15:05: Bhargav Bhatt (Harvard NT Seminar)
-
One can take etale cohomology of varieties, and later refine to schemes, and thus take it for the base field even when it’s not algebraically closed and extract arithmetically interesting information.
-
prismatic cohomology, meant to relate a number of other cohomology theories
-
Prism : a pair \((A, I)\) where \(A\) is a commutative ring with a derived Frobenius lift \(\phi:A\to A\), i.e. a \(\delta{\hbox{-}}\)structure.
- \(I {~\trianglelefteq~}A\) is an ideal defining a Cartier divisor.
- \(A\) is \((P, I){\hbox{-}}\)complete.
- Any ideal generator \(d\in I\) satisfies \(\phi(d) = d^p + p\cdot u, u\in A^{\times}\).
-
Fix a scheme and study prisms over it. Need these definitions to have stability under base-change.
-
Examples:
- \(A \coloneqq{\mathbb{Z}}_p\) and \(\phi = \operatorname{id}\) with \(I = \left\langle{ p }\right\rangle\) yields Crystalline cohomology.
- \(A \coloneqq{\mathbb{Z}}_p{\left[\left[ u \right]\right] }, \phi(u) = u^p\). Then \(I = \left\langle{ E(u) }\right\rangle\) is generated by an Eisenstein polynomial. Here \(A/I = {\mathcal{O}}_K\)
-
Prismatic site : fix a base prism \((A, I)\) for \(X\) a \(p{\hbox{-}}\)adic formal scheme over \(A/I\). Define \begin{align*} (X/A)_\prism = \left\{{ (A, I) \to (B, J) \in \mathop{\mathrm{Mor}}(\mathsf{Prism}), \operatorname{Spf}(B/J) \to X \text{ over } A/I }\right\} ,\end{align*} topologized via the flat topology on \(B/J\).
-
There is a structure sheaf \({\mathcal{O}}_\prism\) where \((B, J) \to B\). Take \(\mathop{\mathrm{{\mathbb{R} }}}\Gamma\), which receives a Frobenius action, to define a cohomology theory. Why is this a good idea?
-
Absolute prismatic sites: for \(X\in {\mathsf{Sch}}(p{\hbox{-}}\text{adic})\), define \begin{align*} X_\prism \coloneqq\left\{{ (B, J) \in \mathsf{Prism},\, \operatorname{Spf}(B/J) \to X }\right\} .\end{align*} Take sheaf cohomology to obtain \(\mathop{\mathrm{{\mathbb{R} }}}\Gamma_\prism(X) \coloneqq\mathop{\mathrm{{\mathbb{R} }}}\Gamma(X_\prism, {\mathcal{O}}_\prism) {\circlearrowleft}_\phi\).
-
The category \(\mathsf{Prism}\) doesn’t have a final object, so has interesting cohomology. Relates to Algebraic K theory of \({\mathbb{Z}}_p\)?
-
Questions: let \(X_{/{\mathbb{Z}}_p}\) be a smooth formal scheme.
- What is the cohomological dimension of \(\mathop{\mathrm{{\mathbb{R} }}}\Gamma_\prism(X)\)?
-
What are the \(F{\hbox{-}}\)crystals on \(X_\prism\)?
- Produce finite flat \(B{\hbox{-}}\)modules?
-
Bhatt and Lurie: found a stacky way to understand the absolute prismatic site of \({\mathbb{Z}}_p\). Drinfeld found independently.
-
Construction due to Simpson: take \(X\in {\mathsf{Var}}({\mathsf{Alg}})\), define a de Rham presheaf \begin{align*} X_{\mathrm{dR}}: {\mathsf{Alg}_{\mathbb{C}} }^{\operatorname{fp}} &\to {\mathsf{Set}}\\ R &\mapsto X(R_{ \text{red} }) .\end{align*}
- Translates other cohomology theories into something about coherent sheaves..?
- Can reduce to studying e.g. a vector bundle on a more complicated object.
