ag.algebraic geometry – Frobenius action on cohomology

Let define the absolute Frobenius action on cohomology,

Let $ X $ be a scheme over a field $ K$. Fix a separable closure $ K^s $ of $ K $. Let $ G_K = operatorname{Gal} ( K^s / K ) $ be the absolute Galois group of $ K $. Let $ mathcal{F} $ an abelian sheaf on $ X_{ mathrm{et} } $

I try to define properly a left $ G_K $-module structure cohomology group $ H^{j} ( X_{K^{s}} , mathcal{F}_{|K^{s} } ) $ as follows,

If $ sigma in G_K $, then, we can set, for $ xi in H^{j} ( X_{K^{s}} , mathcal{F}_{| X_{K_{s}}} ) $, $$ sigma cdot xi = : ( mathrm{Spec} ( sigma ) times mathrm{id}_X )^* xi in H^{j} ( X_{K^{s}} , ( mathrm{Spec} ( sigma ) times mathrm{id}_X )^{-1} mathcal{F}_{|X_{K^{s}}} ) = H^{j} ( X_{K^{s}} , mathcal{F}_{| X_{K^{s}}} ) $$

So, my question is,

How is defined explicitly $ ( mathrm{Spec} ( sigma ) times mathrm{id}_X )^* xi $ in $ H^{j} ( X_{K^{s}} , ( mathrm{Spec} ( sigma ) times mathrm{id}_X )^{-1} mathcal{F}_{|X_{K^{s}}} ) $ ?

To be more concrete, if $ X = mathrm{Spec} (A) $, and $ A = K (t_1 , dots , t_n ) $ is a finitely generated $ K $ – algebra. then, $ X_{K^{s}} = mathrm{Spec} ( K^s otimes_K A ) = mathrm{Spec} ( K^{s} (t_1 , dots , t_n ) ) $.

So, how to express explicitly $ sigma cdot xi = : ( mathrm{Spec} ( sigma ) times mathrm{id}_X )^* xi in H^{j} ( X_{K^{s}} , ( mathrm{Spec} ( sigma ) times mathrm{id}_X )^{-1} mathcal{F}_{|X_{K^{s}}} ) $ with, $ X_{K^{s}} = mathrm{Spec} ( K^s (t_1 , dots , t_n ) ) $ ?

Other questions:

We defined above the first case of absolute Frobenius action on cohomology, so now, how is defined, the arithmetic, geometric, and relative Frobenius actions on cohomology ?

Thank you.

ag.algebraic geometry – Lowest weight of compactly supported cohomology with coefficients

Let $X_0/mathbb F_q$ be a variety, and let $mathcal F$ be a Weil sheaf on $X := (X_0)_{overline{mathbb{F}_q}}$ that is pure of weight $n$. If $j < n$, does the weight $j$ piece of $H^i_c(X,mathcal F)$ necessarily vanish for all $i$?

I thought this was true, but I have made some computations that seem to contradict it.

ag.algebraic geometry – All Galois characters showing up in cohomology of one family of varieties

Fix a prime $p$.

Can we find a smooth proper map $Xto Y$ of $mathbb{Q}_p$-varieties such that any given representation $mathrm{Gal}(overline{mathbb{Q}_p}/mathbb{Q}_p)to mathrm{GL}_1(mathbb{F}_p)$ embeds in $oplus_{igeq 0} H^i_{mathrm{acute{e}t}}(X_ytimes overline{mathbb{Q}_p}, mathbb{F}_p)$ for some $yin Y(mathbb{Q}_p)$?

ag.algebraic geometry – Cohomology of singular projective cubic surface

Let $Xsubset mathbb{P}_{mathbb{C}}^3$ be a projective singular cubic surface with two singular points. Is the cohomology of such objects known? As an example of the type of surfaces I’d be interested in one can take the hypersurface defined by the homogenous cubic $$x^2w+y^2w+z^2w+xyz-zw^2-w^3=0 $$

I thijnk that the cohomology should be known for projective smooth cubic surfaces which should be realized as blowup of $mathbb{P}^2_{mathbb{C}}$ in six points but I’m not sure about singular ones.

at.algebraic topology – What are the stable cohomology classes of the “orthogonal groups” of finite abelian groups?

Let $A$ be a finite abelian group, and equip it with a nondegenerate symmetric bilinear form $langle,rangle : A times A to mathrm{U}(1)$. Then you can reasonably talk about the “orthogonal group” $O(A,langle,rangle)$, i.e. the automorphisms of $A$ which preserve $langle,rangle$.

Actually, when $A$ has even order, the most standard version of “orthogonal group” is to choose a quadratic refinement $q : A to mathrm{U}(1)$ of $langle,rangle$, i.e. a quadratic function such that $q(a_1 a_2) / q(a_1)q(q_2) = langle a_1,a_2rangle$, and to ask for the group $O(A,q)$ of automorphisms preserving $q$. (When $A$ has odd order, there is a unique quadratic refinement, and so the two notions agree. In general, the quadratic refinements form a torsor for $hom(A, mathbb{Z}/2)$.) I can ask my question for both versions of $O(A)$, but in fact the one I care about is the “nonstandard” $O(A,langle,rangle)$ when $A$ has order a power of $2$.

