User isac hed&#233;n - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T23:20:06Z http://mathoverflow.net/feeds/user/17085 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/124897/is-every-graded-subalgebra-of-gra-equal-to-grb-for-some-subalgebra-b-of-a Is every graded subalgebra of gr(A) equal to gr(B) for some subalgebra B of A? Isac Hedén 2013-03-18T22:01:05Z 2013-03-25T12:39:03Z <p>Let $A$ be a finitely generated reduced (i.e. affine) $\mathbb C$-algebra, and $D:A\longrightarrow A$ a locally nilpotent derivation (i.e. $D$ is $\mathbb C$-linear, satisfies Leibniz' rule and $\forall f\in A\,\exists n\in\mathbb N: D^n(f)=0$). Then there is a filtration on $A$ given by $A_i:=\ker D^{i+1}$, and an associated graded algebra $$gr_D(A):=\bigoplus_{i=0}^\infty A_i/A_{i-1}.$$ (it is understood that $A_{-1}=\{0\}$). Given a $D$-invariant subalgebra $B\subset A$, we get an induced filtration (by taking $B_i=B\cap A_i$), and a subalgebra $gr_D(B)\subset gr_D(A)$. I can show that $B$ is uniquely determined by $gr_D(B)$, i.e. if $B_1$ and $B_2$ are two $D$-invariant subalgebars of $A$, then $gr(B_1)=gr(B_2)$ implies $B_1=B_2$. My question is: which graded subalgebras $C\subset gr_D(A)$ are of the form $gr_D(B)$ for some $D$-invariant subalgebra $B\subset A$, and also, how should $B$ be chosen?</p> <p><strong>Motivation:</strong> I am trying to understand affine extensions of $\mathbb G_a$-principal bundles over $\mathbb A_* ^2$ as in <a href="http://mathoverflow.net/questions/119702/normality-condition-on-graded-algebra" rel="nofollow">http://mathoverflow.net/questions/119702/normality-condition-on-graded-algebra</a>. In the question above, $spec(A)\rightarrow \mathbb A_* ^2$ is a non-trivial $\mathbb G_a$-principal bundle -- it is known that all non-trivial $\mathbb G_a$-bundles on $\mathbb A_* ^2$ are affine. Affine extensions of this principal bundle are going to correspond to subalgebras of $A$ (containing $\mathcal O(\mathbb A^2_*)=\mathbb C[x,y]$). Now the structural $\mathbb G_a$-action on $spec(A)$ corresponds to a locally nilpotent derivation on $A$, and my hope is that it will be fruitful in one way or another to consider the graded algebra corresponding to this derivation (locally nilpotent derivations on $A$ correspond to $\mathbb G_a$-actions on $spec(A)$ in a natural way).</p> <p><strong>PS:</strong> Thanks for the downvote (which helped me realize that the formulation of the question was not so great at first)! I hope it is better now.</p> <p><strong>Edit:</strong> Added "graded" in the title, as suggested.</p> <p><strong>Edit2:</strong> Paul is right in his comment below! And I think we have the following description of $gr_D(B)$ as a $B_0$-module: The sequence of ideals $\mathfrak b_n:=D^n(B_n)\hookrightarrow B_0$ is decreasing and satisifies $\mathfrak b_n\mathfrak b_m\subset \mathfrak b_{n+m}$, and furthermore $$gr_D(B)\cong B_0\oplus\bigoplus_{n=1}^\infty \mathfrak b_ns^n\hookrightarrow B_0[s].$$ The isomorphism is induced by $$gr_D(B)_n\rightarrow \mathfrak b_ns^n,\quad b+B_{n-1}\mapsto \frac {D^n(b)}{n!}s^n.$$ The question remains: Is every graded submodule of $gr_D(A)$ of this form for some subalgebra $B\subset A$?</p> <p><strong>Example:</strong> One algebra for which I am particularily interested in the answer to this question would be $A=\mathbb C[x,y,u,v]/(xv-yu-1)$, i.e. $\mathcal O(SL_2)$, with locally nilpotent derivation given by $D(x)=D(y)=0$ and $D(u)=x$, $D(v)=y$. Then $$gr_D(A)\cong \mathbb C[x,y]\oplus\bigoplus_{n=1}^\infty (x,y)^n s^n$$</p> <p><strong>Edit3:</strong> Finally I am not so sure that $B$ is uniquely determined by $gr_D(B)$ -- the argument I had in mind seems not to work.</p> http://mathoverflow.net/questions/119702/normality-condition-on-graded-algebra Normality condition on graded algebra Isac Hedén 2013-01-23T22:28:46Z 2013-03-08T00:22:00Z <p>Let $\mathbb G_a$ denote the additive group of complex numbers.</p> <p><strong>Definition:</strong> Let $V \subset Y$ be a dense open subset of the affine variety $Y$ and $\pi : P \longrightarrow V$ a $\mathbb G_a$-principal bundle. An <em>affine extension</em> is an affine $\mathbb G_a$-variety $\hat P$ together with a morphism $\hat \pi: \hat P \longrightarrow Y$ and an equivariant open embedding $\iota: P \hookrightarrow \hat P$, such that the diagram</p> <p>$$\begin{array}{ccc} P &amp; \hookrightarrow &amp; \hat P \\ \downarrow &amp; &amp; \downarrow \\ V &amp; \hookrightarrow &amp; Y \end{array}$$ is commutative and $\iota (P)= \hat \pi^{-1}(V).