User neha - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T03:14:54Z http://mathoverflow.net/feeds/user/9492 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43311/sheaves-and-cosheaves sheaves and cosheaves Neha 2010-10-23T15:56:13Z 2012-10-02T21:06:20Z <p>I am struggling hard to understand the pushforwards and pullbacks of cosheaves. Are they also cosheaves? And what are quasicoherent cosheaves? Is there anything like coquasicoherent cosheaves? Please tell me a good refernce on theses topics, if there is some.</p> http://mathoverflow.net/questions/65789/glueing-oriented-manifolds-with-boundary glueing oriented manifolds with boundary Neha 2011-05-23T19:33:31Z 2011-05-23T19:33:31Z <p>Can someone please explain glueing two oriented manifolds along a common boundary but with opposite orientations. What happens at the boundary? Which orientation do we keep? Please suggest some good references.</p> http://mathoverflow.net/questions/61772/pure-submodules Pure submodules Neha 2011-04-15T01:25:50Z 2011-04-15T01:30:21Z <p>Is the dual of a pure module also pure?</p> <p>Suppose $H$ is a Hopf algebra over a field $K$ and $S$ is a ring which has a right $H$-action.</p> <p>If ${}_HA$ is a pure left $H$-submodule, will $S\otimes_H A$ be a pure $S$-submodule ? If not, then what are the minimum conditions required (on $H$ or $S$) to make it so? </p> http://mathoverflow.net/questions/47701/coaction-on-a-finite-dual-of-a-hopf-algebra Coaction on a finite dual of a Hopf algebra Neha 2010-11-29T18:01:21Z 2010-11-29T18:01:21Z <p>If $H$ is a central hopf subalgebra of a Hopf algebra $A$. Let $\phi:A\rightarrow A\otimes H$ be a coaction of $H$ on $A$. When does $A^\circ$ becomes a $H^{*}$-submodule i.e. is $A^\circ$ an $H$-subcomodule? Basically I want to realize given a coaction of $H$ on $A$, does it imply a coaction of $H$ on the finite dual of $A$ as well: $$A^\circ\rightarrow A^\circ\otimes H \hspace{8 pt}?$$</p> http://mathoverflow.net/questions/44738/finite-dual-of-an-algebra-morphism Finite dual of an algebra morphism. Neha 2010-11-03T22:09:20Z 2010-11-04T09:48:20Z <p>Is finite dual of an algebra morphism a morphism of coalgebras? Does taking finite dual preserves exactness of an exact sequence of algebra morphisms? When is this possible?</p> http://mathoverflow.net/questions/41711/quasi-coherent-sheaf Quasi coherent sheaf Neha 2010-10-10T20:54:23Z 2010-10-11T12:26:48Z <p>One really stupid, trivial question: A Quasi coherent sheaf $F$ on an affine group scheme(Spec R) is simply an R-module. What happens in case R is a Hopf algebra? Will the Q.coherent sheaf $F$ be an algebra in this case?</p> http://mathoverflow.net/questions/41713/external-tensor-product-of-quasi-coherent-sheaf External tensor product of quasi coherent sheaf Neha 2010-10-10T21:37:30Z 2010-10-10T21:37:30Z <p>When is the following map possible?</p> <p>$A\boxtimes A(\mathcal{G}\times\mathcal{G})\rightarrow A(\mathcal{G}) \otimes A(\mathcal{G})$; where $\mathcal{G}$ is a group scheme, $A$ is a quasi coherent sheaf(of algebras) over $\mathcal{G}$ and $\boxtimes$ is the external tensor product of a sheaf given by $\pi_1^*A\otimes\pi_2^* A$.</p> <p>In general, for any open set $U$, $A\boxtimes A(U\times U)\rightarrow A(U)\otimes A(U)$ doesnt hold true. What are the conditions required for this to hold? Will generation of $A$ by global sections suffice?</p> <p>Please help.</p> http://mathoverflow.net/questions/41254/external-tensor-product-of-sheaves/41423#41423 Answer by Neha for External tensor product of sheaves Neha 2010-10-07T14:51:51Z 2010-10-07T14:51:51Z <p>Thanks Sasha for the answer.</p> <p>Yes, $A\boxtimes A$ is the sheaf given by $p_1^∗A⊗p_2^∗A$ on $G\times G$. Can you please tell me if $V\otimes O_G\to A$ is surjective, then how and why should $V\otimes O_{G\times G}\to p_{i}^{∗} A$ be surjective? And finally how will we get the last step i.e. why would this map become $V\otimes V\otimes O_{G\times G}\to A\boxtimes A$ is surjective? I am sure it is trivial for you, but I am not able to figure out the correct reason, so please help.