We introduce the notion of a prism, which may be regarded as a "deperfection" of the notion of a perfectoid ring. Using prisms, we attach a ringed site -the prismatic site -to a p-adic formal scheme. The resulting cohomology theory specializes to (and often refines) most known integral p-adic cohomology theories.As applications, we prove an improved version of the almost purity theorem allowing ramification along arbitrary closed subsets (without using adic spaces), give a co-ordinate free description of q-de Rham cohomology as conjectured by the second author in [Sch17], and settle a vanishing conjecture for the p-adic Tate twists Zp(n) introduced in [BMS19].(2) (BMS1) Let C/Q p be an algebraically closed complete extension with ring of integers O C , and letwe defined a complex of A inf -modules RΓ A inf (X) equipped with a "Frobenius" operator φ. We shall prove in §17 that there exists a canonical φ-equivariant isomorphismNote that in this case φ A is an isomorphism, so the pullback appearing above merely twists the A-module structure. The comparison results from [BMS18] are then immediate consequences of the comparison results above (except for the results on de Rham-Witt theory in [BMS18] that we have not taken up here); note that the A crys -comparison theorem from [BMS18] follows from Theorem 1.8 (1) thanks to Theorem 1.8 (5) applied to the map (A inf , ker(θ)) → (A crys , (p)) of bounded prisms.Remark 1.10. Our search for the prismatic site was substantially motivated by a desire to obtain a site-theoretic construction of the A inf -cohomology. The "Frobenius" operator on this cohomology had slightly mysterious origins: it came, rather indirectly, via the tilting equivalence for perfectoids. Thanks to the relation to prismatic cohomology, the origins of this operator are now clear: it comes from the Frobenius lift on O ∆ .(3) (Breuil-Kisin, BMS2) Let K/Q p be a complete discretely valued field with perfect residue field k and uniformizer π, and let A = S = W (k) [[u]] with I ⊂ A the kernel of the map A → O K sending u to π. Then in [BMS19] we defined a complex of A-modules RΓ S (X) equipped with a Frobenius, using topological Hochschild homology. In §15, we construct a canonical φ-equivariant isomorphism RΓ S (X) ∼ = RΓ ∆ (X/S) . The above comparison results recover the results of [BMS19] on Breuil-Kisin cohomology.3 Beware that in general the completion of M is not given by R limn M ⊗ L A A/I n ; however, this happens if I is generated by a regular sequence, or if A is noetherian (by Artin-Rees).

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We prove that the Witt vector affine Grassmannian, which parametrizes W (k)-lattices in W (k)[ 1 p ] n for a perfect field k of charactristic p, is representable by an ind-(perfect scheme) over k. This improves on previous results of Zhu by constructing a natural ample line bundle. Along the way, we establish various foundational results on perfect schemes, notably h-descent results for vector bundles.Theorem 1.2. Any vector bundle E on a perfect F p -scheme X gives a sheaf for the h-topology on perfect schemes over X via pullback, and one has H i h (X, E) ≃ H i (X, E) for all i. Moreover, one has effective descent for vector bundles along h-covers of perfect schemes.The first part of this theorem is due to Gabber, cf. [BST13, §3]; the second part, in fact, extends to the full derived category (see §11). Using this descent result, we can informally describe our first construction of L, which is K-theoretic. The idea here is simply that one can often (e.g., when R is a field, or always after an h-cover) filter p a W (R) n /M in such a way that all gradeds Q i are finite projective R-modules, and then defineThe existence of this line bundle was already conjectured by Zhu.2 the MSRI, and the authors would like to thank him for asking the question on the existence of L. They would also like to thank Akhil Mathew for enlightening conversations related to §11.2. Moreover, they wish to thank all the participants of the ARGOS seminar in Bonn in the summer term 2015 for their careful reading of the manuscript, and the many suggestions for improvements and additions. The first version of this preprint contained an error in the proof of Lemma 4.6, tracing back to an error in [GD61, Corollary 3.3.2]; we thank Christopher Hacon, and Linquan Ma (via Karl Schwede) and the anonymous referee for pointing this out. The authors are also indebted to the referee for providing numerous other comments that improved the readability of this paper. Finally, they would like to thank the Clay Mathematics Institute, the University of California (Berkeley), and the MSRI for their support and hospitality. This work was done while B. Bhatt was partially supported by NSF grant DMS 1340424 and P. Scholze was a Clay Research Fellow. h-SHEAVESIn this section, we recall some general facts about sheaves on the h-topology defined by Voevodsky, [Voe96, §3]. We use results of Rydh, [Ryd10], in the non-noetherian case. In the following, all schemes are assumed to be qcqs for simplicity. Let us start by recalling the notion of universally subtrusive morphisms.Definition 2.1. A morphism f : X → Y of qcqs schemes is called universally subtrusive or a v-cover if for any map Spec(V ) → Y , with V a valuation ring, there is an extension V ֒→ W of valuation rings and a commutative diagram Spec(W )/ / X f

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