User michele torielli - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:37:39Z http://mathoverflow.net/feeds/user/4821 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91520/determinantal-rings-are-cohen-macaulay Determinantal rings are Cohen-Macaulay Michele Torielli 2012-03-18T10:40:45Z 2012-03-25T23:37:45Z <p>Consider a $n\times n$ matrix $M$ with entries in $R=\mathbb{C}[x_1,\dots,x_n]$. Let $I$ be the ideal of $(n-1)\times(n-1)$ minors of $M$. Is $\mathcal{O}_{\mathbb{C}^n}/I$ Cohen-Macaulay?If not, what additional assumptions we need for an affirmative answer?</p> http://mathoverflow.net/questions/24110/locally-free-modules Locally free modules Michele Torielli 2010-05-10T14:43:52Z 2011-08-24T21:25:51Z <p>Consider M a locally free $\mathcal{O}_{\mathbb{C}^{n}}$-module. does exist a theory of deformation for that type of object? I would like to know, which conditions has to satisfy the total space of a one (but also higher) parameter deformation of M in such way that each fiber is locally free.</p> http://mathoverflow.net/questions/62264/relative-invariants-of-prehomogeneous-vector-space Relative invariants of prehomogeneous vector space Michele Torielli 2011-04-19T13:10:22Z 2011-08-16T19:22:12Z <p>Let $(G,\rho,V)$ be a prehomogeneous vector spaces with $f_1,\dots,f_N$ the basic irreducible relative invariants. Suppose that $(G',\rho',V')$ is a second prehomogeneous that is in the same castling class of the previous one. Is there a way to compute the basic irreducible relative invariants of the second one starting from $f_1,\dots,f_N$? and if they are just castling transforms of each other?</p> http://mathoverflow.net/questions/69935/phi-hom-ra-b-to-hom-ra-r-otimes-r-b $\Phi: Hom_R(A,B) \to Hom_R(A,R)\otimes_R B$ Michele Torielli 2011-07-10T13:56:38Z 2011-07-11T19:02:05Z <p>Let $R$ be a commutative ring and $A$ and $B$ two $R$-module. Suppose that $A$ is free of rank $n$ with basis $a_1,\dots,a_n$. Then there is an isomorphism $\Phi: Hom_R(A,B) \to Hom_R(A,R)\otimes_R B$ defined by $\Phi(\sigma)=\sum_{i=1}^n \psi_i\otimes \sigma_i$, where $\sigma_i=\sigma(a_i)$ and $\psi_i$ is the map such that $\psi_i(a_j)=\delta_{ij}$. Is there another way to describe $\Phi$?</p> http://mathoverflow.net/questions/61673/prehomogeneous-vector-space-gl-12-times-sl-2 prehomogeneous vector space $GL_1^2 \times SL_2$ Michele Torielli 2011-04-14T09:25:22Z 2011-04-14T09:25:22Z <p>How can I prove that the following 2 prehomogeneous vector spaces are not isomorphic? 1) $(GL_1^2 \times SL_2,2\Lambda_1\oplus\Lambda_1, \mathbb{C}^3\oplus\mathbb{C}^2)$ 2) $(GL_1^2 \times SL_2,2\Lambda_1\oplus\Lambda_1^*, \mathbb{C}^3\oplus\mathbb{C}^2)$? can I use the same type of proof as in: <a href="http://mathoverflow.net/questions/61377/prehomogeneous-vector-spaces/61450#61450" rel="nofollow">http://mathoverflow.net/questions/61377/prehomogeneous-vector-spaces/61450#61450</a> does anyone has reference about that?</p> http://mathoverflow.net/questions/61377/prehomogeneous-vector-spaces Prehomogeneous vector spaces Michele Torielli 2011-04-12T10:14:42Z 2011-04-12T20:57:02Z <p>How can I prove that the following 2 prehomogeneous vector spaces are not isomorphic? 1)$(GL_n,\Lambda_1\oplus \Lambda_1,\mathbb{C}^n \oplus \mathbb{C}^n)$ 2)$(GL_n,\Lambda_1\oplus \Lambda_1^*,\mathbb{C}^n \oplus \mathbb{C}^n)$ where $\Lambda_1$ is the standard representation of $GL_n$ on $\mathbb{C}^n$. in the case of prehomogeneous vector spaces the notion of isomorphism is given by:</p> <p>Two triplets $(G, \rho, V)$ and $(G', \rho', V')$ are isomorphic if there exist a rational isomorphism $\sigma : \rho(G) \to \rho'(G')$ and an isomorphism $\tau : V \to V'$, both defined over $\mathbb{C}$, such that $$\tau(\rho(g)x)=\sigma\rho(g)(\tau(x))$$ for all $g\in G$ and $x\in V$. That is the following diagram is commutative for all $g\in G$: $$$\xymatrix{V \ar[d]_{\rho(g)} \ar[r]^\tau &amp;V' \ar[d]^{\sigma \rho(g)} \ V \ar[r]^\tau &amp;V'}$$$</p> http://mathoverflow.net/questions/58063/semisimple-elements-of-a-lie-algebra Semisimple elements of a lie algebra Michele Torielli 2011-03-10T12:07:24Z 2011-03-10T13:18:59Z <p>Let $G\subset GL_n(\mathbb{C})$ be an algebraic group of dimension n, and let $\mathfrak{g}$ its Lie algebra.Is there a relations between the maximal number of independent semisimple elements of $G$ and the maximal number of independent semisimple elements $\mathfrak{g}$?