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Jan
22
comment Why do people say DG-algebras behave badly in positive characteristic?
Outside characteristic zero, you have to use $E_\infty$-dg-algebras instead of strictly commutative dg-algebras. See Tyler Lawson's answer here: mathoverflow.net/a/23885/2503. (Oops, Denis beat me to it.)
Jan
9
comment (really) basic intuition for $\mathbb A^1$-homotopy theory
I would recommend taking a look at these notes: dropbox.com/s/6a9jnkqf3f55cs8/…
Jan
1
comment Characterization of closed immersions at the level of perfect complexes
@EldenElmanto, that's an interesting approach (and $f$ being a (homotopy) monomorphism is indeed equivalent to $f^*$ being fully faithful).
Dec
14
comment Motivation and potential applications of spectral algebraic geometry
As for your question, I believe that connective DAG (built out of connective E_oo-ring spectra or simplicial commutative rings), is the right generality to deal with questions about algebraic geometry, while nonconnective DAG (= SAG) was really developed with topological applications in mind.
Dec
14
comment Motivation and potential applications of spectral algebraic geometry
The term "derived algebraic geometry" is often used synonymously with what you call "spectral algebraic geometry". The work of Gaitsgory-Lurie, as far as I know, does not use any derived or spectral geometry (though it does use $(\infty,1)$-category theory in an essential way).
Dec
13
comment Derived equivalent varieties with differing integral Mukai-Hodge structures?
@MikhailBondarko, it's not quite so easy, I'm afraid. You would need to be able to define Betti cohomology and Hodge structures at the level of dg-categories, something which is most likely impossible in my opinion. The best you have are noncommutative Hodge structures on Hochschild homology and variants, studied by Kaledin and others.
Dec
13
comment Generalized Euler characteristics of non-motivic origin
@MikhailBondarko, that's right, it was also proved directly in the note of Orlov (the proof essentially amounts to the same thing as the comparison of noncommutative motives modulo Tate twists with usual motives, namely Grothendieck-Riemann-Roch).
Dec
10
comment Derived equivalent varieties with differing integral Mukai-Hodge structures?
This is related to my question: mathoverflow.net/questions/152345/…
Dec
10
comment Generalized Euler characteristics of non-motivic origin
It's actually a conjecture of Orlov, see this paper. (The conjecture is for rational coefficients as it is clearly false integrally. I agree that it is not really expected to be true even rationally, though.)
Dec
10
comment teaching higher algebra
In 2013 in Berlin, Timo Schürg gave a course on a low-tech (model categorical) introduction to the deformation theory of simplicial commutative rings. The notes are available here. It was attended by algebraic geometry students mostly, and was pretty successful I think.
Dec
9
comment reference request for mod p and p-adic K-theory
See also these notes of Mitchell which are an exposition of Quillen's proof.
Dec
1
comment Is Carlos Simpson's Descent available online?
Here is a link: dropbox.com/s/j0ft4tiop740wjv/SIMPSON%20Descent.pdf?dl=0 (I don't remember where I found this.)
Nov
26
comment K theory long exact sequence
@DmitryVaintrob, statements to this effect are in Blumberg-Gepner-Tabuada. They show that K-theory preserves filtered colimits and split exact sequences.
Nov
23
comment Categorical or simplicial introduction to modern homotopy theory
For Blakers-Massey at least, there is a nice model independent proof described in a note of Rezk.
Oct
31
comment Algebraic geometry introduction for homotopy theorists/algebraic topologists
At a more basic level, I think that the "functor of points" approach to algebraic geometry is much clearer to students comfortable with homotopy theory or category theory. So one might prefer to read something like Toen's course notes, before reading a scheme theory book.
Oct
15
comment A tensor product for triangulated categories?
There is some discussion of this question in an old paper of Bondal-Larsen-Lunts (arXiv:math/0401009).
Oct
14
comment Is dgCat a category or a 2-category?
Any model category (in fact, even just a category with weak equivalences) gives rise to an $(\infty,1)$-category. For example, if one takes simplicially enriched categories as a model of $(\infty,1)$-categories, this is given by the Dwyer-Kan simplicial localization.
Sep
29
comment Is dgCat a category or a 2-category?
@ZhenLin: Gepner-Haugseng show that the $\infty$-category of $V$-enriched $\infty$-categories has an enrichment over itself, for $V$ a presentably $E_2$-monoidal $\infty$-category. In particular this gives the enrichment of DGCat over itself, and hence via Dold-Kan an enrichment over $\infty$-categories. (I'm using implicitly the rectification result of Haugseng, which implies that DGCat is equivalent to the $\infty$-category of $\infty$-categories enriched over the $\infty$-category of chain complexes.)
Sep
26
comment $E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras
Yes, the functor $\phi$ just sends a simplicial algebra to its "normalized cochain complex".
Sep
23
comment Why do the model structures on dg-algebras and on dg-categories are not compatible?
A small remark: the functor of $\infty$-categories from (dg-algebras) to (dg-categories) factors through a functor to the $\infty$-category of pointed dg-categories which is fully faithful. As you point out, the mapping spaces in (pointed dg-categories) are obtained from the mapping spaces in (dg-categories) by quotienting out by the action of the simplicial group of units.