Notes on prismatic cohomology

P. Achinger, “Wild ramification and \(K(\pi, 1)\) spaces”, Inventiones Mathematicae 210 (2017), 453–499.

G. Almkvist, “The Grothendieck ring of the category of endomorphisms”, Journal of Algebra 28 (1974), 375–388.

V. Angeltveit, “The norm map of Witt vectors”, Comptes Rendus Mathematique 353 (2015), 381–386.

Y. André, “Le lemme d'Abhyankar perfectoïde”, Publications Mathématiques de l’IHÉS 127 (2018), 1–70.

Y. André, “La conjecture du facteur direct”, Publications Mathématiques de l’IHÉS 127 (2018), 71–93.

J. Anschütz and A.C. Le Bras, “Prismatic Dieudonné theory”, arXiv:1907.10525v3⁶ (2021).

K. Aomoto, “\(q\)-analogue of de Rham cohomology associated with Jackson integrals”, Proceedings of the Japan Academy Series A 66 (1990), 161–164.

K. Aomoto, “\(q\)-analogue of de Rham cohomology associated with Jackson integrals, II”, Proceedings of the Japan Academy Series A 66 (1990), 240–244.

A. Arabia, “Relèvements des algèbres lisses et de leurs morphismes”, Commentarii Mathematici Helvetici 76 (2001), 607–639.

S. Ariotta, “Coherent cochain complexes and Beilinson \(t\)-structures, with an appendix by Achim Krause”, arXiv:2109.01017v1⁷ (2021).

J. Ax, “Zeros of polynomials over local fields—the Galois action”, Journal of Algebra 15 (1970), 417–428.

C. Barwick, “Euler's Gamma function and the field with one element”, lecture notes (MIT, 2017)⁸.

L. Berger, “Limites de représentations cristallines”, Compositio Mathematica 140 (2004), 1473–1498.

P. Berthelot, “Generalités sur les \(\lambda\)-anneaux”, in Séminaire de Géométrie Algébrique (SGA 6): Théorie des Intersections et Théorème de Riemann-Roch, Lecture Notes in Mathematics 225, Springer-Verlag, Berlin, 1971.

P. Berthelot and A. Ogus, Notes on Crystalline Cohomology, Princeton University Press, Princeton (1978).

B. Bhatt, “On the direct summand conjecture and its derived variant”, Inventiones Mathematicae 212 (2018), 297–317.

B. Bhatt, “Cohen-Macaulyness of absolute integral closures”, arXiv:2008.08070v1⁹ (2020).

B. Bhatt, “Geometric aspects of prismatic cohomology”, Eilenberg lectures at Columbia University (fall 2018)¹⁰.

B. Bhatt and A.J. de Jong, “Crystalline cohomology and de Rham cohomology”, www.math.columbia.edu/~dejong/papers/crystalline-comparison.pdf.

B. Bhatt, S. Iyengar, and L. Ma, “Regular rings and perfect(oid) algebras”, Communications in Algebra 47 (2019), 2367–2383.

B. Bhatt and A. Mathew, “The arc-topology”, Duke Mathematical Journal 170 (2021), 1899–1988.

B. Bhatt, M. Morrow, and P. Scholze, “Integral \(p\)-adic Hodge theory”, Publications Mathématiques de l'IHÉS 128 (2018), 219–397.

B. Bhatt, M. Morrow, and P. Scholze, “Topological Hochschild homology and integral \(p\)-adic Hodge theory”, arXiv:1802.03261v2¹¹ (2019).

B. Bhatt and P. Scholze, “Projectivity of the Witt vector affine Grassmannian”, Inventiones Mathematicae 209 (2017), 329–423.

B. Bhatt and P. Scholze, “Prisms and prismatic cohomology”, arXiv:1905.08229v2¹² (2019).

B. Bhatt and P. Scholze, “Prismatic \(F\)-crystals and crystalline Galois representations”, arXiv:2106.14735v1¹³ (2021).

B. Bhatt, K. Schwede, and S. Takagi, “The weak ordinarity conjecture and \(F\)-singularities”, Advanced Studies in Pure Mathematics 74 (2017), 11–39.

J. Borger, “The basic geometry of Witt vectors, I: The affine case”, Algebra and Number Theory 5 (2011), 231–285.

J. Borger, “The basic geometry of Witt vectors, II: Spaces”, Mathematische Annalen 351 (2011), 871–933.

J. Borger, “Witt vectors, lambda-rings, and arithmetic jet spaces”, lecture notes (Copenhagen, spring 2016)¹⁴.

J. Borger and B. Wieland, “Plethystic algebra”, Advances in Mathematics 194 (2005), 246–283.

M. Borovoi and Y. Cornulier, “Conjugate complex homogeneous spaces with non-isomorphic fundamental groups”, C. R. Acad. Sci. Paris 353 (2015), 1001–1005.

