Prismatic Stable Homotopy Theory, c.f. Lurie, J. (2026)

In the 2026 ICM lecture, Jacob Lurie discussed developments in last one decade or so in $p-$ adic geometry and the conjectures initiated/inspired by those developments. Lurie offers two lenses: algebraic topology and algebraic geometry to understand these lenses, starting first the the algebraic topology.

Complex $K-$ theory is an example of the invariant of topological spaces. Let $X$ be a “nice” topological space (such as, compact manifold, finite simplicial complex) with a complex vector bundle $\epsilon$ defined on it having an addition law gives by the formation of direct sums which makes a collection $[\epsilon]$ of isomorphism classes into a commutative monoid with addition $[\epsilon] + [F] = [\epsilon \oplus F]$ (but not subtract them). However, still, it’s possible to make a group by formally adjoining the inverses; definition given by the Atiyah-Hirzebruch 1960, where the complex K-theory of $X$ is the abelian group $KU^{0}(X)$ one gets after the procedure above. For example, if $X$ consists of a single point, then its vector bundle is just a vector space, and up to an isomorphism, it is classified by a dimension which is natural number and upon formally adjoining its inverses, one gets the group of integers ( $KU^{0}(\{x\}) \simeq Z$ ).

One can define a sequence of invariants for all spaces $X$ and integer $n$ given by

$X \mapsto KU^{n}(x)$

which is an example of generalised cohomology theory (properties like, long exact sequences, excisions, torus); $K$ -theory has all features except that $K$ -theory of a point is not in degree zero and it is a copy of integers in every even degree, given by the Bott periodicity (1960)

$KU^{n}(\{x\}) = Z$ if $n$ is even and it is = 0 if $n$ is odd.

In 1930s, Hopf introduced fibrations $S^{3} \mapsto S^{2}$ , $S^{7} \mapsto S^{4}$ , and $S^{15} \mapsto S^{8}$ using division algebras over $R$ , and an immediate question that appeared in 1935 was:

when does there exist a map of spheres $S^{2n-1} \mapsto S^{n}$ behaving like a Hopf fibration? This was answered by Adams 1960 in negative except for $n=2, 4, 8$ . While the original solution used secondary operations in ordinary cohomology (85 pages) but was reduced to 8 pages using complex K-theory by Adams-Atiyah in beautiful 8 pages.

The counterpart of K-theory in the world of algebraic geometry is to look at the finitely-generated projective modules over a ring (analogue of vector bundles). Let $R$ be a ring and $M$ be a finitely generated projective $R-$ module with its isomorphism class $[M]$ forming a commutative monoid with addition law $[M]+[N]= [M \oplus N]$ . This can be made a group by formally adjoining the inverses, which is the abelian group, given by $K_{0}(R)$ . For the ring which is a field, the K-group is just integers,

Algebraic K-theory is given by higher K-groups of R, $\{K_{n}(R)\}_{n>=0}$ introduced by Quillen in 1970s and appears in generalization of Riemann-Roch, K-groups of rings of integers (L-functions), surgery classification of manifolds and obstructions.

A broad question is to compute or understand the groups $K_{n}(R)$ ; for example, the higher K-groups of $Z/4 Z$ are not completely known yet (except the order of these groups due to Antieau-Krause-Nikolaus). Higher K-groups in this example are finite groups so they split into sum of odd and 2-torsion part; the latter (2-adic part) is difficult.

The cyclotomic trace of Bokstedt-Hsiang-Madsen is given by

$tr_{cyc}: K_{*}(R) \mapsto TC_{*}(R)$

which is useful because the approximation is good by some measures are computable by some measures. Note that $TC_{*}(R)$ denotes the p-complete topological cyclic homology of R.

On a bad note: the first pointer is that $TC_{*}(R)$ is p-complete and depend on p-completion of $R$ , which is not shared by algebraic K-theory. Secondly, TCH (topological cyclic homology) satisfies a strong local-to-global principle (etale descent) which K-theory does not.

On a good note: Clausen-Mathew-Morrow 2020 proves that if $R$ is p-complete, the cyclotomic trace $tr_{cyc}: K(R) \mapsto TC(R)$ exhibits as TCH as what one gets from algebraic K-theory by forcing it to have three properties (in previous paragraph).

Therefore, TCH is a good approximation to K-theory. How computable is $TC(R)$ ?

The groups $TC_{*}(R)$ extracts its mathematical content from THH(R) which is always p-complete and is an example of cyclotomic spectrum. About cyclotomic spectra, they form a tensor category (algebras, modules over them) and a triangulated category (Ext groups).

For a ring R, one can regard it over a ring of Z, and hence, $THH(R)$ is a module over $THH(Z)$ . In the world of cyclotomic modules over THH(Z),

$TC_{*}(R) := Ext^{-*}_{THH(Z)}(THH(Z), THH(R))$ .

One would be interested in understanding $THH(R)$ plus some extra work to understand $TC_{*}(R)$ .

Bokstedt Periodicity 1985 showed that $THH_{*}(F_{p})=F_{p}[b]$ with $deg(b)=2$ . Thus, if R is an $F_{p}-$ algebra (any ring where p=0), then, $THH(R)/b = HH(R/F_{p})$ . So THH is reduced modulo a generator to get a more concrete ingredient called HH (Hochschild Homology; at the level of chain complex and understod well). A key theorem here is Hochschild-Kostant-Rosenberg 1962 given by