</p> http://mathoverflow.net/questions/41254/external-tensor-product-of-sheaves External tensor product of sheaves Neha 2010-10-06T11:08:22Z 2010-10-07T14:51:51Z <p>Suppose $\mathcal{A}$ is a quasi coherent sheaf of algebras over a group scheme $\mathcal{G}$. Suppose it is generated by global section. Then , what can we say about the external tensor product $\mathcal{A}\boxtimes\mathcal{A}$? Will this sheaf also be generated by the tensor product of the global section with itself? Or is it bigger than this?</p> http://mathoverflow.net/questions/39858/localization-of-a-polynomial-ring-at-a-prime-ideal Localization of a polynomial ring at a prime ideal. Neha 2010-09-24T13:16:44Z 2010-09-24T14:00:34Z <p>If $R=\mathbb{C}[x,y]$ is the polynomial ring in two variables $x$ and $y$ then we know that the localization of R at the multiplicative set $S=[1,x,x^2,x^3,...]$ is given by $R_x=\mathbb{C}[x,x^{-1},y]$. Now, what will be the localization of $R$ at the prime ideal $(x)$. i.e. what will $R_{(x)}$ be?</p> http://mathoverflow.net/questions/65789/glueing-oriented-manifolds-with-boundary Comment by Neha Neha 2011-05-25T10:35:58Z 2011-05-25T10:35:58Z Ryan, I understand the differential case and the topological case in general. My question was about topological manifolds with boundary and orientation on them. What I am really asking about is some \textbf{good reference} that contains induced orientation on the boundary and gluing.. Thanks. http://mathoverflow.net/questions/41713/external-tensor-product-of-quasi-coherent-sheaf Comment by Neha Neha 2011-04-22T16:22:32Z 2011-04-22T16:22:32Z I meant $\mathcal{G}\times \mathcal{G}$ in the above statement. http://mathoverflow.net/questions/41713/external-tensor-product-of-quasi-coherent-sheaf Comment by Neha Neha 2011-04-22T16:21:44Z 2011-04-22T16:21:44Z Let $(\mathcal{G},\mathcal{O})$ be an affine scheme, and $\mathcal{A}$ a quasi coherent sheaf over $\mathcal{G}$. Is $(\mathcal{A}\boxtimes \mathcal{A})(\mathcal{G}\mathcal{G})=\mathcal{A}(\mathcal{G})\otimes \mathcal{A}(\mathcal{G})$ ? http://mathoverflow.net/questions/61772/pure-submodules Comment by Neha Neha 2011-04-15T12:53:51Z 2011-04-15T12:53:51Z David Roberts: Thanks. Mariano: yes, you are right. Pure is applied to submodules. That was my mistake. here I recall the definition of a Pure submodule. Let M, P be modules over a ring R. If i:P\to M is injective then P is a pure submodule of M if, for any R-module X, the natural induced map on tensor products i\otimes id_X:P\otimes X \to M\otimes X is injective. My question is if A is an H-algebra which is a pure submodule of $H^A$, is the dual H-algebra $A^*$ with values in $H$ also a pure submodule of some extension? David: Thanks, but I dont what any condition on $S$. http://mathoverflow.net/questions/43311/sheaves-and-cosheaves Comment by Neha Neha 2010-10-27T12:09:30Z 2010-10-27T12:09:30Z If $\mathcal{A}$ is a sheaf of algebras over $\mathbb{K}$, then by the dual of $\mathcal{A}$, I mean $\mathcal{A}^∗:=\text{Hom}_{\mathbb{K}}(\mathcal{A},\mathbb{K})$, which I think is a coseaf!!! http://mathoverflow.net/questions/43311/sheaves-and-cosheaves/43312#43312 Comment by Neha Neha 2010-10-26T13:07:55Z 2010-10-26T13:07:55Z Thanks everybody. Daniel, Can you please elaborate a bit more on localizing a coalgebra or a comodule. Where can I read about this more? http://mathoverflow.net/questions/43311/sheaves-and-cosheaves/43419#43419 Comment by Neha Neha 2010-10-26T13:04:28Z 2010-10-26T13:04:28Z Many thanks Rodrigues. What I am still thinking is how to define quasicoherent cosheaves! It seems that it is natural to think of them as cosheaves of $\mathcal{O}^\circ_X$-comodules. and call them quasicoherent if their dual