Lecture 21: Crystalline cohomology and the de Rham-Witt complex
|
|
- Jeffry Vincent Reynolds
- 5 years ago
- Views:
Transcription
1 Lecture 21: Crystalline cohomology and the de Rham-Witt complex Paul VanKoughnett November 12, 2014 As we ve been saying, to understand K3 surfaces in characteristic p, and in particular to rediscover the amazing theory of periods, we need some way of doing differential calculus. A weird obstruction arises here, though: if is a connection on a vector bundle in characteristic p, v is a horizontal section of the vector bundle, and x is a function, then we have (x p v) = px p 1 v dx + x p (v) = 0. (1) In a certain sense, this means there are too many horizontal sections on a vector bundle with connection. This is connected to a lot of pathologies in characteristic p key among them is the fact that the de Rham cohomology of a variety X, defined naïvely as H (Ω X/k ), vanishes above degree d rather than above degree 2d. (Elden adroitly asks how pathology 1 and pathology 2 above are connected. I don t know.) The historical work-around to this problem, and also to the problem of defining l-adic cohomology when l = p, is crystalline cohomology. The most basic idea is that the issue with (1) is resolved if we can divide x p by p. This might look like straight-up reversion to characteristic zero, but it s actually weaker! For starters, not everything is a pth power; moreover, we typically don t even need to do this for all x e. g. thinking about a relative de Rham complex Ω X/Y for X Y a closed immersion, we ll only need to worry about x coming from the sheaf of ideals cutting out X from Y. 1 Divided powers Definition 1. Let (A, I) be a pair of a ring A and an ideal I in A. A divided power structure or PD-structure (puissances divisées) γ on (A, I) is a set of functions γ n : I A for n N satisfying the following properties (for a A, x, y I, m, n 0): 1. γ 0 (x) = 1, and γ n (x) I for n γ 1 (x) = x. 3. γ n (x + y) = i+j=n γ i(x)γ j (y). 4. γ n (ax) = a n γ n (x). 5. γ n (x)γ m (x) = ( m+n n ) γm+n (x). 6. γ m (γ n (x)) = (mn)! m!() m γ mn (x). We re supposed to think that γ n (x) = xn. One should note that, by 3 or 5 above, we do have γ n (x) = x n. As a result, in a Q-algebra, γ n (x) = xn is the unique divided power structure on any ideal. In a ring of characteristic p, on the other hand, this fact means that we can only define a divided power structure on an ideal I which is p-nil, meaning that x p = 0 for x I. 1
2 Example 2. Let R be a mixed-characteristic DVR with maximal ideal m this means that 0 p m. If m has uniformizer π, then p has some π-adic valuation e. A PD-structure on m is entirely specified by what it does to π, by 4 above. We can (uniquely) define it iff π n 1 is divisible by for each n (since we must have π n (π)). One quickly reduces to checking this when n = p r. In this case, v p (p r!) = pr 1 p 1. So v π (p r!) = e pr 1 p 1. This is less than or equal to pr 1 iff e p 1. Thus, there is a unique PD-structure on a mixed-characteristic DVR with ramification index at most p 1. In number theory, these are called tamely ramified. Example 3. Given an A-module M, we can define a free PD-algebra (Γ(M), Γ + (M)) on M. This is the A-algebra generated by symbols {γ n (m) : n 1, m M} subject to the above relations. It is graded, with γ n (m) in degree n, and the ideal of elements of positive degree has a PD-structure. In particular, letting M = A r on a basis {x 1,..., x r } we get a PD-polynomial algebra A x 1,..., x r. One defines morphisms of PD-structures in the obvious way. There s a slightly more opaque idea of two PD-structures on the same ring being compatible, but I won t go over this. It s not hard to check that PDstructures localize well, so we can define them on sheaves of ideals on schemes. This allows us to introduce the crystalline site. 2 The crystalline site Definition 4. Let S be a base scheme on which p is locally nilpotent, together with a sheaf of ideals I and a PD-structure γ on I, and let X S be an S-scheme. The crystalline site Cris(X/S) has, as objects, the pairs U V, γ X where U X is an open immersion, U V is a closed immersion with locally nilpotent sheaf of ideals J, γ is a PD-structure on J, and γ is compatible with the PD-structure on S. (In practice, S will be the finite-length Witt vectors over a perfect field, and this compatibility condition will be trivial to check.) I ll typically write just [U V ] for such an object, but remember that the PD-structure is part of the data, too. Such objects are called PD-thickenings. The morphisms are the maps of diagrams that preserve the PD-structure. Finally, a set of maps {[U i V i ] [U V ]} is a cover