O. Brinon and B. Conrad, CMI Summer School Notes on \(p\)-adic Hodge Theory, authors' draft¹⁵.

A. Buium, “Differential characters of abelian varieties over \(p\)-adic fields”, Inventiones Mathematicae 122 (1995), 309–340.

A. Buium, “Arithmetic analogues of derivations”, Journal of Algebra 198 (1997), 290–299.

B. Cais and C. Davis, “Canonical Cohen rings for norm fields”, International Mathematics Research Notices (2014), 10.1093/imrn/rnu098.

B. Cais and T. Liu, “On \(F\)-crystalline representations”, Documenta Mathematica 21 (2016), 223–270.

K. Česnavičius and T. Koshikawa, “The \(A_{\mathrm{inf}}\)-cohomology in the semistable case”, Compositio Mathematica 155 (2019), 2039–2128.

F. Cherbonnier and P. Colmez, “Représentations \(p\)-adiques surconvergentes”, Inventiones Mathematicae 133 (1998), 581–611.

C. Davis and K.S. Kedlaya, “On the Witt vector Frobenius”, Proceedings of the American Mathematical Society 142 (2014), 2211–2226.

V. Drinfeld, “Coverings of \(p\)-adic symmetric regions” (in Russian), Functional Analysis and Its Applications 10 (1976), 29–40.

V. Drinfeld, “Prismatization”, arXiv:2005.04746v2¹⁶ (2021).

V. Drinfeld, “A stacky approach to crystalline (and prismatic) cohomology”, video¹⁷ (2019).

A. Ducros, “About Hrushovski and Loeser's work on the homotopy type of Berkovich spaces”, in Nonarchimedean and Tropical Geometry, Simons Symposia, Springer International (2016), 99–131.

R. Elkik, “Solutions d'équations à coefficients dans un anneau hensélien”, Annales Scientifiques de l'École Normale Superieure 6 (1973), 553–603.

L. Fargues and J.-M. Fontaine, “Courbes et fibrés vectoriels en théorie de Hodge \(p\)-adique”, Astérisque 406 (2018).

L. Fargues and P. Scholze, “Geometrization of the local Langlands correspondence”, arXiv:2102.13459v1¹⁸ (2021).

N.J. Fine, “Basic Hypergeometric Series and Applications”, Mathematical Series and Monographs 27, American Mathematical Society, Providence (1988).

J.-M. Fontaine, “Représentations \(p\)-adiques des corps locaux, I”, in The Grothendieck Festschrift, volume II, Progress in Mathematics 87, Birkhäuser, Boston (1990), 249–309.

J.-M. Fontaine, “Perfectoïdes, presque pureté et monodromie-poids (d’après Peter Scholze)”, Séminaire Bourbaki, volume 2011/2012, Astérisque 352 (2013).

J.-M. Fontaine and Y. Ouyang, Theory of \(p\)-adic Galois Representations, authors' draft¹⁹.

J.-M. Fontaine and J.-P. Wintenberger, “Le “corps des normes” de certaines extensions algébriques de corps locaux”, C. R. Acad. Sci. Paris Sér. A B 288 (1979), 367–370.

J. Fresnel and B. de Mathan, “Algèbres \(L^1\) \(p\)-adiques”, Bulletin de la Société Mathématique de France 106 (1978), 225–260.

O. Gabber and L. Ramero, Almost Ring Theory, Lecture Notes in Mathematics 1800, Springer-Verlag, Berlin (2003).

O. Gabber and L. Ramero, “Foundations for almost ring theory–Release 7.5”, arXiv:0409584v13²⁰ (2018).

R. Ghrist, “Three examples of applied and computational homology”, Nieuw Archief voor Wiskunde 9 (2008), 122–125.

D. Grayson, “Grothendieck rings and Witt vectors”, Communications in Algebra 6 (1978), 249–255.

D. Grayson, “The \(K\)-theory of endomorphisms”, Journal of Algebra 48 (1977), 439–446.

P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, New York (1978).

A. Grothendieck, “On the de Rham cohomology of algebraic varieties”, Publications Mathématiques de l’IHÉS 29 (1966), 95–103.

A. Grothendieck, “Crystals and the de Rham cohomology of schemes”, in Dix Exposés sur la Cohomologie des Schemas, North-Holland, Amsterdam (1968), 306–358.