-
Def: Cartier-Witt stack, a.k.a. the prismatization of \({\mathbb{Z}}_p\)
- Define \(\mathsf{WCart}\) to be the formal stack on \(p{\hbox{-}}\)complete rings.
-
Plug in a \(p{\hbox{-}}\)nilpotent ring \(R\) to extract all (derived) prism structure on \(W(R)\).
-
Prisms aren’t base-change compatible without the derived part.
-
This is a groupoid.
-
An explicit presentation: \(\mathsf{WCart}_0(R)\) are distinguished Witt Vectors in \(W(R)\). Given by \([a_0, a_1, \cdots ]\) where \(a_0\) is nilpotent and \(a_1\) is a unit. This is a formal affine scheme. \(\mathsf{WCart}= \mathsf{WCart}_0 / W^*\) is a presentation as a stack quotient.
- Receives a natural Frobenius action, which is a derived Frobenius lift.
-
Start by understanding its points, suffices to evaluate on fields of characteristic \(p\).
-
If \(k\in \mathsf{Field}(\mathsf{Perf})_{\operatorname{ch. p}}\), \(\mathsf{WCart}(k) = \left\{{ {\operatorname{pt}}}\right\}\), with the point represented by \((W(k), ?)\).
- Yields a (geometric?) point \(x_{{\mathbb{F}_p}}: \operatorname{Spec}({\mathbb{F}_p}) \to \mathsf{WCart}\).
-
Analogy to understanding Hodge-Tate cohomology. Similar easy locus in this stack.
-
Take 0th component of distinguished Witt Vectors to get a diagram
-
The bottom-left is this Hodge-Tate stack
-
Now has a better chance of being an algebraic stack instead of a formal stack. Bottom arrow kills the formal direction.
-
Will be classifying stack of a group scheme : need to produce a point and take automorphisms.
-
Take the distinguished element \(V(?) \in W({\mathbb{Z}}_p)\). Produces a map \begin{align*} \operatorname{Spf}({\mathbb{Z}}_p) \xrightarrow{\pi_{\operatorname{HT}}} \mathsf{WCart}^{\operatorname{HT}} .\end{align*}
- Fact: \(\pi_{\operatorname{HT}}\) is a flat cover and \(\mathop{\mathrm{Aut}}(\pi_{\operatorname{HT}}) = W^*[ F]\).
-
Upshot: \(\mathsf{WCart}^{\operatorname{HT}}= {\mathbf{B}}W^* [F]\) is a classifying stack. quasicoherent sheaves on the left and representations of the (classifying stack of the) group scheme on the right. I.e. \({ \mathsf{D} }_{\operatorname{qc}}(\mathsf{WCart}^{\operatorname{HT}}) = \mathop{\mathrm{{\mathbb{R} }}}(W^*[F])\).
-
Teichmüller lift yields a \({\mathbb{Z}}/p\) grading on the LHS.
-
Something about Deligne-Illusie? Hodge-to-deRham degeneration
-
Upshot: a divisor inside is easy to understand.
-
-
Fact: \({ \mathsf{D} }_{\operatorname{qc}}(\mathsf{WCart})\) are equivalent to \begin{align*} \varprojlim_{(A, I)\in \mathsf{Prism}} { \mathsf{D} }_{(P, I)-?}(A) .\end{align*}
-
Diffracted Hodge cohomology: let \(X\in {\mathsf{Schf}}_{{\mathbb{Z}}_p}\). Get a prismatic structure sheaf using the assignment \((A, I) \to \mathop{\mathrm{{\mathbb{R} }}}\Gamma_\prism \qty{ (X\otimes A/I) / A}\).
-
Heuristic: \(\operatorname{Spec}{\mathbb{Z}}_p\) should be 1-dimensional over something.
-
Get an absolute comparison: \(\operatorname{cohdim}\mathop{\mathrm{{\mathbb{R} }}}\Gamma_\prism (X) \leq d+1\) where \(d = \operatorname{reldim}X_{/{\mathbb{Z}}_p}\).