if {V i V } is a Zariski cover. Remark 5. Note that the Zariski site is of X is used as an underlying site on which the crystalline site is built. As far as I m aware, PD-structures are similarly well-behaved with respect to étale localization and so on, and so one could build étale-crystalline sites and so on. I don t know what you d do with them. The crystalline topos (X/S) cris is the category of sheaves on the above site. In practice, one can handle a crystalline sheaf F as follows. By the above definition of cover, F defines a Zariski sheaf F V on V in each PD-thickening [U V ], via S F V (W V ) = F ([U V W W ]). These Zariski sheaves are compatible: given a map of PD-thickenings U V g U V, 2
3 there s a map of Zariski sheaves on V, g F V F V. Finally, if the above diagram is a pullback, one can check that the map g F V F V is an isomorphism. Conversely, a system of Zariski sheaves on PD-thickenings with this compatibility property precisely gives you a crystalline sheaf. A key example of a crystalline sheaf: the structure sheaf O X/S : [U V ] O V (V ). The associated Zariski sheaf on U V is just O V. There s a functor u : Cris(X/S) X from the crystalline site to the Zariski site of X, sending [U V ] to the Zariski open U. A functor of sites defines a geometric morphism of toposes, under certain conditions: specifically, u should preserve sheaves, and u should preserve finite limits. In this case, for a Zariski sheaf G on X, we have u G [U V ] = G (u[u V ]) = G (U), which certainly defines a crystalline sheaf. For [U V ], the Zariski sheaf (u G ) V is just G U, regarded as a sheaf on V, which has the same underlying topological space. To check that u preserves finite limits, we have to check that forgetting PD-structures preserves finite limits, which is true because PD-structures are algebraic. Thus, we get a morphism of toposes: u : X Zar (X/S) cris : u. I just described the pullback functor u. For the pushforward, one has (u F )(U) = F [U U] with the trivial PD-structure on the zero ideal. The point of all this is the following. The cohomology of an object in a topos say, O X/S in (X/S) cris can be computed as the right derived pushforward of that object to the terminal topos, that is, the point. In this case, the pushforward map to the terminal topos factors through the Zariski site of X. Thus, we can compute crystalline cohomology as H cris(x/s) = H (O X/S ) = H Zar(Ru (O X/S ), the hypercohomology of a complex of sheaves on the ordinary Zariski site X. So in practice, we ll be spending much of our time trying to understand the functor Ru. As it turns out, there are at least two nice ways to compute this. One is via the de Rham-Witt complex, which Dylan introduced briefly and which I ll talk about some more. The second is the great theorem that, if X has a smooth lift to characteristic zero, then the crystalline cohomology of X is the de Rham cohomology of that lift. Better yet, if X even has a closed immersion into an object Z which is a smooth lift to characteristic zero, then Ru (O X/S ) = O Z OZ Ω Z/S, where Z is a scheme called the divided-power envelope of X in Z. All this in due time. 3 The Witt vectors I ll end by talking about the Witt vectors and the de Rham-Witt complex. Let me briefly describe the Witt vectors. For a ring A, W (A) is the set A N with a weird ring structure. We have (a 0, a 1,... ) + (b 0, b 1,... ) = (S 0 (a 0, b 0 ), S 1 (a 0, a 1, b 0, b 1 ),... ) where S 0 = a 0 + b 0, S 1 = a 1 + b p (ap 0 + bp 0 (a 0 + b 0 ) p ), and so on. Likewise, (a 0, a 1,... )(b 0, b 1,... ) = (P 0 (a 0, b 0 ), P 1 (a 0, a 1, b 0, b 1 ),... ) 3
4 where P 0 = a 0 b 0, P 1 = a p 0 b 1 + a 1 b p 0 + pa 1b 1, and so on. One finds these polynomials via the Witt polynomials n Φ n (a 0, a 1,... ) = p i a pn i i. Over Q, these polynomials define a bijection (Φ 0, Φ 1,... ) : W (A) A N, and one defines the ring structure on W (A) so that this is an isomorphism. It s a surprising theorem of Witt that the S i and P i polynomials one gets this way have coefficients in Z, allowing one to define Witt vectors in positive characteristic, where they re more interesting. Again, one can sheafify all this, and define sheaves W O X on a scheme X. Moving to the slightly less well-known, in characteristic p, the Witt vectors have more structure: there s a ring homomorphism, the Frobenius, given by and an additive map, the Verschiebung, given by These satisfy some relations: i=0 F (a 0, a 1,... ) = (a p 0, ap 1,... ), V (a 0, a 1,... ) = (0, a 0, a 1,... ). F V = V F = p, xv (y) = V (F (x)y), V (1) = p. Define W n O X = W O X /V n (W O X ), so that W O X = lim W n O X. These are the finite-length Witt vectors, and can be represented by finite sequences of sections of O X, just as the full Witt vectors can be represented by infinite sequences. Finally, there are Teichmüller representatives, a multiplicative but not additive map O X W O X, x [x] = (x, 0, 0,... ). One can write any Witt vector as a formal sum i=0 V i [x i ]. 