M. Hazewinkel, Formal Groups and Applications, Pure and Applied Mathematics 78, Academic Press, New York (1978).

R. Heitmann and L. Ma, “Extended plus closure in complete local rings”, arXiv:1708.05761v3²¹ (2018).

E. Heine, “Über die Reihe \(1 + \frac{(q^\alpha-1)(q^\beta-1)}{(q-1)(q^\gamma-1)} x + \frac{(q^\alpha-1)(q^{\alpha+1}-1)(q^\beta-1)(q^{\beta+1}-1)}{(q-1)(q^2-1)(q^\gamma-1)(q^{\gamma+1}-1)} x^2 + \cdots\)”, Journal für de reine und angewandte Mathematik 32 (1846), 210–212.

L. Hesselholt, “On the topological cyclic homology of the algebraic closure of a local field”, in An Alpine Anthology of Homotopy Theory, Contemporary Mathematics 399, American Mathematical Society, Providence (2006), 133–162.

M. Hochster, “Homological conjectures, old and new”, Illinois Journal of Mathematics 51 (2007), 151–169.

R. Huber, Étale Cohomology of Rigid Analytic Varieties and Adic Spaces, Springer, Wiesbaden (1996).

L. Illusie, Complexe Cotangent et Déformations I, Lecture Notes in Mathematics 239, Springer-Verlag, Berlin (1971).

L. Illusie, Complexe Cotangent et Déformations II, Lecture Notes in Mathematics 283, Springer-Verlag, Berlin (1972).

R. Ishizuka, “A calculation of the perfectoidization of semiperfectoid rings”, Nagoya Mathematical Journal, to appear.

F.H. Jackson, “On \(q\)-functions and a certain difference operator”, Transactions of the Royal Society of Edinburgh 46 (1908), 253–281.

F.H. Jackson, “\(q\)-difference equations”, American Journal of Mathematics 32 (1910), 305–314.

N. Jacobson, Basic Algebra, II, W. H. Freeman, San Francisco, (1980).

A. Joyal, “\(\delta\)-anneau et vecteurs de Witt”, C.R. Math. Rep. Acad. Sci. Canada 7 (1985), 177–182.

A. Joyal, “\(\delta\)-anneau et \(\lambda\)-anneaux, II”, C.R. Math. Rep. Acad. Sci. Canada 7 (1985), 227–232.

R. Jurrius and R. Pellikaan, “Defining the \(q\)-analogue of a matroid”, arXiv:1610.09250v1²² (2016).

K. Kato, “Logarithmic structures of Fontaine-Illusie”, in Algebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore (1989), 191–224.

N.M. Katz, “Nilpotent connections and the monodromy theorem: applications of a result of Turrittin”, Publications Mathématiques de l’I.H.É.S 39 (1970), 175–232.

K.S. Kedlaya, “Nonarchimedean geometry of Witt vectors”, Nagoya Mathematical Journal 209 (2013), 111–165.

K.S. Kedlaya, “New methods for \((\varphi, \Gamma)\)-modules”, Research in the Mathematical Sciences 2 (2015).

K.S. Kedlaya, “Sheaves, stacks, and shtukas”, in Perfectoid Spaces: Lectures from the 2017 Arizona Winter School, Mathematical Surveys and Monographs 242, American Mathematical Society, Providence (2019).

K.S. Kedlaya and R. Liu, “Relative \(p\)-adic Hodge theory: Foundations”, Astérisque 371 (2015).

The Kerodon Authors, Kerodon, kerodon.net.

R. Kiehl and R. Weissauer, Weil Conjectures, Perverse Sheaves, and \(\ell\)-adic Fourier Transform, Ergebnisse der Mathematik 42, Springer-Verlag, Berlin (2001).

M. Kisin, “Crystalline representations and \(F\)-crystals”, in Algebraic geometry and number theory, Progress in Mathematics 253, Birkhäuser Boston, Boston, MA (2006), 459–496.

D. Knutson, \(\lambda\)-Rings and the Representation Theory of the Symmetric Group, Lecture Notes in Mathematics 308, Springer, Berlin (1973).

M. Kontsevich and D. Zagier, “Periods”, in Mathematics Unlimited—2001 and Beyond, Springer, Berlin (2001), 771–808.

T. Koshikawa, “Logarithmic prismatic cohomology I”, arXiv:2007.14037v2²³ (2021).

E. Lau, “Dieudonné theory over semiperfect rings and perfectoid rings”, arXiv:1603.07831v2²⁴ (2016).

E. Lau, “Divided Dieudonné crystals”, arXiv:1811.09439v1²⁵ (2018).

V. Lafforgue, “Chtoucas pour les groups réductifs et paramétrisaton de Langlands globale”, Journal of the American Mathematical Society 31 (2018), 719–891.

W.E. Lang, “Two theorems on de Rham cohomology”, Compositio Mathematica 40 (1980) 417–423.

J. Lurie, Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press, Princeton (2009).

S. Mac Lane, Categories for the Working Mathematician, second edition, Graduate Texts in Mathematics 5, Springer, New York (1978).

