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bio website preschema.com
location Essen, Germany
age 23
visits member for 5 years
seen 3 hours ago

May
13
comment Universal property of gluing [collage, cograph] of dg-categories
@FrancescoGenovese, they are the same for pretriangulated dg-categories, and otherwise the version of Orlov is the pretriangulated envelope of the version of Kuznetsov-Lunts.
May
4
comment Morphism between Fourier-Mukai functors implies the morphism between kernels?
However, the corresponding statement at the level of dg-categories is true: $\mathbf{D}(X \times Y) = \underline{\mathbf{Hom}}(\mathbf{D}(X), \mathbf{D}(Y))$, so that morphisms of integral kernels correspond to morphisms of the induced dg-functors (note though that the Hom means the internal Hom in the localization of the category of dg-categories with respect to the quasi-equivalences).
May
2
comment Comparison of model structures
@user38585, see section 1.3.3. In particular Corollary 1.3.16 is a useful way to check if a given Quillen adjunction is a Quillen equivalence.
Apr
17
comment Could one recover the relative K-theory from the quotient derived category?
Given dg enhancements of the derived categories, the Verdier quotient may be described as the homotopy category of the homotopy cofibre in the category of small dg categories with the Morita model structure. As a functor on the category of dg-categories, K-theory is known to preserve homotopy cofibres (this is part of being a localizing invariant in the sense of Tabuada).
Mar
6
awarded  Nice Answer
Mar
6
awarded  Yearling
Mar
6
comment Why is the derived tensor product only defined for bounded above derived categories?
An even more modern treatment using the language of model categories is the paper "Local and stable homological algebra in Grothendieck abelian categories" by Cisinski-Deglise (arxiv.org/abs/0712.3296). One gets an unbounded derived tensor product mostly by abstract nonsense, see Example 2.2 and Proposition 2.3.
Mar
6
answered Why is “naive” definition of non-commutative spectrum bad?
Feb
14
comment The most important facts, modern surveys, and readable introductions to p-adic cohomology theories (crystalline cohomology and the mysterious functor)
We had a seminar on this paper in Berlin recently, you might like to take a look at the program: mi.fu-berlin.de/en/math/groups/arithmetic_geometry/…
Feb
10
revised Motivic derived algebraic geometry
Added tags
Feb
10
comment Motivic derived algebraic geometry
@DylanWilson, that sounds pretty interesting, though I know nothing about tmf.
Feb
10
comment Motivic derived algebraic geometry
@FernandoMuro, right this is what I'm wondering about: whether such a construction could have any interesting applications for which E-infinity derived geometry would be insufficient.
Feb
9
asked Motivic derived algebraic geometry
Feb
8
answered For what varieties do we have results on the category of singularities?
Jan
24
comment Reasons for the use of Nisnevich topology in motivic homotopy theory
@SimonPepinLehalleur, why don't you write this as an answer?
Jan
22
comment Push-outs of fully faithful (enriched) functors
This looks relevant: faculty.fortlewis.edu/Scull_L/pushouts.pdf
Jan
4
answered It looks so coKleisli, but it's not. What is it?
Jan
4
comment unique enhancement for derived categories
let us continue this discussion in chat
Jan
3
comment Reconstruction of noncommutative scheme
For stacks a reconstruction theorem was proved in arxiv.org/abs/1004.3087, though this is more in the spirit of Balmer than Rosenberg. In the noncommutative setting, the first question is how to define the categories of quasi-coherent sheaves; this is done in the setting of fibred categories, or categories over a category in Kontsevich-Rosenberg's paper "Noncommutative stacks". I haven't seen any reconstruction theorems in this setting, but this unpublished preprint you mention sounds promising. Wouldn't that be a satisfactory answer to your question? (Is it available online btw?)
Jan
3
comment unique enhancement for derived categories
@FernandoMuro, ok. I am just wondering what possible difference there could be, since the various models of stable linear (infty,1)-categories, including pretriangulated dg-categories, are known to be equivalent, as you of course know.