4 The de Rham-Witt complex The de Rham-Witt complex is a pro-complex W Ω X with some extra structure. The key point is that its (n, i) term is not honestly W n O X Ω i X, or even the Witt vectors applied in a derived fashion to the complex, as the notation might suggest. Ω X Definition 6. A V -pro-complex E over X is an inverse system of graded commutative DGAs E n, with restriction maps R : E n E n 1, such that E 0 = W O X, and there s an additive map V : E i n E i+1 n extending the Verschiebung on E 0, and satisfying V (x dy) = V x d(v y), (d[x])v y = V ([x] p 1 y d[x]), and V R = RV. The de Rham-Witt complex is the initial V -pro-complex. (I got this wrong in the lecture, adding a bunch of extra properties to the definition of this category. I don t know if my definition worked or not.) Don t worry if this is a little opaque. There s much more that one can say about the de Rham-Witt complex. For starters, W 1 Ω X is actually just Ω X. There s also a unique ring homomorphism F : W nω X W n 1 Ω X, extending the usual Frobenius on W no X and satisfying F dv = d and F (d[x]) = [x] p 1 d[x]. 4
5 It also satisfies F V = V F = p, xv (y) = V (F (x)y), df = pf d. From these more familiar relations, one can deduce the ones defining V above. Here are a few things worth noting: V (dy) = V (1 dy) = V (1) d(v y) = p d(v y). And my argument for one of the properties of F : but also so dividing by p, one gets d([x] p ) = d(f x) = pf d[x], d([x] p ) = p[x] p 1 d[x], F d[x] = [x] p 1 d[x]. Interestingly enough, by functoriality, the geometric Frobenius Frob : X X induces a map on the de Rham-Witt complex. On W Ω i, this map is precisely p i F. This is the first suggestion of a very important theme: that analytic information can be recovered in positive characteristic from the algebraic behavior of the Frobenius. In this case, an analogue of the Hodge filtration shows up as the power of p dividing the Frobenius. I ll end by just stating the comparison theorem. Theorem 7. If k is a perfect field of characteristic p and X is a smooth k-scheme, then Ru O X/Wn(k) W n Ω X. 5
Non characteristic finiteness theorems in crystalline cohomology
Non characteristic finiteness theorems in crystalline cohomology 1 Non characteristic finiteness theorems in crystalline cohomology Pierre Berthelot Université de Rennes 1 I.H.É.S., September 23, 2015
More informationLifting the Cartier transform of Ogus-Vologodsky. modulo p n. Daxin Xu. California Institute of Technology
Lifting the Cartier transform of Ogus-Vologodsky modulo p n Daxin Xu California Institute of Technology Riemann-Hilbert correspondences 2018, Padova A theorem of Deligne-Illusie k a perfect field of characteristic
More informationBRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,
CONNECTIONS, CURVATURE, AND p-curvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and p-curvature, in terms of maps of sheaves
More informationSPEAKER: JOHN BERGDALL
November 24, 2014 HODGE TATE AND DE RHAM REPRESENTATIONS SPEAKER: JOHN BERGDALL My goal today is to just go over some results regarding Hodge-Tate and de Rham representations. We always let K/Q p be a
More informationBEZOUT S THEOREM CHRISTIAN KLEVDAL
BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar
More informationAlgebraic v.s. Analytic Point of View
Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,
More informationINTERSECTION THEORY CLASS 19
INTERSECTION THEORY CLASS 19 RAVI VAKIL CONTENTS 1. Recap of Last day 1 1.1. New facts 2 2. Statement of the theorem 3 2.1. GRR for a special case of closed immersions f : X Y = P(N 1) 4 2.2. GRR for closed
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationSome remarks on Frobenius and Lefschetz in étale cohomology
Some remarks on obenius and Lefschetz in étale cohomology Gabriel Chênevert January 5, 2004 In this lecture I will discuss some more or less related issues revolving around the main idea relating (étale)
More informationThe geometric Satake isomorphism for p-adic groups
The geometric Satake isomorphism for p-adic groups Xinwen Zhu Notes by Tony Feng 1 Geometric Satake Let me start by recalling a fundamental theorem in the Geometric Langlands Program, which is the Geometric
More informationPeter Scholze Notes by Tony Feng. This is proved by real analysis, and the main step is to represent de Rham cohomology classes by harmonic forms.