</p> http://mathoverflow.net/questions/55846/differential-of-a-nilpotent-or-semisimple-element Differential of a nilpotent or semisimple element Michele Torielli 2011-02-18T12:33:04Z 2011-02-23T10:39:29Z <p>Let $G$ be an algebraic connected subgroup of $GL_n(\mathbb{C})$ and let $\chi : G \to \mathbb{C}^*$ a character. Consider $d_e\chi : \mathfrak{g} \to \mathbb{C}$ the differential of $\chi$ at the identity of $G$. Is it true that $d_e\chi(\nu)=0$ if $\nu$ is a nilpotent element?If not, is it true under some assumptions? Can we say something if $\nu$ is semisimple?</p> http://mathoverflow.net/questions/23254/reductive-lie-algebra Reductive Lie algebra Michele Torielli 2010-05-02T12:27:52Z 2011-01-20T20:55:52Z <p>Does it exist a Lie algebra $\mathfrak{g}$ that is reductive but if we consider the inclusion of Lie agebras $\mathfrak{g} \subset \mathfrak{h}$ then $\mathfrak{g}$ is not reductive in $\mathfrak{h}?$</p> http://mathoverflow.net/questions/43665/compute-lie-algebra-cohomology Compute Lie algebra cohomology Michele Torielli 2010-10-26T13:51:13Z 2011-01-05T15:26:02Z <p>Is there a computer algebra system that is able to compute the Lie algebra cohomology in a given representation?what if the Lie algebra is finite dimensional? In my case I would like to be able to compute the the cohomology in the following situation: let $\mathfrak{g}\subset \mathfrak{h}$ be an inclusion of finite dimensional complex Lie albebras, I'd like to compute the cohomology of $Hom(\mathfrak{g}, \mathfrak{h}/\mathfrak{g})$. </p> http://mathoverflow.net/questions/47428/cm-for-radical-ideal CM for radical ideal Michele Torielli 2010-11-26T14:17:53Z 2010-11-27T09:47:38Z <p>Let R the polynomial ring in n variables with complex coefficients and I an ideal of R. Is it true that if R/I is CM also R/J is CM (where J is the radical of I)? Is there a relations between a resolution of R/J and one of R/I? What if I suppose that proj.dim(R/I)=2?</p> http://mathoverflow.net/questions/30872/formally-versal-deformation Formally versal deformation Michele Torielli 2010-07-07T11:03:12Z 2010-07-07T11:03:12Z <p>Is it true that for any deformation theory, the deformation constructed from a basis of $T^1$ is formally versal (let's assume that $T^1$ is finite dimensional)? Moreover, does anyone has a reference about the subject?</p> http://mathoverflow.net/questions/27818/formal-deformation-theory Formal deformation theory Michele Torielli 2010-06-11T14:08:41Z 2010-06-24T08:22:14Z <p>I found the following argument in more than one article: Let $X_0$ be a complex space and define the functor $F:Art \to Sets$ s.t. $F(A)={\text{Isomorphism classes of deformations of X_0 over A}}$. Let $(X",A"),(X',A'), (X,A)$ deformations of $X$ and consider 2 maps $A"\to A$ and $A' \to A$ such that the first one is surjective, so we can consider $A' \times_{A} A"$. Then a deformation over $A' \times_{A} A"$ is <code>$Y=(X_0,\mathcal{O}_{X'} \times_{\mathcal{O}_X} \mathcal{O}_{X"})$</code>. Can someone explain to me what is Y? Suppose that $X=V(f)$ for some <code>$f\in \mathbb{C}[x_1, \dots, x_n]$</code>, how I compute Y? </p> http://mathoverflow.net/questions/27453/integrability-of-vector-fields Integrability of vector fields Michele Torielli 2010-06-08T11:39:52Z 2010-06-08T11:39:52Z <p>Where can I find some reference about integrability of vector fields and Frobenius's theorem?</p> http://mathoverflow.net/questions/25468/proof-of-saito-criterion Proof of Saito criterion Michele Torielli 2010-05-21T10:24:17Z 2010-05-21T10:24:17Z <p>Does it exist another proof of saito's criterion for free divisors, other than the one in "K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, pp. 265-291."