fib. and w.e. are level-wise).</p> <p>Then one defines the class $S$ of local w.e. to be that of some maps of simplicial presheaves which induce isomorphisms on homotopy groups etc...</p> <p>Then one takes the left Bousfield localization of the projective model structure along $S$, to get the projective local model structure (that which models "homotopy" sheaves).</p> <p>I don't understand much in this things, so I have two questions:</p> <blockquote> <p>1) In general, given a set $S$ of maps, we define the set of $S$-local equivalences (those which satisfy some left property w.r.t. $S$-local objects, which are those which satisfy some right property w.r.t. $S$...). For our $S$, will $S$-local equivalences coincide with $S$?</p> <p>2) If I take $T$ to be the set of hypercovers, will $T$-local equivalences be $S$?</p> </blockquote> <p>Thank you very much, Sasha</p> http://mathoverflow.net/questions/90367/reference-request-jacquet-module-and-asymptotic-of-matrix-coefficients Reference request - Jacquet module and asymptotic of matrix coefficients Sasha 2012-03-06T15:04:29Z 2012-03-06T15:04:29Z <p>Hello,</p> <p>I would like to know some nice references about the relation between asymptotics of matrix coefficients of representations of reductive groups over local fields, and the pairing between the Jacquet module of the representation and the Jacquet module of its dual.</p> <p>I would like to know reference for the p-adic case, as well as for the real case (where one uses Jacquet-Casselmann functor instead of Jacquet functor).</p> <p>As I understand, both cases are due to Casselmann.</p> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/89584/about-g-modules-versus-lieg-modules-for-algebraic-groups About $G$-modules versus $Lie(G)$-modules for algebraic groups Sasha 2012-02-26T14:01:49Z 2012-03-01T19:30:40Z <p>Hello,</p> <p>I would like to know clear references about the following facts:</p> <p>Let $G$ be a connected algebraic group (over alg. closed field in char. 0), $Lie(G)$ its Lie algebra, $M$ a $G$-module. I don't assume that $G$ is affine, but if there is a nice simple reference with $G$ affine, then I'll like it too.</p> <blockquote> <p>If $v \in M$ is a vector killed by $Lie(G)$, then it is fixed by $G$.</p> </blockquote> <p>This will establish that $G-mod \to Lie(G)-mod$ is fully faithful (by the usual method of interpreting a morphism as an element of the inner $Hom$).</p> <blockquote> <p>For tori, solvable, nilpotent, semi-simple groups, the expected characterizations of the essential image of $G-mod \to Lie(G)-mod$.</p> </blockquote> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/89263/my-first-question-on-affine-schemes-in-algebraic-geometry/89265#89265 Answer by Sasha for My first question - on Affine Schemes in Algebraic Geometry Sasha 2012-02-23T09:29:47Z 2012-02-23T09:29:47Z <p>An affine scheme can be characterized in the category of locally ringed spaces (one needs the "locally" if I remember correctly). A l.r.s. $X$ is an affine scheme i.f.f. $Hom(Y,X)$ functorially equals $Hom(\Gamma(X,\mathcal{O}_X),\Gamma(Y,\mathcal{O}_Y))$, for $Y$ a l.r.s.</p> <p>In other words, the affine scheme construction is the construction of a right adjoint to $\Gamma: ( l.r.s. ) \to ( rings )^{op}$.</p> http://mathoverflow.net/questions/87985/reference-wanted-etale-sheaves-on-x-versus-on-overlinex Reference wanted - etale sheaves on $X$ versus on $\overline{X}$ Sasha 2012-02-09T13:47:48Z 2012-02-09T19:00:50Z <p>Hello,</p> <p>Let $X$ be a scheme of finite type over a field $k$. Let $l$ be an Galois extension of $k$ with Galois group $\Gamma$, and $\overline{X}$ be the base change of $X$ from $k$ to $l$. Then If I understand correctly (I don't understand much), I have a functor $Sh(X_{et}) \to Sh(\overline{X}_{et})^{\Gamma}$, from etale sheaves on $X$ to etale sheaves on $\overline{X}$, equivariant w.r.t. the action of $\Gamma$ on $\overline{X}$.</p> <p>My question is whether this is an equivalence of categories, and whether you can give a reference for this.</p> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/88020/sheaves-for-which-the-derived-compact-or-not-pushforward-is-zero/88022#88022 Answer by Sasha for sheaves for which the derived (compact or not) pushforward is zero Sasha 2012-02-09T18:45:02Z 2012-02-09T18:45:02Z <p>Inspired by Borel-Weil-Bott, The following comes to mind (I don't know enough to put it in context):</p> <p>We can consider a circle $S^1$, and take the local system which corresponds to the sign character of $\mathbb{Z}$. Then its zero'th cohomology is zero, and the first by duality is also zero.</p> <p>Sasha</p> http://mathoverflow.net/questions/86696/reference-wanted-preservation-of-constructible-sheaves-in-classical-topology Reference wanted - preservation of constructible sheaves (in classical topology) by all functors Sasha 2012-01-26T08:06:46Z 2012-01-26T09:06:44Z <p>Hello,</p> <p>Can anybody point to me a reference about the preservation of the derived bounded category of sheaves with constructible cohomology on the underlying classical (anayltic) space of a complex algebraic variety, with respect to the functors Verdier duality and push-forward (probably "!").</p> <p>Note that I am aware of Kashiwara and Schapira book, but I would like some other reference which does not use this microlocalization things which I do not know.