p-adic Hodge Theory Peter Scholze Notes by Tony Feng 1 Classical Hodge Theory Let X be a compact complex manifold. We discuss three properties of classical Hodge theory. Hodge decomposition. Hodge s theorem
More informationThe Steenrod algebra
The Steenrod algebra Paul VanKoughnett January 25, 2016 References are the first few chapters of Mosher and Tangora, and if you can read French, Serre s Cohomologie modulo 2 des complexes d Eilenberg-MacLane
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24
FOUNDATIONS OF ALGEBRAIC GEOMETR CLASS 24 RAVI VAKIL CONTENTS 1. Normalization, continued 1 2. Sheaf Spec 3 3. Sheaf Proj 4 Last day: Fibers of morphisms. Properties preserved by base change: open immersions,
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves
More information6. DE RHAM-WITT COMPLEX AND LOG CRYS- TALLINE COHOMOLOGY
6. DE RHAM-WITT COMPLEX AND LOG CRYS- TALLINE COHOMOLOGY α : L k : a fine log str. on s = Spec(k) W n (L) : Teichmüller lifting of L to W n (s) : W n (L) = L Ker(W n (k) k ) L W n (k) : a [α(a)] Example
More informationThree Descriptions of the Cohomology of Bun G (X) (Lecture 4)
Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and
More informationON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS
ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS RYAN E GRADY 1. L SPACES An L space is a ringed space with a structure sheaf a sheaf L algebras, where an L algebra is the homotopical
More informationA p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1
A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 ALEXANDER G.M. PAULIN Abstract. The (de Rham) geometric Langlands correspondence for GL n asserts that to an irreducible rank n integrable connection
More informationNotes on p-divisible Groups
Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about
More informationHomotopy types of algebraic varieties
Homotopy types of algebraic varieties Bertrand Toën These are the notes of my talk given at the conference Theory of motives, homotopy theory of varieties, and dessins d enfants, Palo Alto, April 23-26,
More informationVERY ROUGH NOTES ON SPECTRAL DEFORMATION THEORY
VERY ROUGH NOTES ON SPECTRAL DEFORMATION THEORY 1. Classical Deformation Theory I want to begin with some classical deformation theory, before moving on to the spectral generalizations that constitute
More informationA mini-course on crystalline cohomology
A mini-course on crystalline cohomology June 15, 2018 Haoyang Guo Abstract This is the lecture notes for the mini-course during June 11-15, 2018 at University of Michigan, about the crystalline cohomology.
