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1d
comment Deligne's theorem on the characterisation of Tannakian categories
Sorry, by "dualizable" I meant "finite free" in the last sentence.
1d
comment Deligne's theorem on the characterisation of Tannakian categories
$A\otimes -$ is not strong monoidal as an endofunctor of $Ind(T)$ (your second isomorphism is incorrect!). It is strong monoidal as a functor to $A$-modules. Then $\Gamma$ is lax monoidal from $A$-modules to $\Gamma(A)$-modules, but it is strong when restricted to the full subcategory of dualizable $A$-modules, where $A\otimes -$ lands.
Apr
14
comment (Geometric) Proof for the projective bundle formula in K-theory
@SaalHardali $H$ is the dual of the tautological bundle. The tautological bundle is tautologically a subbundle of the pullback of $E$ to $\mathbb P(E)$. It seems to me that $\lambda_E$ being a generator is equivalent to Bott periodicity, by the way.
Apr
14
comment (Geometric) Proof for the projective bundle formula in K-theory
Over $\mathbb P(E)$ your Koszul complex starts with $\mathbb C\hookrightarrow E\otimes H$, that's why you have powers of $[H]$.
Apr
5
comment Is $MGL$ an $H\mathbb{Z}$-algebra?
Right, that's an interesting point. This doesn't hold in general: for $n>0$, $\pi_n\Sigma^{0,-n}(S^0_{\mathbb C})=W(\mathbb C)=\mathbb Z/2$ has nothing to do with $\pi_n(S^0)$. But I think it might hold for any slice-complete spectrum, at least if the slices have the form $\Sigma^{p,q}HA$ with $A$ an abelian group, because we know it for such slices. In particular it should hold for $MGL$ (my guess about the image of $MGL$ above was way off...).
Apr
4
comment Is $MGL$ an $H\mathbb{Z}$-algebra?
No, $K(\mathbb C)\neq KU$.
Apr
4
comment Is $MGL$ an $H\mathbb{Z}$-algebra?
Yeah, there's always an alternative realization functor $SH(k)\to SH$ given by the spectral enrichment of $SH(k)$. It's lax monoidal and sends $H\mathbb Z$ to $H\mathbb Z^{top}$ and $KGL$ to $K(k)$. I think it also sends $MGL$ to $H\mathbb Z^{top}$, actually.
Apr
2
answered Is $MGL$ an $H\mathbb{Z}$-algebra?
Apr
1
comment Is $MGL$ an $H\mathbb{Z}$-algebra?
The $b_i$'s in your first formula do not come from the homotopy groups of $MGL$. In particular, they are not the same guys that appear in the Hopkins-Morel iso!
Mar
31
comment A Kunneth formula for the etale cohomology of the product of ('simple') varieties over not (necessarily) algebraically closed field
So the argument is as follows: the Künneth formula from SGA4.5 gives an equivalence of $l$-profinite spaces with $Gal(k)$-action $Et(\bar X \times \bar Y)^\wedge_l\simeq (Et(\bar X)\times Et(\bar Y))^\wedge_l$. Taking homotopy $Gal(k)$-orbits, we get $Et(X\times_k Y)^\wedge_l \simeq (Et(X)\times_{Et(k)}Et(Y))^\wedge_l$.
Mar
31
comment A Kunneth formula for the etale cohomology of the product of ('simple') varieties over not (necessarily) algebraically closed field
If you're referring to the part about the etale homotopy type, it's true that $Et(X\times_kY)=Et(X)\times_{Et(k)}Et(Y)$, after $l$-completion, for any $X$ and $Y$ of finite type over $k$. What I was missing above is that taking homotopy orbits does in fact commute with homotopy pullbacks (see mathoverflow.net/questions/836/…).
Mar
31
comment Algebraic cobordism (of a point) outside the geometric diagonal
I mean that your conjecture is true for finite fields but false in general.
Mar
29
awarded  Enlightened
Mar
29
awarded  Nice Answer
Mar
25
comment Useful, non-trivial general theorems about morphisms of schemes
In your statement of Chow's lemma you could replace noetherian by qcqs.
Mar
25
comment Useful, non-trivial general theorems about morphisms of schemes
You could add Stein factorization to the list: (proper) = (integral) $\circ$ (proper with connected geometric fibers). stacks.math.columbia.edu/tag/03GX
Mar
23
comment Algebraic cobordism (of a point) outside the geometric diagonal
If this were true you'd have an analogous result for K-theory, by tensoring with $\mathbb Z[\beta^{\pm 1}]$. But it's true for finite fields, by the slice spectral sequence.
Mar
23
comment Algebraic cobordism (of a point) outside the geometric diagonal
I have only quoted 7.6 and 7.7 in Spitzweck.
Mar
23
comment Algebraic cobordism (of a point) outside the geometric diagonal
They are zero if $n$ is positive!
Mar
23
answered Algebraic cobordism (of a point) outside the geometric diagonal