# Prismatization and the image of the J-homomorphism

# Prismatization and the image of the J-homomorphism

For p a prime number, Bhatt–Lurie and Drinfeld defined the prismatization of a p-complete commutative ring R. I will overview a different perspective on prismatization, using homotopy theory and in particular topological Hochschild homology. This perspective enlarges the scope of the definition: namely, we may consider the prismatization of any cyclotomic E-infinity ring A; for sufficiently nice p-complete commutative rings R, we recover the previously known object by considering the case A = THH(R). I will then discuss a description of the cyclotomic E-infinity ring THH(Z_p) for odd primes p in terms of a classical object of homotopy theory, the image of the J-homomorphism, and (via the aforementioned generalization of prismatization) the relation between this description and Lurie's theory of the prismatization of the field with one element. This is all joint work, partially with Devalapurkar and partially with Devalapurkar, Hahn, and Yuan.