More informationDerived intersections and the Hodge theorem
Derived intersections and the Hodge theorem Abstract The algebraic Hodge theorem was proved in a beautiful 1987 paper by Deligne and Illusie, using positive characteristic methods. We argue that the central
More informationGauß-Manin Connection via Witt-Differentials
Gauß-Manin Connection via Witt-Differentials Andreas anger Thomas Zink October 20, 2004 We generalize ideas of Bloch [Bl] about de Rham-Witt connections to a relative situation, and apply this to construct
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More information1 Notations and Statement of the Main Results
An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main
More informationToric coordinates in relative p-adic Hodge theory
Toric coordinates in relative p-adic Hodge theory Kiran S. Kedlaya in joint work with Ruochuan Liu Department of Mathematics, Massachusetts Institute of Technology Department of Mathematics, University
More information3. The Sheaf of Regular Functions
24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice
More informationAlgebraic Geometry
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationTunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society
Tunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society Grothendieck Messing deformation theory for varieties of K3 type Andreas Langer and Thomas Zink
More informationDuality, Residues, Fundamental class
Duality, Residues, Fundamental class Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu May 22, 2011 Joseph Lipman (Purdue University) Duality, Residues, Fundamental class
More informationTHE SHIMURA-TANIYAMA FORMULA AND p-divisible GROUPS
THE SHIMURA-TANIYAMA FORMULA AND p-divisible GROUPS DANIEL LITT Let us fix the following notation: 1. Notation and Introduction K is a number field; L is a CM field with totally real subfield L + ; (A,
More informationCohomology theories III
Cohomology theories III Bruno Chiarellotto Università di Padova Sep 11th, 2011 Road Map Bruno Chiarellotto (Università di Padova) Cohomology theories III Sep 11th, 2011 1 / 27 Contents 1 p-cohomologies
More informationREVISITING THE DE RHAM-WITT COMPLEX
REVISITING THE DE RHAM-WITT COMPLEX BHARGAV BHATT, JACOB LURIE, AND AKHIL MATHEW Abstract. The goal of this paper is to offer a new construction of the de Rham-Witt complex of smooth varieties over perfect
More information1 Existence of the Néron model
Néron models Setting: S a Dedekind domain, K its field of fractions, A/K an abelian variety. A model of A/S is a flat, separable S-scheme of finite type X with X K = A. The nicest possible model over S
More informationApplications to the Beilinson-Bloch Conjecture
Applications to the Beilinson-Bloch Conjecture Green June 30, 2010 1 Green 1 - Applications to the Beilinson-Bloch Conjecture California is like Italy without the art. - Oscar Wilde Let X be a smooth projective
More informationBASIC MODULI THEORY YURI J. F. SULYMA
BASIC MODULI THEORY YURI J. F. SULYMA Slogan 0.1. Groupoids + Sites = Stacks 1. Groupoids Definition 1.1. Let G be a discrete group acting on a set. Let /G be the category with objects the elements of
More informationLECTURE 1: OVERVIEW. ; Q p ), where Y K
LECTURE 1: OVERVIEW 1. The Cohomology of Algebraic Varieties Let Y be a smooth proper variety defined over a field K of characteristic zero, and let K be an algebraic closure of K. Then one has two different
More information0.1 Spec of a monoid
These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 26
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 26 RAVI VAKIL CONTENTS 1. Proper morphisms 1 Last day: separatedness, definition of variety. Today: proper morphisms. I said a little more about separatedness of
More informationHomology and Cohomology of Stacks (Lecture 7)
Homology and Cohomology of Stacks (Lecture 7) February 19, 2014 In this course, we will need to discuss the l-adic homology and cohomology of algebro-geometric objects of a more general nature than algebraic
More informationPICARD GROUPS OF MODULI PROBLEMS II
PICARD GROUPS OF MODULI PROBLEMS II DANIEL LI 1. Recap Let s briefly recall what we did last time. I discussed the stack BG m, as classifying line bundles by analyzing the sense in which line bundles may
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationCrystalline Cohomology and Frobenius
Crystalline Cohomology and Frobenius Drew Moore References: Berthelot s Notes on Crystalline Cohomology, discussions with Matt Motivation Let X 0 be a proper, smooth variety over F p. Grothendieck s etale
More informationLevel raising. Kevin Buzzard April 26, v1 written 29/3/04; minor tinkering and clarifications written