?if yes, where?</p> http://mathoverflow.net/questions/23706/reductive-lie-algebra-of-a-lie-group Reductive Lie algebra of a Lie group Michele Torielli 2010-05-06T13:14:05Z 2010-05-13T18:47:51Z <p>In the answer of my question:</p> <p><a href="http://mathoverflow.net/questions/23085/on-the-full-reducibility-of-representations-of-reductive-lie-algebras" rel="nofollow">http://mathoverflow.net/questions/23085/on-the-full-reducibility-of-representations-of-reductive-lie-algebras</a></p> <p>James E. Humphreys replied to me saying that:"the notion of "reductive" for a Lie algebra in characteristic 0 has no intrinsic interest, unless you study the Lie algebra of a Lie (or algebraic) group and relate their representations carefully."</p> <p>Can please someone explain that to me or give to me any reference? thank you!</p> http://mathoverflow.net/questions/24345/representations-of-reductive-lie-group Representations of reductive Lie group Michele Torielli 2010-05-12T08:03:08Z 2010-05-12T10:49:30Z <p>Let $G$ be a reductive algebraic group and $\varrho$ a representation of $G$ in $GL(n)$. Is it true that $\varrho$ is completely reducible? Moreover, how are related the representations of the Lie algebra $\mathfrak{g}$ of $G$ with the one of $G$?Finally, the centre of the identity component of $G$ consists of semisimple transformations, is it true also for $\mathfrak{g}$?</p> http://mathoverflow.net/questions/23414/centre-of-a-lie-algebra Centre of a Lie algebra Michele Torielli 2010-05-04T11:22:18Z 2010-05-05T18:51:14Z <p>Let $f\in \mathbb{C}[x_1,\dots, x_n]$ be a reduced homogeneous polynomial of degree n.</p> <p>Let <code>$\mathfrak{g}=\{ \delta \in Der_{\mathbb{C}^n}|\delta(f)\in (f)\mathcal{O}_{\mathbb{C}^n} \text{ and weight}(\delta) =0 \}$</code> be a (reductive) complex Lie algebra with minimal system of generators $\langle \sigma_1, \dots, \sigma_s, \delta_1, \dots, \delta_r\rangle$ such that:</p> <ol> <li>$\sigma_1, \dots, \sigma_s$ are simultaneously diagonalizable,</li> <li>$\delta_1, \dots, \delta_r$ are nilpotent,</li> <li>$[\sigma_i,\delta_j]\in \mathbb{Q} \cdot \delta_j$ for all i,j.</li> </ol> <p>Is it true that the centre of $\mathfrak{g}$ is made only of diagonalizable elements?</p> http://mathoverflow.net/questions/23312/textbooks-on-singular-and-macaulay-2/23328#23328 Answer by Michele Torielli for Textbooks on SINGULAR and Macaulay 2 Michele Torielli 2010-05-03T08:01:36Z 2010-05-03T08:10:09Z <p>Also in the book: "Introduction to singularities and deformations" by Gruel, Lossen and Shustin, you can find a lot of material on Singular.</p> http://mathoverflow.net/questions/23168/deformations-of-free-modules Deformations of free modules Michele Torielli 2010-05-01T10:27:06Z 2010-05-01T11:08:07Z <p>Where can I found a description of the deformation theory for modules?Is it possible to deform a free module in such way that each fibre of the deformation is still free?</p> http://mathoverflow.net/questions/23085/on-the-full-reducibility-of-representations-of-reductive-lie-algebras On the full reducibility of representations of reductive Lie algebras Michele Torielli 2010-04-30T08:18:47Z 2010-05-01T00:58:20Z <p>I would like to find a reference for the following fact: every finite dimensional complex representation of a reductive Lie algebra is semisimple.</p> http://mathoverflow.net/questions/21960/reductive-lie-subalgebra reductive Lie subalgebra Michele Torielli 2010-04-20T14:14:26Z 2010-04-20T16:09:09Z <p>Suppose to have a Lie algebra L with a reductive lie subalgebra G. Let l an element of L such that [l,g] is in G for every g in G, is it true that l is an element of G?if not, there are some restriction on G that makes it true?</p> http://mathoverflow.net/questions/19104/deformations-for-complex-space-germs Deformations for complex space germs Michele Torielli 2010-03-23T10:59:54Z 2010-03-23T15:08:22Z <p>Is there a space such that it doesn't have any deformation but its space of first order infinitesimal deformations is non trivial?