</p> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/86469/number-of-affines-needed-to-cover-a-variety Number of affines needed to cover a variety Sasha 2012-01-23T18:37:52Z 2012-01-24T08:44:14Z <p>Hello,</p> <p>I am aware of the related question <a href="http://mathoverflow.net/questions/13478/minimal-size-of-an-open-affine-cover" rel="nofollow">"Minimal size of an open affine cover"</a>, but would like to ask more specifically:</p> <p>Do you have some elementary (i.e. not using hard things like compactification and such) proof for one of the following (here "variety" is separated over alg. closed field):</p> <blockquote> <p>(1) Let $X$ be a variety; Can you show that $X$ can be covered by $C \cdot dim(X) + D$ open affines, where $C,D$ are universal constants?</p> <p>(2) Let $X$ be quasi-projective; Can you show (1) for it with $C=1,D=1$?</p> <p>(3) Let $X$ be smooth quasi-projective, and char. = 0; Can you show (2) for it?</p> </blockquote> <p>It is easy for a variety $X$ to find an open affine whose complement is of smaller dimension than $X$. But I don't see how given $Y$ closed in $X$, to find an affine open $U$ in $X$ such that $Y-U$ is of smaller dimension than $Y$.</p> <p>Sasha</p> http://mathoverflow.net/questions/123535/a-submodule-of-a-constant-d-module-is-constant/123558#123558 Comment by Sasha Sasha 2013-03-05T14:30:31Z 2013-03-05T14:30:31Z Thank you, now I understand. http://mathoverflow.net/questions/123535/a-submodule-of-a-constant-d-module-is-constant/123558#123558 Comment by Sasha Sasha 2013-03-05T09:25:50Z 2013-03-05T09:25:50Z Thank you for the answer; I did not understand your final conclusion, that $E$ is constant. You got that $E$ is constant as an $O$-module, but what about the $D$-module structure? http://mathoverflow.net/questions/116105/on-the-l-function-of-unique-subrepresentation-of-induced-representation Comment by Sasha Sasha 2012-12-13T10:14:59Z 2012-12-13T10:14:59Z Hi, maybe not to the point, but do you have a reference for the definition of the &quot;standard L-function $L(s,\tau)$&quot;? Thank you. http://mathoverflow.net/questions/115880/duality-in-category-o-vs-duality-of-d-modules Comment by Sasha Sasha 2012-12-09T14:32:38Z 2012-12-09T14:32:38Z OK, thank you, I found the paper: <a href="http://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/duality.pdf" rel="nofollow">kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/&hellip;</a> http://mathoverflow.net/questions/110101/compact-objects-in-triangulated-and-infinity-categories Comment by Sasha Sasha 2012-10-20T18:52:44Z 2012-10-20T18:52:44Z @Akhil and @Marc - Thank you, I will think about your comments. http://mathoverflow.net/questions/110101/compact-objects-in-triangulated-and-infinity-categories Comment by Sasha Sasha 2012-10-20T10:14:18Z 2012-10-20T10:14:18Z I meant that I don't see it personally. Could you hint how can I see it? http://mathoverflow.net/questions/106370/gandhis-quote-formalized Comment by Sasha Sasha 2012-09-04T21:23:40Z 2012-09-04T21:23:40Z @quid: No, I did not. http://mathoverflow.net/questions/105827/quiver-description-of-blocks-in-category-o-for-sl-2 Comment by Sasha Sasha 2012-08-29T12:55:14Z 2012-08-29T12:55:14Z Thank you, but I don't have this book in my local library unfortunately.. http://mathoverflow.net/questions/90638/can-one-prove-vanishing-of-higher-direct-images-fiber-wise/93711#93711 Comment by Sasha Sasha 2012-08-03T15:03:40Z 2012-08-03T15:03:40Z I am sorry, I forgot to mention that one has $K$ if the map $X \to Y$ is flat.. http://mathoverflow.net/questions/100509/decomposition-of-a-proper-morphism/100515#100515 Comment by Sasha Sasha 2012-06-24T13:37:40Z 2012-06-24T13:37:40Z @Dan: Maybe I am wrong, but I think that: Any morphism having a left inverse is a closed embedding, since it is a base change of diagonal (I assume for simplicity that everything is separated). Then our $X \to Xbar \times Y$ is a locally closed embedding, since it is a composition of the closed embedding $X \to X \times Y$ (it has left inverse) with the open embedding $X \times Y \to Xbar \times Y$. On the other hand, $X \to Xbar \times Y$ is proper, since it is the composition $X \to Xbar \times X \to Xbar \times Y$, first arrow being a closed embedding, and second being proper. http://mathoverflow.net/questions/100509/decomposition-of-a-proper-morphism/100515#100515 Comment by Sasha Sasha 2012-06-24T10:46:24Z 2012-06-24T10:46:24Z I see, Thank you! http://mathoverflow.net/questions/99361/the-anticanonical-bundle-on-a-flag-variety-is-ample/99364#99364 Comment by Sasha Sasha 2012-06-15T07:08:49Z 2012-06-15T07:08:49Z Now I looked at Jantzen, II.4.4. This is exactly to the point! Thank you. http://mathoverflow.net/questions/99361/the-anticanonical-bundle-on-a-flag-variety-is-ample Comment by Sasha Sasha 2012-06-13T07:15:21Z 2012-06-13T07:15:21Z Thank you for your answers. http://mathoverflow.net/questions/99361/the-anticanonical-bundle-on-a-flag-variety-is-ample/99364#99364 Comment by Sasha Sasha 2012-06-13T07:14:56Z 2012-06-13T07:14:56Z Thank you. I looked into this article, &quot;A very SIMPLE proof of Bott's theorem&quot;. I did not see ampleness there. Am I wrong and it can be deduced from there? http://mathoverflow.net/questions/95026/connective-spectra-versus-simplicial-abelian-groups-very-basic-question Comment by Sasha Sasha 2012-04-26T16:55:22Z 2012-04-26T16:55:22Z @Tom: OK, Thank you, I changed &quot;multiplication&quot; to &quot;an operation&quot;.