-
There is a deRham comparison: \begin{align*} X_{{\mathbb{F}_p}}^* H_\prism(X) \cong \mathop{\mathrm{{\mathbb{R} }}}\Gamma_\mathrm{dR}(X_{{\mathbb{F}_p}}) .\end{align*}
-
There is a Hodge-Tate comparison: the object \(H_\prism(X)\) restricted to \(\mathsf{WCart}^{\operatorname{HT}}\) has an increasing filtration with \({\mathsf{gr}\,}_i = \mathop{\mathrm{{\mathbb{R} }}}\Gamma(X, \Omega^i_X)[-i]\).
- Use representation interpretation, then \(\mu_p \curvearrowright{\mathsf{gr}\,}_i\) by weight \(-i\).
-
Combine these comparisons to get Deligne-Illusie : if \(\operatorname{reldim}X < p\), then \begin{align*} \mathop{\mathrm{{\mathbb{R} }}}\Gamma_\mathrm{dR}(X_{{\mathbb{F}_p}}) \cong \bigoplus_{i} \mathop{\mathrm{{\mathbb{R} }}}\Gamma(X_{{\mathbb{F}_p}}, \Omega^i[-i]) .\end{align*} Get a lift to characteristic zero, yields Hodge-to-deRham degeneration there.
-
An \(F{\hbox{-}}\)crystal on \(X_\prism\) is a vector bundle \(\mathcal{E} \in { \mathsf{Vect} }(X_\prism, {\mathcal{O}}_\prism)\)? Plus some extra data.
-
Infinite tensor product: \begin{align*} I_\prism \otimes F^* I_\prism \otimes(F^2)^* I_\prism \otimes\cdots .\end{align*} Converges to some object \({\mathcal{O}}_\prism \left\{{ 1 }\right\} \in {\operatorname{Pic}}(X_\prism, {\mathcal{O}}_\prism )\), twisted? Yields isomorphism of sheaves after inverting \(I_\prism\), \begin{align*} F^* {\mathcal{O}}_\prism \left\{{ 1 }\right\} \cong I_\prism^{-1}\otimes{\mathcal{O}}_\prism \left\{{ 1 }\right\} .\end{align*}
- Convergence: this is a formal stack, any thickening are identified with something… On each finite approximation, most terms are \({\mathcal{O}}_X\).
-
Some analog of Artin-Schreier here, taking fixed points?
-
There is a natural functor from \(F{\hbox{-}}\)crystals on \(X\) to local \({\mathbb{Z}}_p\) systems on a geometric fiber \(X_?\)?
-
Main theorem: produces descent data, uses work on Beilinson fiber sequence (Benjamin Antieau, Morrow, others?)
-
Can say \begin{align*} H^i(\Delta_{{\mathbb{Z}}_p}) = \begin{cases} {\mathbb{Z}}_p & i=0 \\ \prod_{{\mathbb{N}}} {\mathbb{Z}}_p & i=1. \end{cases} \end{align*} Can compute using topological Hochschild homology? \(\pi_{-1}( {\operatorname{TP}}({\mathbb{Z}}_p) )\) is where the \(i=1\) part comes from.
-
topological periodic cyclic homology corresponds to prismatic cohomology
- topological Hochschild homology corresponds to Hodge-Tate cohomology.
-
topological periodic cyclic homology corresponds to prismatic cohomology
-
Prismatic is filtered by things that look like Hodge-Tate
-
Absolute = arithmetic (take eigenspaces, related to motivic cohomology, relative = geometric?
- Link to K-Theory comes from eigenspaces somehow.
-
Similar to situation in Étale cohomology : need absolute and relative to compute either.
Why are triangulated categories and derived category important?
- Homological algebra lives in the derived category
- In AG, tight link between birational equivalence (of say smooth projective varieties and equivalence of \({\mathsf{DCoh}}\), the derived categories of coherent sheaves
-
See the Bondal-Orlov conjecture.
- birational is a weakening of isomorphism between varieties
- Being derived equivalent is a weakening of having equivalent \({\mathsf{DCoh}}\)
- Both recover actual isomorphisms in the case of smooth projective varieties
-
Rep theory: equivalent derived categories is called Morita equivalence.