</p> http://mathoverflow.net/questions/91520/determinantal-rings-are-cohen-macaulay Comment by Michele Torielli Michele Torielli 2012-03-19T15:23:30Z 2012-03-19T15:23:30Z the case that I'm more interested in is when I is of codimension 2. http://mathoverflow.net/questions/91520/determinantal-rings-are-cohen-macaulay/91527#91527 Comment by Michele Torielli Michele Torielli 2012-03-18T13:29:56Z 2012-03-18T13:29:56Z What is the condition on the height of $I$? http://mathoverflow.net/questions/69935/phi-hom-ra-b-to-hom-ra-r-otimes-r-b/69936#69936 Comment by Michele Torielli Michele Torielli 2011-07-10T14:34:28Z 2011-07-10T14:34:28Z I agree too. That what I was thinking but I was hoping to be wrong.Thank you. http://mathoverflow.net/questions/69935/phi-hom-ra-b-to-hom-ra-r-otimes-r-b/69936#69936 Comment by Michele Torielli Michele Torielli 2011-07-10T14:20:54Z 2011-07-10T14:20:54Z When I said another way, what I had in mind was something like the inverse map, i.e. not involving explicitely $a_1,\dots,a_n$. In fact $\Phi^{-1}(\nu\otimes b)=(a\mapsto \nu(a)b)$. Instead, if I'm not wrong, identifying $Hom(A,X)$ with $X^n$ require the basis of $A$. http://mathoverflow.net/questions/61377/prehomogeneous-vector-spaces Comment by Michele Torielli Michele Torielli 2011-04-13T10:09:52Z 2011-04-13T10:09:52Z it was RHS. sorry again http://mathoverflow.net/questions/61377/prehomogeneous-vector-spaces/61450#61450 Comment by Michele Torielli Michele Torielli 2011-04-13T08:05:00Z 2011-04-13T08:05:00Z thanks. can I ask you what is the definition of isotypic? http://mathoverflow.net/questions/61377/prehomogeneous-vector-spaces/61450#61450 Comment by Michele Torielli Michele Torielli 2011-04-12T21:01:38Z 2011-04-12T21:01:38Z Thank you. just one question, when you say that $\Lambda_1$ and $\Lambda^*_1$ are not isomorphic, do you mean as representation?because as triplets $(GL_n,\Lambda_1,\mathbb{C}^n)$ and $(GL_n,\Lambda^*_1,\mathbb{C}^n)$ are isomorphic. http://mathoverflow.net/questions/61377/prehomogeneous-vector-spaces Comment by Michele Torielli Michele Torielli 2011-04-12T20:58:06Z 2011-04-12T20:58:06Z sorry I forgot a $g$ in the LHS. http://mathoverflow.net/questions/58063/semisimple-elements-of-a-lie-algebra Comment by Michele Torielli Michele Torielli 2011-03-10T14:21:49Z 2011-03-10T14:21:49Z independent = linearly independent. sorry. http://mathoverflow.net/questions/55846/differential-of-a-nilpotent-or-semisimple-element/55857#55857 Comment by Michele Torielli Michele Torielli 2011-02-18T14:40:20Z 2011-02-18T14:40:20Z where can I found material about that? http://mathoverflow.net/questions/55846/differential-of-a-nilpotent-or-semisimple-element/55857#55857 Comment by Michele Torielli Michele Torielli 2011-02-18T14:38:55Z 2011-02-18T14:38:55Z $\chi$ is a character attached to a semi-invariant of the action of $G$ on $\mathbb{C}^n$. thank you. http://mathoverflow.net/questions/47428/cm-for-radical-ideal Comment by Michele Torielli Michele Torielli 2010-11-27T18:16:20Z 2010-11-27T18:16:20Z sorry, you are right. In my case I is the jacobian ideal of a polynomial f such that R/I is CM of codim 2 and hence proj.dim 2 http://mathoverflow.net/questions/47428/cm-for-radical-ideal Comment by Michele Torielli Michele Torielli 2010-11-27T17:19:50Z 2010-11-27T17:19:50Z because I'm looking at a case in which the projective dimension is 2. Sorry, I'm not confident with Hartshorne's example....but I work in char p=0. http://mathoverflow.net/questions/47428/cm-for-radical-ideal/47434#47434 Comment by Michele Torielli Michele Torielli 2010-11-26T16:07:18Z 2010-11-26T16:07:18Z what if R is local?like power series? http://mathoverflow.net/questions/47428/cm-for-radical-ideal/47434#47434 Comment by Michele Torielli Michele Torielli 2010-11-26T15:20:12Z 2010-11-26T15:20:12Z ok, is it true adding some restriction on I?