sheaf is quasicoherent. What do you say? http://mathoverflow.net/questions/43311/sheaves-and-cosheaves/43312#43312 Comment by Neha Neha 2010-10-23T17:36:41Z 2010-10-23T17:36:41Z Thanks Peter, I want to learn all the operations that one can do with a sheaf. Like taking their external tensor product. How about the structural cosheaf? is it a cosheaf of corings? So, I have a quasicoherent sheaf $A$ of algebras over a scheme $X$. That means it is essentially $\mathcal{O}_X$-modules. The dual of $A$ is a cosheaf. Will this dual cosheaf be also quasicoherent? And will it be a quasicoherent cosheaf of coalgebras? I mean, will it be $\mathcal{O}_X$-comodules. http://mathoverflow.net/questions/43311/sheaves-and-cosheaves Comment by Neha Neha 2010-10-23T16:37:11Z 2010-10-23T16:37:11Z All the possible operations that one can do with a sheaf. Like taking their external tensor product. How about the structural cosheaf? is it a cosheaf of corings? So, I have a quasicoherent sheaf $A$ of algebras over a scheme $X$. That means it is essentially $\mathcal{O}_X$-modules. The dual of $A$ is a cosheaf. Will this dual cosheaf be also quasicoherent? And will it be a quasicoherent cosheaf of coalgebras? I mean, will it be $\mathcal{O}_X$-comodules. Please suggest some good literature on this if you know some! http://mathoverflow.net/questions/41711/quasi-coherent-sheaf Comment by Neha Neha 2010-10-12T10:56:49Z 2010-10-12T10:56:49Z Thanks so much. So what I understand finally is that the quasi coherent sheaf F over an affine scheme, Spec R is simply an R-module. But if F is a quasi coherent sheaf of algebras, then F would be an R-algebra, say M ?!? Do I think of it as a constant sheaf with all its sections as M ?!? http://mathoverflow.net/questions/41711/quasi-coherent-sheaf/41762#41762 Comment by Neha Neha 2010-10-12T10:56:30Z 2010-10-12T10:56:30Z Thanks so much. So what I understand finally is that the quasi coherent sheaf $F$ over an affine scheme, Spec $R$ is simply an R-module. But if F is a quasi coherent sheaf of algebras, then $F$ would be an $R$-algebra, say $M$ ?!? Do I think of it as a constant sheaf with all its sections as $M$ ?!? http://mathoverflow.net/questions/41711/quasi-coherent-sheaf/41762#41762 Comment by Neha Neha 2010-10-11T10:58:42Z 2010-10-11T10:58:42Z Thanks Scott. So I understand that the quasi coherent sheaf F need not be an algebra, even if we start with a Hopf Algebra H. But when would F be an algebra, What extra conditions would be required for defining multiplication? http://mathoverflow.net/questions/41713/external-tensor-product-of-quasi-coherent-sheaf Comment by Neha Neha 2010-10-11T10:56:27Z 2010-10-11T10:56:27Z Yes, I need a non-zero map. (isomorphism even better!) I know there is always a map in other direction. But when would the inverse(non-zero) map exist? I need to know the conditions that one require for this inverse to exist. Will generation of $\mathcal{A}$ by global sections be sufficient condition for this map to exist? http://mathoverflow.net/questions/41711/quasi-coherent-sheaf Comment by Neha Neha 2010-10-11T10:50:26Z 2010-10-11T10:50:26Z Thanks to all : Martin Brandenburg, Donu Arapura, Lennart Meier,Scott Carnahan. So I understand that the quasi coherent sheaf $F$ need not be an algebra, even if we start with a Hopf Algebra H. But when would $F$ be an algebra, What extra conditions would be required for defining multiplication? http://mathoverflow.net/questions/41254/external-tensor-product-of-sheaves/41423#41423 Comment by Neha Neha 2010-10-08T15:48:19Z 2010-10-08T15:48:19Z Then in such a case, when $A$ is generated by global sections, will the following diagram commute(why/why not)? $\xymatrix { A\boxtimes A(U\times U) \ar@{.&gt;}[rrd]^{\exists !}\\ A(G) \boxtimes A(G) \ar@{^{(}-&gt;}[u] \ar@{^{(}-&gt;}[rr]^-{} &amp;&amp; A(U) \otimes A(U). }$ for any open set $U\in G$, Please suggest. Thanks.