$</p> <p>I am interested in affine extensions of the trivial $\mathbb G_a$-bundle over the affine plane punctured at the origin, i.e. $\mathbb A^2_*:=Sp(\mathbb C[x,y])\setminus{\mathbf o}$ with $\mathbf o:=(x,y)$, and I have the following description:</p> <p><strong>Proposition:</strong> If $P \longrightarrow \mathbb A_*^2$ is the trivial bundle, then any affine extension $\hat P \longrightarrow \mathbb A^2$ is of the form $$\hat P = Sp (A),$$ where $$A= \bigoplus_{\nu=0}^\infty \mathfrak m_\nu t^\nu \subset \mathbb C [x,y,t],$$ with a decreasing sequence $(\mathfrak m_\nu)$, $\nu \in \mathbb N$, of ideals $\mathfrak m_\nu \subset \mathbb C [x,y]$, such that</p> <p>1. $\mathfrak m_\nu \cdot \mathfrak m_\lambda \subset \mathfrak m_{\nu + \lambda}$ for all $\nu, \lambda\in \mathbb N$,</p> <p>2. $\mathfrak m_0= \mathbb C [x,y]$, and </p> <p>3. $V(\mathfrak m_\nu) \subset \mathbf o$ for $\nu > 0$.</p> <p>On the other hand, every finitely generated $\mathbb C$-algebra of that form defines an affine extension of the trivial bundle.</p> <p><strong>Question:</strong> Does somebody know of a criterion on the sequence $(\mathfrak m_\nu)$$_{\nu \in \mathbb N}$ so that $A$ becomes normal?</p> <p><strong>Examples:</strong> a) If $\mathfrak m_\nu= \mathbb C [x,y]$ for all $\nu$ we have $\hat P \cong \mathbb A^2 \times \mathbb G_a$.</p> <p>b) If $\mathfrak m_\nu =(x^m,y^n)^\nu$, we have $A = \mathbb C [x,y,x^mt, y^nt].$</p> <p>In the second example, one can see that $A$ is normal for instance if $\mathfrak m_\nu=(x^2,y)^\nu$, but not if $\mathfrak m_\nu=(x^2,y^2)^\nu$ -- in the latter case I think the normalization would be defined by the sequence $\mathfrak m_\nu=(x^2,xy,y^2)^\nu$. Since $A = \mathbb C [x,y,x^mt, y^nt]\cong\mathbb C[x,y,u,v]/(x^mv-y^nu)$ in example b), $Sp(A)$ is a hypersurface in $\mathbb C^4$, so normality is equivalent to singularities being of codimension at least two.</p> <p><strong>Edit:</strong> If the question is difficult in general, I am also interested in the following special case: For which monomial ideals $\mathfrak m\subset\mathbb C[x,y]$, is the ring defined by the sequence $\mathfrak m^\nu$ normal? In this situation I would expect something like: $A$ is normal iff the support of $\mathfrak m^\nu$ consists of all lattice points in the convex hull of the support of $\mathfrak m^\nu$ in $\mathbb R^2$. Here the support of $\mathfrak m^\nu$ is the set of pairs $(k,l)\in\mathbb N^2$ such that $x^ky^l\in\mathfrak m^\nu$.</p> http://mathoverflow.net/questions/68421/simplest-examples-of-nonisomorphic-complex-algebraic-varieties-with-isomorphic-an/72831#72831 Answer by Isac Hedén for Simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytifications Isac Hedén 2011-08-13T12:51:02Z 2011-08-14T05:08:37Z <p>I've been asking around a little, and nobody seems to be able to tell whether or not Russell's hypersurface is analytically $\mathbb C^3$. Adrien Dubouloz shows in <a href="http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.4085v1.pdf" rel="nofollow">this</a> article that the Makar-Limanov invariant of its product with $\mathbb C$ is trivial. Maybe that helps.</p> http://mathoverflow.net/questions/124897/is-every-graded-subalgebra-of-gra-equal-to-grb-for-some-subalgebra-b-of-a Comment by Isac Hedén Isac Hedén 2013-03-21T16:19:18Z 2013-03-21T16:19:18Z Thanks Paul! You are right - that is a necessary condition! See Edit2 for what I think is &quot;the precise&quot; statement. http://mathoverflow.net/questions/119702/normality-condition-on-graded-algebra/119784#119784 Comment by Isac Hedén Isac Hedén 2013-01-26T18:08:48Z 2013-01-26T18:08:48Z Thanks for the answer! It looks like a good criterion for the case where $A$ is a Rees algebra. Unfortunately this is not always the case though, since we for instance could modify A by repeating every ideal once, i.e. take the ideal sequence $m_1,m_1,m_2, m_2,m_3,m_3,\ldots$ instead of $m_1,m_2,m_3,\ldots$ where all the $m_i$ are powers of a monomial ideal $m_1$ -- without violating the three conditions that the ideal sequence should satisfy. Could there be a way of associating a Rees algebra to $A$ which has the property of being normal exactly when $A$ is, e.g. by means of a Rees valuation? http://mathoverflow.net/questions/119702/normality-condition-on-graded-algebra/119707#119707 Comment by Isac Hedén Isac Hedén 2013-01-24T14:09:08Z 2013-01-24T14:09:08Z ... Did you have a particular article by Kei-ichi Watanabe i mind? Thanks again for your answer! http://mathoverflow.net/questions/119702/normality-condition-on-graded-algebra/119707#119707 Comment by Isac Hedén Isac Hedén 2013-01-24T14:07:28Z 2013-01-24T14:07:28Z Thanks for the answer! The condition seems to be very strong, as formulated, and I don't really see how it is related to normality. Did you mean the condition should hold for all elements with $\nu(f)\geq 1$? Otherwise, if for instance we take $\mathfrak m$ to be a monomial ideal which contains a power of $x$, but not $x$ itself, and take the ideal sequence to be the powers of $\mathfrak m$, the condition fails (with $f=x$, since $\nu(f)=0$) although $A$ may very well be normal (as in example b)). What I'd really like is of course a condition which is both necessary and sufficient...