Level raising Kevin Buzzard April 26, 2012 [History: 3/6/08] v1 written 29/3/04; minor tinkering and clarifications written 1 Introduction What s at the heart of level raising, when working with 1-dimensional
More information3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection
3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection is called the objects of C and is denoted Obj(C). Given
More informationIdentification of the graded pieces Kęstutis Česnavičius
Identification of the graded pieces Kęstutis Česnavičius 1. TP for quasiregular semiperfect algebras We fix a prime number p, recall that an F p -algebra R is perfect if its absolute Frobenius endomorphism
More informationarxiv: v1 [math.ag] 25 Feb 2018
ON HIGHER DIRECT IMAGES OF CONVERGENT ISOCRYSTALS arxiv:1802.09060v1 [math.ag] 25 Feb 2018 DAXIN XU Abstract. Let k be a perfect field of characteristic p > 0 and W the ring of Witt vectors of k. In this
More informationChern classes à la Grothendieck
Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces
More informationLast week: proétale topology on X, with a map of sites ν : X proét X ét. Sheaves on X proét : O + X = ν O + X ét
INTEGRAL p-adi HODGE THEORY, TALK 3 (RATIONAL p-adi HODGE THEORY I, THE PRIMITIVE OMPARISON THEOREM) RAFFAEL SINGER (NOTES BY JAMES NEWTON) 1. Recap We x /Q p complete and algebraically closed, with tilt
More informationÉTALE π 1 OF A SMOOTH CURVE
ÉTALE π 1 OF A SMOOTH CURVE AKHIL MATHEW 1. Introduction One of the early achievements of Grothendieck s theory of schemes was the (partial) computation of the étale fundamental group of a smooth projective
More informationThe absolute de Rham-Witt complex
The absolute de Rham-Witt complex Lars Hesselholt Introduction This note is a brief survey of the absolute de Rham-Witt complex. We explain the structure of this complex for a smooth scheme over a complete
More informationNOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY
NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY RUNE HAUGSENG The aim of these notes is to define flat and faithfully flat morphisms and review some of their important properties, and to define the fpqc
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Where we were 1 2. Yoneda s lemma 2 3. Limits and colimits 6 4. Adjoints 8 First, some bureaucratic details. We will move to 380-F for Monday
More informationABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY
ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY ARDA H. DEMIRHAN Abstract. We examine the conditions for uniqueness of differentials in the abstract setting of differential geometry. Then we ll come up
More information1. Motivation: search for a p-adic cohomology theory
AN INTRODUCTION TO RIGID COHOMOLOGY CHRISTOPHER LAZDA 1. Motivation: search for a p-adic cohomology theory Let k be a perfect field of characteristic p > 0, W = W (k) the ring of Witt vectors of k, and
More informationCurves on an algebraic surface II
Curves on an algebraic surface II Dylan Wilson October 1, 2014 (1) Last time, Paul constructed Curves X and Pic X for us. In this lecture, we d like to compute the dimension of the Picard group for an
More informationMath 248B. Applications of base change for coherent cohomology
Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).
More informationA GEOMETRIC CONSTRUCTION OF SEMISTABLE EXTENSIONS OF CRYSTALLINE REPRESENTATIONS. 1. Introduction
A GEOMETRIC CONSTRUCTION OF SEMISTABLE EXTENSIONS OF CRYSTALLINE REPRESENTATIONS MARTIN OLSSON Abstract. We study unipotent fundamental groups for open varieties over p-adic fields with base point degenerating
More informationDiscussion Session on p-divisible Groups
Discussion Session on p-divisible Groups Notes by Tony Feng April 7, 2016 These are notes from a discussion session of p-divisible groups. Some questions were posed by Dennis Gaitsgory, and then their
More informationAlgebraic varieties and schemes over any scheme. Non singular varieties
Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two
More informationSHIMURA VARIETIES AND TAF
SHIMURA VARIETIES AND TAF PAUL VANKOUGHNETT 1. Introduction The primary source is chapter 6 of [?]. We ve spent a long time learning generalities about abelian varieties. In this talk (or two), we ll assemble
More informationBasic results on Grothendieck Duality
Basic results on Grothendieck Duality Joseph Lipman 1 Purdue University Department of Mathematics lipman@math.purdue.edu http://www.math.purdue.edu/ lipman November 2007 1 Supported in part by NSA Grant
More informationp-divisible GROUPS 1. p-divisible groups and finite group schemes Here are two related problems with the algebraic geometry of formal groups.
