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Mar
21
comment What are some standard operations on the set of localizations of a triangulated category?
Thanks Dragos. I'm not sure that $\langle D_1,D_2\rangle$ contains only those objects that are either extensions of $d_1$ by $d_2$ or that are extensions of $d_2$ by $d_1$. It could also contain an object that carries a filtration whose 1st, 3rd, 5th,... graded pieces belong to $D_1$ and whose 2nd, 4th, 6th,... graded pieces belong to $D_2$. I also don't think that $x$ and $y$ are $D$-equivalent only if there is either a $D$-equivalence $x \to y$ or a $D$-equivalence $y \to x$. But I still don't know whether $D_1 \rtimes D_2 = \langle D_1,D_2\rangle$, maybe my two objections cancel out.
Mar
20
comment Topology on the space of constructible sheaves
Hi Semyon. How about chapter VIII, section 1 of Kashiwara and Schapira's "Sheaves on Manifolds"?
Mar
19
revised Topology on the space of constructible sheaves
added 285 characters in body
Mar
19
revised Topology on the space of constructible sheaves
added 285 characters in body
Mar
19
answered Topology on the space of constructible sheaves
Mar
17
comment What are some standard operations on the set of localizations of a triangulated category?
I think I misunderstood your first comment, and also maybe now I'm not sure what you mean by "symmetric closure." By "D_1-equivalent" in the second paragraph I mean, "is connected by a chain of D_1-equivalences, $x \to x' \leftarrow \cdots \to y$," which I hope is the same as "$x$ and $y$ are isomorphic in a quotient category $C/D$." If that's enough to prove $D_1 \rtimes D_2$ is subcategory generated by $D_1$ and $D_2$, can you spoon feed me the explanation?
Mar
17
comment What are some standard operations on the set of localizations of a triangulated category?
Hi Dragos. I think a $D$-equivalence $x \to y$ does not guarantee a $D$-equivalence $y \to x$. For example when $C$ is the derived category of abelian groups and $D$ is the subcategory of torsion groups, then $\mathbf{Z} \to \mathbf{Q}$ is a $D$-equivalence. I don't know if this disproves your conclusion, though.
Mar
17
asked What are some standard operations on the set of localizations of a triangulated category?
Mar
17
comment On the ordered set of real numbers, does sheaf+cosheaf imply constant?
Thanks, Dmitri.
Mar
17
accepted On the ordered set of real numbers, does sheaf+cosheaf imply constant?
Mar
15
comment On the ordered set of real numbers, does sheaf+cosheaf imply constant?
Thanks Dmitri. I think I haven't understood your criterion for isomorphism of spectra yet, in the second paragraph. But before I dig in let me check something: in the third paragraph you take a supremum over a set of numbers $r$ that obey $r > q \geq t$, which implies $r > t$. Then in parentheses you argue that this set of numbers is nonempty because it contains $r = t$. Is that a problem?
Mar
14
asked On the ordered set of real numbers, does sheaf+cosheaf imply constant?
Feb
9
accepted What's in the genus of the cubic lattice?
Feb
9
comment What's in the genus of the cubic lattice?
I think I didn't put it together until now, that odd unimodular lattices make a genus, and that's the genus containing $\mathbf{Z}^n$.
Feb
4
comment What's in the genus of the cubic lattice?
Thanks, Noam. -
Feb
4
comment What's in the genus of the cubic lattice?
Thanks for the O'Meara reference. The last footnote on the last page, referring to that Kneser work, is: "See M. Kneser (1957). For an example of the classical approach using 'reduction theory' we refer to BW Jones (1950)." I'd be interested to find out what O'M means, but I didn't find the Jones book yet.
Feb
3
comment What's in the genus of the cubic lattice?
Thanks Jeremy. How does magma do it? And is it the kind of thing that was known, or could have been, to Zolotareff?
Feb
3
asked What's in the genus of the cubic lattice?
Jan
31
revised Is there an approach to Gabber's theorem from the singular support of coherent sheaves?
"I heard" from somebody specific