All unlabeled numbers refer to the Bhatt-Lurie paper. Notes from talks can be found by clicking the drop down boxes. For the most part, we will try to follow the sample schedule of the 22/23 Regensburg seminar on prismatic cohomology.
Motivation for the seminar, review of Witt vectors, definition of delta-rings, definition of a prism and different types of prisms, examples of prisms given in [BS, 3], prismatic site
Notes (notes taken by Giacomo, minor corrections and edits by Elmo). For more on the Witt vectors, see e.g. these notes by Emerton.
Introduce Breuil-Kisin twists, explain the prismatic logarithm and transversal approximation, extend the latter to general prisms, show the example of the q-de Rham prism, discuss the prismatic logarithm on Tate modules
Notes by Giacomo
Define Cartier-Witt divisors and the Cartier-Witt stack, prove Proposition 3.3.5, define Breuil-Kisin twists on WCart, introduce Hodge-Tate ideal sheaf and the Hodge-Tate divisor, prove Theorem 3.4.13
Notes by Christina
Discuss complexes on the Hodge-Tate divisor, identify complexes on the Hodge-Tate divisor with complexes equipped with a Sen operator (3.5.8), prove fully faithfulness (3.5.8), explain why the second part of (3.5.8) is true, introduce the Frobenius on WCart, relate the Frobenius and the Hodge-Tate divisor, compare the q-de Rham prism and WCart
Notes by David K.
Review classical p-adic cohomology theories on schemes: review derived Hodge cohomology (Appendix A,B), introduce the Beilinson t-structure and filtered complexes (Appendix D), review the de Rham complex and its Hodge and conjugate filtrations (Appendix E), review crystalline cohomology (Appendix F)
Notes + this lecture by Kedlaya for lemmas on Witt-vector tilting adjunctions.
Construct relative prismatic cohomology (4.1.3), construct the Hodge-Tate complex (4.1.6), prove étale descent for relative prismatic cohomology, discuss the global case (4.2.1), prove derived descent (4.2.8), review quasisyntomic morphisms (Appendix C), explain quasisyntomic descent and prove (4.3.13)
Notes (we didn't have time to cover the quasisyntomic topology in the lecture)
Introduce absolute prismatic cohomology (4.4), define the absolute prismatic site of a scheme (4.4.27), discuss (4.4.30), introduce absolute Hodge-Tate cohomology (4.5), prove (4.5.8) and (4.5.10)
Notes by Michele
Introduce (4.6.1), discuss its implications and sketch a proof, construct the diffracted Hodge complex (4.7.1, 4.7.2), discuss (4.7.5), introduce the topics of (4.8) and prove (4.8.8)
Skipped: the case for schemes (4.9), the role of the Sullivan fracture square and the various descent results (4.9.6, 4.9.7, 4.9.13)
Introduce the Nygaard filtration (5.0, 5.1), compare the Nygaard filtration and the Hodge filtration (5.2.3, 5.2.5), introduce the crystalline Nygaard filtration (5.3), introduce the absolute de Rham comparison (5.4), construct the absolute Nygaard filtration and describe the associated graded and its properties (5.5), define the relative Frobenius (5.7.1, 5.7.5) and define Nygaard completion (5.8)
Review the construction in (6.1), discuss and sketch proofs of (6.2.4, 6.2.8) and along the way discuss (6.2.6, 6.2.9, 6.2.10, 6.2.11), introduce the HKR filtration (6.3), discuss the material in (6.4)