Sasha
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Registered User
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May 18 |
answered | Terminology: complex of sheaves with cohomology sheaves concentrated in degree zero |
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May 17 |
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Bondal counter example to the Jordan-Holder property in derived categories The reason for nonextendability is the fact that the Euler form $\chi(-,-)$ on the orthogonal subcategory $P^\perp$ is skew-symmetric, and so $P^\perp$ does not have exceptional objects. |
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May 15 |
answered | Bondal counter example to the Jordan-Holder property in derived categories |
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Apr 30 |
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Derived equivalence of two varieties which are isomorphism over certain open subvarieties If two line bundles on a smooth variety agree on an open subset with complement of codimension at least two, then they are isomorphic. This is the Hartogs theorem. |
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Apr 30 |
accepted | Derived equivalence of two varieties which are isomorphism over certain open subvarieties |
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Apr 30 |
answered | Are all Equivariant Bundles of a Total Flag Manifold Constructable from Line Bundles? |
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Apr 30 |
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Derived equivalence of two varieties which are isomorphism over certain open subvarieties arxiv.org/pdf/math.AG/0205287, Conjecture 1.2. |
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Apr 30 |
accepted | Are these two definitions of $\mathcal{O}(1)$ over a ruled surface closely related? |
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Apr 30 |
answered | Derived equivalence of two varieties which are isomorphism over certain open subvarieties |
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Apr 29 |
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Defining Equations of a Flag Variety Equations of the flag variety are useless (almost always). The important thing is the universal property --- given a scheme with a vector bundle and a (complete) flag of subbundles in it there is a unique map to the flag variety such that the flag is the pullback of the tautological flag. |
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Apr 28 |
accepted | Commutativity of nilpotents in minuscule case |
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Apr 28 |
answered | Commutativity of nilpotents in minuscule case |
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Apr 21 |
accepted | Flatness and tensor product of rings |
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Apr 21 |
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Flatness and tensor product of rings Use free resolution $$ 0 \to {\mathbb Z}[x_1,x_2] \xrightarrow{x_1-x_2} {\mathbb Z}[x_1,x_2] \to {\mathbb Z}[x] \to 0 $$ to compute $Tor_1^{{\mathbb Z}[x_1,x_2]}({\mathbb Z}[x],{\mathbb Z}[x]) = {\mathbb Z}[x] \ne 0$. |
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Apr 21 |
answered | Flatness and tensor product of rings |
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Apr 19 |
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Relationship of additive and triangulated structures in the triangulated cateogry Assuming the category is $k$-linear you can do a more general thing by taking the cone of $\lambda f + \mu g$ for $\lambda,\mu \in k$. It gives a family of objects specializing to $C_f$, $C_g$, and $C_{f+g}$. So, one can say that all these objects are in the same deformation family. |
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Apr 19 |
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A hypersurface in the Grassmannian of endomorphisms. Do you know such an interpretation for $q = 1$? |
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Apr 15 |
answered | Are these two definitions of $\mathcal{O}(1)$ over a ruled surface closely related? |
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Apr 13 |
answered | Locally free resolution of sheaves on finite group quotient |
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Apr 13 |
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on rational singularities Of course not. For example, let $X$ be a quadratic cone, $X'$ its blowup, and $Y$ the vertex of the cone. |
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Apr 13 |
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Resolutions of higher-dimensional Kummer varieties You can rewrite the construction as first blowing up all torsion point and then taking the quotient. In this case the fixed point locus of the involution is the union of exceptional divisors, so the quotient is smooth. Alternatively, you can look at the local model --- the blowup of ${\mathbb C}^n/\pm1$ and check it smoothness. |
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Apr 12 |
accepted | Resolutions of higher-dimensional Kummer varieties |
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Apr 12 |
answered | Resolutions of higher-dimensional Kummer varieties |
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Apr 12 |
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question on spectral sequence Derived pushforward is not compatible with tensor products. So, I would not expect such a relation. |
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Apr 12 |
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question on spectral sequence $H^0(Y,R^1p_*F)$ is not a summand of $H^1(F)$. The spectral seuence only gives an exact sequence $$ 0 \to H^1(Y,R^0p_*F) \to H^1(F) \to H^0(Y,R^1p_*F) \to H^2(Y,R^0p_*F). $$ and the last map is quite often nontrivial. |
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Apr 11 |
revised |
Determinant of coherent sheaves added 2 characters in body |
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Apr 9 |
answered | cohomology of exterior powers of tangent bundle |
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Apr 9 |
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Additive functors and Derived Categories It depends on what do you want from the derived functor. If you want it to be a natural extension to the derived category of the initial functor, you have to be careful. As far as I understand in case of a nonexact functor the answer is that you have to modify the derived category appropriately. I think you can find an explanation in Positselki papers. |
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Apr 9 |
answered | Additive functors and Derived Categories |
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Apr 9 |
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Exterior and symmetric powers of external tensor products of representations I learned this from Kapranov's papers, and if I remember right he referred to the book of Barut and Ronchka. But I guess you can find this in many places. For example I would try Weyman's book (Cohomology of vector bundles and syzygies). In finite characteristic I don't know the answer (why don't you ask this as a question?), but definitely you have to be more careful. |
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Apr 2 |
answered | How to define intersection of coherent sheaf and 1-cycle? |
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Apr 2 |
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Name for a class of parabolic subgroups I would suggest to call this $G/P$ the adjoint Grassmannian. Just because $G/P$ for maximal $P$ is called a $G$-Grassmannian and when $P$ corresponds to the highest root one has $G/P \subset P({\mathfrak{g}})$. |
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Apr 2 |
answered | Exterior and symmetric powers of external tensor products of representations |
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Mar 30 |
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Almost-Lie Algebras? Is there a compatible notion of a Hom-Lie group? |
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Mar 29 |
accepted | is the orthogonal complement of a saturated sequence saturated? |
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Mar 29 |
answered | is the orthogonal complement of a saturated sequence saturated? |
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Mar 29 |
accepted | How much of a variety can be reconstructed from codimension-zero data? |
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Mar 28 |
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A statement for a subset generated a triangulated category Ben, my counterexample seems to contradict to what you claimed. |
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Mar 28 |
answered | A statement for a subset generated a triangulated category |
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Mar 27 |
awarded | ● Nice Answer |
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Mar 27 |
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How much of a variety can be reconstructed from codimension-zero data? Dmitry, you are completely right. I edited the answer, now you see that in some cases you can reconstruct a lot of geometry! |
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Mar 27 |
revised |
How much of a variety can be reconstructed from codimension-zero data? added 913 characters in body |
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Mar 25 |
answered | How much of a variety can be reconstructed from codimension-zero data? |
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Mar 24 |
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How much of a variety can be reconstructed from codimension-zero data? How do you define $Hom$'s in your category? Is it a full subcategory of $Coh(X)$? |
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Mar 21 |
answered | Stability of $T_{\mathbb{P}^2}$ and $\Omega_{\mathbb{P}^2}$? |
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Mar 16 |
accepted | Curvilinear subschemes and Ext functors. |
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Mar 15 |
answered | Curvilinear subschemes and Ext functors. |
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Mar 15 |
accepted | Is this an embedding of $S^{[2]}$? |
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Mar 15 |
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Is this an embedding of $S^{[2]}$? @Barbara: You are completely right, sorry. I added an explanation to the answer. |
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Mar 15 |
revised |
Is this an embedding of $S^{[2]}$? added 838 characters in body |