p-divisible GROUPS PAUL VANKOUGHNETT 1. p-divisible groups and finite group schemes Here are two related problems with the algebraic geometry of formal groups. Problem 1. The height of a formal group is
More informationwhich is a group homomorphism, such that if W V U, then
4. Sheaves Definition 4.1. Let X be a topological space. A presheaf of groups F on X is a a function which assigns to every open set U X a group F(U) and to every inclusion V U a restriction map, ρ UV
More informationWhat is a motive? Johan M. Commelin March 3, 2014
What is a motive? Johan M Commelin March 3, 2014 Abstract The question in the title does not yet have a definite answer One might even say that it is one of the most central, delicate, and difficult questions
More informationTHE MAIN CONSTRUCTION I
THE MAIN CONSTRUCTION I Contents 1. Introduction/ Notation 1 2. Motivation: the Cartier isomorphism 2 3. Definition of the map 3 4. Small algebras and almost purity 3 5. Cohomology of Z d p 5 6. Décalage
More informationSeminar on Crystalline Cohomology
Seminar on Crystalline Cohomology Yun Hao November 28, 2016 Contents 1 Divided Power Algebra October 31, 2016 1 1.1 Divided Power Structure.............................. 1 1.2 Extension of Divided Power
More informationGeometry 9: Serre-Swan theorem
Geometry 9: Serre-Swan theorem Rules: You may choose to solve only hard exercises (marked with!, * and **) or ordinary ones (marked with! or unmarked), or both, if you want to have extra problems. To have
More informationA Dieudonné Theory for p-divisible Groups
A Dieudonné Theory for p-divisible Groups Thomas Zink 1 Introduction Let k be a perfect field of characteristic p > 0. We denote by W (k) the ring of Witt vectors. Let us denote by ξ F ξ, ξ W (k) the Frobenius
More informationMIXED HODGE MODULES PAVEL SAFRONOV
MIED HODGE MODULES PAVEL SAFRONOV 1. Mixed Hodge theory 1.1. Pure Hodge structures. Let be a smooth projective complex variety and Ω the complex of sheaves of holomorphic differential forms with the de
More informationTheta divisors and the Frobenius morphism
Theta divisors and the Frobenius morphism David A. Madore Abstract We introduce theta divisors for vector bundles and relate them to the ordinariness of curves in characteristic p > 0. We prove, following
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5 RAVI VAKIL CONTENTS 1. The inverse image sheaf 1 2. Recovering sheaves from a sheaf on a base 3 3. Toward schemes 5 4. The underlying set of affine schemes 6 Last
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Proper morphisms 1 2. Scheme-theoretic closure, and scheme-theoretic image 2 3. Rational maps 3 4. Examples of rational maps 5 Last day:
More informationWhat is the Langlands program all about?
What is the Langlands program all about? Laurent Lafforgue November 13, 2013 Hua Loo-Keng Distinguished Lecture Academy of Mathematics and Systems Science, Chinese Academy of Sciences This talk is mainly
More informationDieudonné Modules and p-divisible Groups
Dieudonné Modules and p-divisible Groups Brian Lawrence September 26, 2014 The notion of l-adic Tate modules, for primes l away from the characteristic of the ground field, is incredibly useful. The analogous
More informationAPPENDIX 3: AN OVERVIEW OF CHOW GROUPS
APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss
More informationBeilinson s conjectures I
Beilinson s conjectures I Akshay Venkatesh February 17, 2016 1 Deligne s conjecture As we saw, Deligne made a conjecture for varieties (actually at the level of motives) for the special values of L-function.
