Cyclic and non-commutative geometry in positive characteristic

Devarshi Cyclic homology and non-commutative Mukherjee geometry in positive characteristic

Devarshi Mukherjee

joint with Guillermo Corti˜nasand Ralf Meyer

1Mathematisches Institut Universit¨atG¨ottingen

YMC*A, WWU M¨unster,2021 Cyclic homology Goals of the program and non-commutative geometry in positive characteristic

Devarshi Mukherjee

I Develop non-commutative geometry in positive characteristic, and over Zp and Qp; I Find a non-commutative generalisation of the appropriate de Rham theory for affine varieties over Fp. Cyclic homology Why is this interesting? - I and non-commutative geometry in positive characteristic

Devarshi Mukherjee

I Group actions on affine varieties over Fp crossed product algebras; I Graphs Leavitt path algebras; I Carry over tools from rigid geometry to non-commutative geometry. Cyclic homology Why is this interesting? - II and non-commutative geometry in positive characteristic I A fundamental invariant of a smooth manifold M is its Devarshi Mukherjee de Rham hdR∗(M); I These are the cohomology groups of the cochain complex

0 → C∞(M) →d Ω1(M) → · · · →d Ωn(M),

I is an invariant of associative algebras that generalises differential forms. ∞ ∼ k That is, HHk (C (M)) = Ω (M), I Periodic cyclic homology is a non-commutative generalisation of : HP (C∞(M)) ∼ L hdRn+2j (M), n = 0, 1. n = j∈Z Cyclic homology Why is this interesting? - II (continued) and non-commutative geometry in positive characteristic

Devarshi Mukherjee I Periodic cyclic homology conveys important information about ‘topological’ algebras, through its link with (bivariant) K-theory; I Want similar information about topological algebras over Qp and Zp; I In the commutative world, we want a similar link with de Rham cohomology as in the smooth manifold case, but for affine varieties over Fp; I What are the right topological algebras and invariants in this setting? Cyclic homology What goes wrong in positive characteristic? and non-commutative geometry in positive characteristic

Devarshi Mukherjee

I De Rham cohomology is homotopy invariant - so contractible spaces have no cohomology; I The proof of this relies on integration of differential forms - this leads to denominators; I But in positive characteristic, you are not allowed to divide; I So (algebraic) de Rham cohomology is undesirable in positive characteristic. Cyclic homology The correct de Rham theory in positive and non-commutative geometry in characteristic positive characteristic

Devarshi Mukherjee I Lift a smooth commutative Fp-algebra A to a smooth commutative Zp-algebra R; I Obvious things - take the de Rham cohomology of R or its p-adic completion Rb - these fail; I Instead, take something in between: R ⊆ R† ⊆ Rb. It consists of power series, whose coefficients converge somewhat rapidly; † † I The de Rham cohomology of R := R ⊗Zp Qp is the right thing to consider - called rigid cohomology; I Corti˜nas-Cuntz-Meyer-Tamme, 2017: HP (R†) ∼ L hdRn+2j (R/pR), n = 0, 1 n = j∈Z Cyclic homology Towards local and analytic cyclic homology - I and non-commutative geometry in positive characteristic

Devarshi I In the non-commutative world, it is not obvious what Mukherjee R ⊆ R† ⊆ Rb should mean; I We do this by making sense of convergent power series in non-commuting variables - uses bornological analysis; I It is inspired by the definition of a smooth subalgebra A∞ ⊆ A of a C ∗-algebra, which generalises C∞(M) ⊆ C(M); I We would like an invariant X that satisfies X (R†) =∼ X (Rb); I In the complex case, X = HL(=HA) does the job and provides a powerful invariant for C ∗-algebras. Cyclic homology Towards local and analytic cyclic homology - II and non-commutative geometry in positive characteristic

Devarshi I We additionally want our invariant to be independent of Mukherjee choices of liftings of an Fp-algebra to torsion-free Zp-algebras; I It is unclear that HP has such properties. Theorem (Corti˜nas-Meyer-Mukherjee) We construct a functor   HA: Fp-algebras −→ Qp-vector spaces

that satisfies homotopy invariance, excision, Morita invariance, independence of choices of liftings, etc. Cyclic homology Some computations and non-commutative geometry in positive characteristic

Devarshi Mukherjee

I For smooth curves over Fp, our theory recovers rigid cohomology; I For Leavitt path algebras, we recover the computation of HP get something which only depends on the incidence matrix of the graph. Cyclic homology Summary and non-commutative geometry in positive characteristic

Devarshi Mukherjee I Over C, periodic cyclic homology generalises de Rham cohomology;

I Over Fp, periodic cyclic homology of a Zp-algebra lift of † † the form pR  R  A yields the rigid cohomology of A; I We define a well-behaved invariant that is independent of choices of ‘liftings’ of A to torsion-free Zp-algebras; I When A is the coordinate ring of a smooth curve over Fp, our invariant agrees with rigid cohomology.