Holmstrom Old questions

What about sending questions to other people on the list of mathematicians?

Basic questions

What is a symmetric space?

For Toen

In the note on homotopy types of alg varieties, a schematic homotopy theory is a functor from smooth proj vars over k to schematic homotopy types over K. Why is projective necessary here?

For Deligne, never got the time to ask him

Ask about spaces and representability in char p. How should we interpret the fact that some CTs are not homotopy invar?

Interpret rigid cohomology in terms of homotopy theory?

Are there in general any invariants you can define for schemes but not for simplicial schemes or simplicail sheaves?

General intuition for cohomology of schemes over Z?

Motivation for arithmetic Chow groups? Expected relation to special values???

For Scholl

Sent mail 25th or 26th of Sep 2008

Examples of sheaves without transfers? What about \\Omega^q?

What is the precise expected domain category for Arakelov Chow groups? Why?

What does "homotopy cartesian" mean?

Can it really be true that for any morphism of varieties f: X \\to Y, $\\f: X \\h(Y)$ becomes a direct factor of $h(X)$. Must it not be the image that becomes a direct factor??? See André notes under Pure motives.

Explain Manin's identity principle.

Jannsen's result "abelian semisimple": does it mean that other cats of pure motives are not even abelian? (they are always rigid tensor)

Is it interesting to talk of mixed motives over fields which are not number fields or finite fields? Would Ext groups vanish at all?

What is known unconditionally for the regulator map - for example, is it known that both motivic and Deligne cohomology have the right dimension? Nekovar states that $dim H_D$ is the order of vanishing of the related L-function.

Plan: Read Feliu, check for open problems. Also, try to ask about each formal property: does it hold for Arakelov Chow groups?

Feliu defines and arithmetic variety as a regular, quasi-projective scheme, flat over an arithmetic ring. Is there a precise definition of arithmetic ring? Also, she says that everything she does with higher arithmetic Chow groups works for arithmetic varieties over a field, but not for arithmetic varieties over other arithmetic rings. Is there any relation between higher arithmetic Chow groups of an arithmetic variety over a ring and the same groups of the base change to the quotient field of this ring?

What is a stratification?

Explain why the two defs of real Hodge structure are equiv: A real Hodge structure is a finite-dimensional real vector space $V$ equipped with an action of the real algebraic group $S$. Here $S$ is the group obtained from the multiplicative group $G_m$ by restriction of scalars from \\mathbb{C} to \\mathbb{R}. (So the real points of $\\\\S$ is \\mathbb{C}^*). Such an action is equivalent to giving a bigrading of V_{\\mathbb{C}} on which complex conjugation interchanges the grading indices.

Why only the standard realizations of motives?

What is a specialisation argument. For example, the standard conj of Hodge type is true over any field if it is true over finite fields.

Are there "regulators" to continuous etale cohomology, and are they good for anything? For example for the rational factor.

Clarify everything relating to geometric vs absolute theories. List all, explain representability, etc. What is the geometric theory corresponding to Deligne cohomology? What about grading vs bigrading, and extra structure?

Clarify relative cohomology and supports.

What is a universal extension?

Explain the Yoneda pairing on Ext groups

For Hyland

See concept note.

Why does one always consider simplicial objects? Is the Delta cat distinguished among test cats, or could we always use any test cat? For example: would cubical abelian groups be equivalent to complexes of abelian groups? Could we use cubical schemes to do Hodge structures for singular schemes? Etc

For Totaro

For Levine

Prepare question on whether Deligne cohomology is representable. What exactly would one have to check? Nisnevich descent: check only for squares of schemes, or of a general simplicial scheme? Is there any advantage in being able to define higher arithmetic Chow groups in the stable homotopy cat (as a cofiber or something?), compared to defining it by a cone?

For Grojnowski

Nisnevich topology

Jardine

In notes on simplicial presheaves: End of section 7 (middle of page 25): something is wrong with the initial term of the spectral sequence - should it be $\\V_{\\t$ instead of $s$ somewhere?

Weibel

Errata for the K-book In the general references: Berstein should be Bernstein.

nLab page on Old questions