More information14 Lecture 14: Basic generallities on adic spaces
14 Lecture 14: Basic generallities on adic spaces 14.1 Introduction The aim of this lecture and the next two is to address general adic spaces and their connection to rigid geometry. 14.2 Two open questions
More informationD-manifolds and derived differential geometry
D-manifolds and derived differential geometry Dominic Joyce, Oxford University September 2014 Based on survey paper: arxiv:1206.4207, 44 pages and preliminary version of book which may be downloaded from
More informationCOMPLEX ALGEBRAIC SURFACES CLASS 9
COMPLEX ALGEBRAIC SURFACES CLASS 9 RAVI VAKIL CONTENTS 1. Construction of Castelnuovo s contraction map 1 2. Ruled surfaces 3 (At the end of last lecture I discussed the Weak Factorization Theorem, Resolution
More informationLocally G-ringed spaces and rigid spaces
18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 Rigid analytic spaces (at last!) We are now ready to talk about rigid analytic spaces in earnest. I ll give the
More informationProgram of the DaFra seminar. Unramified and Tamely Ramified Geometric Class Field Theory
Program of the DaFra seminar Unramified and Tamely Ramified Geometric Class Field Theory Introduction Torsten Wedhorn Winter semester 2017/18 Gemetric class field theory gives a geometric formulation and
More informationINTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 23
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 23 RAVI VAKIL Contents 1. More background on invertible sheaves 1 1.1. Operations on invertible sheaves 1 1.2. Maps to projective space correspond to a vector
More informationwhere Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset
Classification of quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring
More informationDeformation theory of representable morphisms of algebraic stacks
Deformation theory of representable morphisms of algebraic stacks Martin C. Olsson School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, molsson@math.ias.edu Received:
More informationTopos Theory. Lectures 17-20: The interpretation of logic in categories. Olivia Caramello. Topos Theory. Olivia Caramello.
logic s Lectures 17-20: logic in 2 / 40 logic s Interpreting first-order logic in In Logic, first-order s are a wide class of formal s used for talking about structures of any kind (where the restriction
More informationIntroduction and preliminaries Wouter Zomervrucht, Februari 26, 2014
Introduction and preliminaries Wouter Zomervrucht, Februari 26, 204. Introduction Theorem. Serre duality). Let k be a field, X a smooth projective scheme over k of relative dimension n, and F a locally
More informationDERIVED CATEGORIES: LECTURE 4. References
DERIVED CATEGORIES: LECTURE 4 EVGENY SHINDER References [Muk] Shigeru Mukai, Fourier functor and its application to the moduli of bundles on an abelian variety, Algebraic geometry, Sendai, 1985, 515 550,
More informationRigid cohomology and its coefficients
Rigid cohomology and its coefficients Kiran S. Kedlaya Department of Mathematics, Massachusetts Institute of Technology p-adic Geometry and Homotopy Theory Loen, August 4, 2009 These slides can be found
More informationOn the Néron-Ogg-Shafarevich Criterion for K3 Surfaces. Submitted By: Yukihide Nakada. Supervisor: Prof. Bruno Chiarellotto
Università degli Studi di Padova Universität Duisburg-Essen ALGANT Master s Thesis On the Néron-Ogg-Shafarevich Criterion for K3 Surfaces Submitted By: Yukihide Nakada Supervisor: Prof. Bruno Chiarellotto
More informationQUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS
QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS SAM RASKIN 1. Differential operators on stacks 1.1. We will define a D-module of differential operators on a smooth stack and construct a symbol map when
More informationSmooth morphisms. Peter Bruin 21 February 2007
Smooth morphisms Peter Bruin 21 February 2007 Introduction The goal of this talk is to define smooth morphisms of schemes, which are one of the main ingredients in Néron s fundamental theorem [BLR, 1.3,
More information1. Differential Forms Let X be a smooth complete variety over C. Then as a consequence of Hodge theory + GAGA: H i (X an, C) = H i (X, Ω X) =
SOME APPLICATIONS OF POSITIVE CHARACTERISTIC TECHNIQUES TO VANISHING THEOREMS DONU ARAPURA To Joe Lipman These are notes to my talk at Lipman s birthday conference. Some details have appeared in [A1, A2].
More informationPARABOLIC SHEAVES ON LOGARITHMIC SCHEMES
PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES Angelo Vistoli Scuola Normale Superiore Bordeaux, June 23, 2010 Joint work with Niels Borne Université de Lille 1 Let X be an algebraic variety over C, x 0 X. What
More informationD-MODULES: AN INTRODUCTION
D-MODULES: AN INTRODUCTION ANNA ROMANOVA 1. overview D-modules are a useful tool in both representation theory and algebraic geometry. In this talk, I will motivate the study of D-modules by describing
More information1 Replete topoi. X = Shv proét (X) X is locally weakly contractible (next lecture) X is replete. D(X ) is left complete. K D(X ) we have R lim
Reference: [BS] Bhatt, Scholze, The pro-étale topology for schemes In this lecture we consider replete topoi This is a nice class of topoi that include the pro-étale topos, and whose derived categories
More information