- Derived equivalence is a weakening of Morita equivalence
- Can replace an algebra by a much simpler derived-equivalent one
- Use to study blocks of group algebra
- Lots of numerical consequences?
A Roadmap to Hill-Hopkins-Ravenel
Some Lurie stuff Content
Lurie’s Seminar on Algebraic Topology
Lurie’s Topics in Geometric Topology
The Relationship Between [[topological Hochschild homology|THH]] and K-Theory
Some remarks on \({\operatorname{THH}}\) and \(K{\hbox{-}}\)Theory, no clue what the original source was:
-
Algebraic K theory is hard, using topological Hochschild homology somehow makes computations easier.
-
\(K{\hbox{-}}\)theory says something about vector bundles, topological Hochschild homology describes monodromy of vector bundles around infinitesimal loops
-
For \(X\) a nice scheme, take \(LX\) the derived free loop space : the derived stacks \(\mathop{\mathrm{Maps}}_{ \operatorname{DSt}}(S^1, X)\).
- Points of \(LX\): infinitesimal loops in \(X\)
-
Identify \({\operatorname{THH}}(X) \xrightarrow{\sim} {\mathcal{O}}(LX)\) (global functions)
- Corollary of a result in Ben-Zvi–Francis–Nadler “Integral Transforms and Drinfeld Centers in derived algebraic geometry”?
-
Dennis trace : a comparison \(K(X) \to {\operatorname{THH}}(X)\), takes \(E\in { {\mathsf{Bun}}\qty{\operatorname{GL}_r} }\) to the canonical monodromy automorphism of the pullback of \(E\) to \(LX\)
- Use the map \(LX\to X\) sending a loop to its basepoint
-
Traces are \(S^1{\hbox{-}}\)equivariant because loops! Just equip \(K(X)\) with the trivial \(S^1\) action.
-
Take homotopy fixed points to get something smaller than \({\operatorname{THH}}\): \({\operatorname{THC}}^-\), topological negative cyclic homology
- See Connes’ negative cyclic homology
-
Dennis trace is invariant under all covering maps of circles, even multisheeted
- Encoded not in a group action by a cyclotomic structure.
- Take homotopy fixed points of the cyclotomic structure on \({\operatorname{THH}}\) to get \({\operatorname{TC}}\), Topological cyclic homology
- There is a map \(K\to {\operatorname{TC}}\)
-
Theorem of Dundas-Goodwillie-McCarthy: whenever \(A\to A'\) is a nilpotent extension of connective ring spectra, \(K(A') \xrightarrow{\sim} K(A) { \underset{\scriptscriptstyle {{\operatorname{TC}}(A)} }{\times} } {\operatorname{TC}}(A')\)
Eilenberg-MacLane spaces
- Some good stuff from Akhil Mathew on EM spaces:
Why Care About References?
- Why should I care about stacks? #unanswered_questions
-
Why should I care about derived stacks? #unanswered_questions
-
Note from Arun: one can get [[Topological modular forms|TMF]] and [[Topological modular forms|tmf]] along with their ring structures without doing Obstruction Theory
Homotopy Theory is Connected to Lie algebra cohomology
schemes and class field theory
- Definitions of schemes and scheme-y curves
- Definitions of schemes and scheme-y curves
- Grothendieck’s fundamental group
- Statement of class field theory in terms of fundamental groups
-
Idele group for arithmetic schemes
-Actual class group for schemes
-
Wiesend’s finiteness theorem is one of the strongest and most beautiful results in higher Global class field theory?
-
The main aim of higher Global class field theory is to determine the abelian fundamental group \(\pi_1^{{\operatorname{ab}}}(X)\) of a regular arithmetic scheme \(X\), i.e. of a connected regular scheme separated scheme scheme flat morphism and of finite type over \({\mathbb{Z}}\), in terms of an arithmetically defined class groups \(C(X)\).
-
Fundamental theorem of class field theory?