I am interested in understanding the low $mathbb{Z}/2$-cohomology of $O(A,langle,rangle)$.

I specifically want to know about classes which are “stable”, but I am having trouble saying precisely what I want “stable” to mean. Certainly if I have two groups $A, A’$ each equipped with nondegenerate symmetric bilinear forms, then I only care about the image of restriction along $O(A) subset O(A) times O(A’) subset O(A times A’)$. But I expect that if there is a stronger notion of “stability”, then I only care about those classes.

ag.algebraic geometry – The multiplicativity of the (complex) geometric realization of motivic cohomology

Consider the (complex) geometric realization of the motivic cohomology theory on simplicial presheaves over complex smooth schemes, which is a functorial homomorphism of $R$-modules, where $R$ is the coefficient ring:
varphi: H^{s,t}_{mot}(X;R)to H^s(X(mathbb{C});R).

When restricted to the Chow ring $varphi:H^{2*,*}_{mot}(X;mathbb{Z})to H^{2*}(X(mathbb{C});mathbb{Z})$ and $X$ a complex algebraic variety, it is a classical result that $varphi$ is the cycle class map ring homomorphism.

Generally, is the map $varphi: H^{*,*}_{mot}(X;R)to H^*(X(mathbb{C});R)$ an R-algebra homomorphism? I was not able to find a direct proof of this result in the literature. Or perhaps this is an obvious fact that I am missing.

reference request – (Co)homology of a directed space with coefficients in a commutative monoid

This is essentially a reference request, or a request for an explanation of why this cannot be done in a useful or interesting way (i.e. an explanation of why no such reference exists!).

If I have a directed space, perhaps modeled by something like a globular complex or complicial set, is there any way to compute “homology” or “cohomology” of it with coefficients in a (commutative) monoid?

A particularly simple example might just be to think of $S^1$ as a directed 1-type, and ask whether or not there’s some homology theory such that $H_1(S^1,mathbb{N})congmathbb{N}$, or something along those lines.

This seems like too simple of an idea to not have been attempted. Does anyone have a reference? If not, does anyone know what goes wrong?

One possibly silly attempt at answering this might be: write down the directed space $X$ as some kind of $(infty,infty)$-category, then write down the sequence of deloopings $(M,BM,B^2M,…)$ where we have to take the $n$th “delooping” here to be something like the $(infty,n)$-category with one object and an $M$‘s worth of $n$-morphisms. Then define $H^n(X,M)$ to be something like functors of $(infty,infty)$-categories $Xto B^nM$ modulo some notion of equivalence of $(infty,infty)$-functors.

ag.algebraic geometry – Calculations regarding the weight one Morphic cohomology

According to the paper “A theory of algebraic cocycles” (this is a summary of the original paper), theorem 8, gives the morphic cohomology of weight one. More precisely for any complex quasi-projective variety $X$ (the theorem mentions projective but it is expected to be true for quasi-projectives too), one has the following isomorphisms:
$$L^1H^0(X)cong mathbb{Z}, L^1H^1(X)cong H^1(X,mathbb{Z})$$
These two morphic cohomology groups can be identified with $pi_2 (coprod_n Mor(X, mathbb{P}^{n}))^+$ and $pi_1 (coprod_n Mor(X, mathbb{P}^{n}))^+$ respectively (you might also need to check the definition of morphic cohomology and also remark 3.6 in the original paper). The plus sign is the group completion of the monoid. Since these are supposed to be true for all quasi projective varieties one can expect that these isomorphisms hold for local ring and function fields by taking colimit. So I just want to verify this for a function field of a variety. Let’s call the field $F$. There is an easy way to describe $Mor(X, mathbb{P}^{n})$ for affine $X$ and especially for $X=text{Spec}(F)$. It amounts to giving a line bundle on $text{Spec}(F)$ which is trivial and elements in $s_1, cdots s_{n+1}$ in the field $F$ that not all are zero and the map to the projective space is given by the class of $(s_1,cdots, s_{n+1})$, where two such tuples are identified if one is a non-zero multiple of the other one.

This seems to identify $Mor(text{Spec}(F), mathbb{P}^{n})$ with $text{Spec}(F)times mathbb{P}^n$. Then what does $(coprod_n Mor(text{Spec}(F), mathbb{P}^{n}))^+$ look like? How one can recover the expected results from this? Is this supposed to be $Sym^{infty}(text{F}times mathbb{P}^1)$? If so then it implies that the first homotopy group is $H^1(text{Spec}(F),mathbb{Z})$ which is the correct answer but the second homotopy group becomes $mathbb{Z}oplus H^2(text{Spec}(F),mathbb{Z})$ which is incorrect.

Note that the singular cohomology of field is defined as the colimit of singular cohomology of Zariski opens of the variety whose function field is $F$.