A.M. Masullo, Arithmetic deformations of crystalline cohomology, PhD thesis, Stanford University, (2019), available via Stanford Digital Repository²⁶.

A. Mathew, “Simplicial commutative rings, I”, math.uchicago.edu/~amathew/SCR.pdf.

M. Matignon and M. Reversat, “Sous-corps fermés d'un corps valué”, Journal of Algebra 90 (1984), 491–515.

J.S. Milne and J. Suh, “Nonhomeomorphic conjugates of connected Shimura varieties”, American Journal of Mathematics 132 (2010), 731–750.

J. Nicaise, “Berkovich Skeleta and Birational Geometry”, in Nonarchimedean and Tropical Geometry, Simons Symposia, Springer International (2016), 173–194.

J.P. Pridham, “On \(q\)-de Rham cohomology via \(\Lambda\)-rings”, Mathematische Annalen 375 (2019), 425–452.

D.G. Quillen, Homotopical Algebra, Lecture Notes in Mathematics 43, Springer-Verlag, Berlin, 1967.

C.S. Rajan, “An example of non-homeomorphic conjugate varieties”, Mathematics Research Letters 18 (2011), 937–942.

M. Raynaud and L. Gruson, “Critères de platitude et de projectivité. Techniques de “platification” d'un module”, Inventiones Mathematicae 13 (1971), 1–89.

C. Rezk, “Etale extensions of \(\lambda\)-rings”, (2019).

P. Roberts, “The root closure of a ring of mixed characteristic”, arXiv:0810.0215v1²⁷ (2008).

D. Rydh, “Submersions and effective descent of étale morphisms”, Bulletin de la Société Mathématique de France 138 (2010), 181–230.

P. Scholze, “Perfectoid spaces”, Publications Mathématiques de l’IHÉS 116 (2012), 245–313.

P. Scholze, “On torsion in the cohomology of locally symmetric spaces”, Annals of Mathematics 182 (2015), 945–1066.

P. Scholze, “Canonical \(q\)-deformations in arithmetic geometry”, Annales de la Faculté des sciences de Toulouse 26 (2017), 1163–1192.

P. Scholze, “Étale cohomology of diamonds”, arXiv:1709.07343v2²⁸ (2021).

P. Scholze, “Prismatic crystals and crystalline Galois representations”, RAMpAGe seminar video²⁹ and notes³⁰ (2020).

P. Scholze and J. Weinstein, “Moduli of \(p\)-divisible groups”, Cambridge Journal of Mathematics 1 (2013), 145–237.

P. Scholze and J. Weinstein, Berkeley Lectures on \(p\)-adic Geometry, Annals of Mathematics Studies 207, Princeton University Press, Princeton (2020).

J.-P. Serre, “Géométrie algébrique et géométrie analytique”, Annales de l’institut Fourier 6 (1956), 1–42.

J.-P. Serre, “Exemples de variétés projectives conjuguées non homéomorphes”, C. R. Acad. Sci. Paris 258 (1964), 4194–4196.

J.-P. Serre, Local Fields, Graduate Texts in Mathematics 67, Springer-Verlag, New York-Berlin, (1979).

The Stacks Project Authors, Stacks Project, stacks.math.columbia.edu.

R. Swan, “On seminormality”, Journal of Algebra 67 (1980), 210–229.

J. Milnor and J.D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton University Press, Princeton (1974).

D.O. Tall and G.C. Wraith, “Representable functors and operations on rings”, Proceedings of the London Mathematical Society 20 (1970) 619–643.

J. Thomae, “Über die höheren hypergeometrische Reihen”, Mathematische Annalen 2 (1870), 427–444.

W. van der Kallen, “Descent for the \(K\)-theory of polynomial rings”, Mathematische Zeitschrift 191 (1986), 405–415.

V. Voevodsky, “Homology of schemes”, Selecta Mathematica 2 (1996), 111–153.

N. Wach, “Représentations \(p\)-adiques potentiellement cristallines”, Bull. Soc. Math. France 124 (1996), 375–400.

C. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press (1994).

C. Wilkerson, “Lambda-rings, binomial domains, and vector bundles over \(\mathbf{C}P(\infty)\)”, Communications in Algebra 10 (1982), 311–328.

E. Witt, “Zyklische Körper und Algebren der Charakteristik \(p\) vom Grad \(p^n\text{.}\) Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik \(p\)”, Journal für die reine und angewandte Mathematik 176 (1937), 126–140.

D. Yau, Lambda-rings, World Scientific, Hackensack (2010).

A. Yekutieli, “On flatness and completion for infinitely generated modules over Noetherian rings”, Communications in Algebra 39 (2011), 4221–4245.

K. Yamamoto, “The Artin-Hasse-Šafarevič function”, Japanese Journal of Mathematics 29 (1959), 165–172.