Special lectures: Frans Oort
Special lecture 1: Finite commutative group schemes annihilated by p (with interlude exercise sessions by Đào Văn Thịnh)
Abstract. Hanspeter Kraft announced in 1975 a result
classifying all finite commutative group schemes in characteristic p annihilated by p.
In particular, this implies that all
isomorphism classes of such group schemes of a fixed rank is finite.
This inspiring idea has been proved to be true. I will carefully describe the classification, will give part of the proof of the theorem in detail, with examples, and I will sketch the rest of the proof in these lectures..
Special lecture 2: Is a finite group scheme annihilated by its rank?
Abstract. The classical Lagrange theorem for abstract groups says that any element in a group is annihilated by the order of that group. Is the analogous true for group schemes?
Consider a group scheme G, finite and flat of constant rank n over an arbitrary base ring R. Is G annihilated by n? In other words, does [n]_G factor through the trivial element of G(R)? This problem has been open for the last 60 years.
I will indicate cases where we know the answer is affirmative, with complete proofs, and I will sketch possible approaches. Examples will illustrate some experience with this question.
Course 1: Basic Hopf algebras (10 hours)
Lecturer: Ngô Đắc Tuấn
- Tensor product of vector spaces
- Algebras, bialgebras
- Convolution and Hopf algebras
- Graduations (graded Hopf algebras)
- Connectedness
References:
[1] Darij Grinberg, Victor Reiner. Hopf Algebras in Combinatorics, arXiv:1409.8356v7
Course 2: Some theorems on structure of Hopf algebras (10 hours)
Lecturer: Phùng Hồ Hải
- Comodules and local finiteness
- Fundamental theorem of Hopf algebras
- Milnor-Moore theorem
- Cartier-Gabriel theorem
References:
[1] J.W. Milnor and J.C. Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965), 211-264.
[2] P. Cartier. A primer of Hopf algebras.
[3] M.E. Sweedler. Hopf Algebras. Mathematics Lecture Note Series, 44. W. A. Benjamin, 1969.
Course 3: Finite group schemes (10 hours)
Lecturers: Ngô Trung Hiếu, Đào Văn Thịnh
- Introduction to group schemes and examples
- Cartier duality
- Étale group schemes over fields
- Étale group schemes over rings
- Connected and étale components
References:
[1] R. Schoof. Introduction to finite group schemes (lecture notes, available online)
[2] R. Pink. Finite group schemes (lecture notes, available online)
Topic 1: Applications of Hopf algebras to combinatorics (10 hours)
The main goal of these lectures is to present some Hopf algebras related to combinatorics and other topics.
Talk 1: Connes-Kreimer algebra I: rooted trees. The following article gives an overview and some references to the topic:
[Foi] L. Foissy. An introduction to Hopf algebras of trees. http://loic.foissy.free.fr/ pageperso/preprint3.pdf.
Talk 2: Connes-Kreimer algebra II: planar trees. The following articles gives an overview and some references to the topic:
[Foi] L. Foissy. An introduction to Hopf algebras of trees. http://loic.foissy.free.fr/ pageperso/preprint3.pdf.
[LR98] Loday, Jean-Louis; Ronco, María O. Hopf algebra of the planar binary trees. Adv. Math. 139 (1998), no. 2, 293–309.
Talk 3: Hopf algebra and multiple zeta values I. The goal is to present the following article:
[Hof97] Hoffman, Michael E. The algebra of multiple harmonic series. J. Algebra 194 (1997), no. 2, 477–495.
Talk 4: Hopf algebra and multiple zeta values II. The goal is to present the following article:
[Hof00] Hoffman, Michael E. Quasi-shuffle products. J. Algebraic Combin. 11 (2000), no. 1, 49–68.
Talk 5: Hopf algebra and multiple zeta values III. The goal is to present the following article:
[IKZ06] Ihara, Kentaro; Kaneko, Masanobu; Zagier, Don Derivation and double shuffle relations for multiple zeta values. Compos. Math. 142 (2006), no. 2, 307–338.
A presentation is already given in Sections 1.6 and 1.7 of the book
[BGF] J. Burgos Gil and J. Fresan. Multiple zeta values: from numbers to motives. to appear, Clay Mathematics Proceedings.
Topic 2: Tannakian duality (12 hours).
The aim of this series of talks is to explain Deligne's celebrated theorem proving the general Tannakian duality, which establishes a one-one correspondence between tensor abelian categories over a field k equipped with an exact, faithful tensor functor to coherent sheaves over a k-scheme S (i.e. a Tannakian category) and affine groupoids acting transitively on S. The hardest part of the proof is to show that, given a Tannakian category, the groupoid reconstructed from it acting faithfully flat on the base (which amounts to say the groupopid schemes is faithfully flat over S ×k S). This is somewhat parallel to the proof of Saavedra that the reconstructed group scheme from a neutral Tannakian category is non-empty.
To show this, Deligne introduces the notion of tensor products of tensor categories and show that the tensor product of Tannakian categories is again a Tannakian category.
The aim of this series of talks is to present some basis properties of Tannakian category and the Tannakian duality established by Deligne in [2].
References:
[1] Deligne, Milne. Tannakian duality, Lecture Notes in Mathematics, 1982
[2] Deligne, Categories tannakiennes, Grothendieck Festschrift, 1990.
Talk 1: Groupoid scheme and general Tannakian duality [1, Section 1], [2, Section 1, 2]
Give a brief review on tensor abelian categories and fiber functors [1, Section 1 and 2, Sections 2.1-2.5].
Introduce the notion of groupoid schemes; affine groupoid schemes and their coordinate ring; Hopf algebroids and their comodules, [2, Section 1].
Formulate the main theorem [2, Theorem 1.12].
Talk 2: Basic properties of tannakian categories [2, Section 2]
Introduce the general notion of Tannakian duality, recall the notion of stacks, gerbs as in [1, Section 3 and Appendix].
Present basis properties of tannakian categories [2, Sections 2.6-2.18], in particular the proof of key proposition [2, Prop 2.14].
Talk 3: Basic properties of transitive affine groupoids their representations [2, Section 3].
Present basic properties of affine groupoids acting transitively on their bases [2, Sections 3.1-3.10]. If time permits, mention and explain Proposition 3.11.
Explain the Barr-Beck theorem and faithfully flat descent. Apply these results to prove Corollary 3.8 of [2], [2, Sections 4.1-4.9].
Talk 4: Tensor product of abelian categories [2, Section 5].
Construct tensor product of module categories over coherent (in particular, finite dimensional) algebras over a field [2, Sections 5.1-5.12]. Point out the missing case when the base field k is not perfect [5.6].
Apply the result above to the case of tensor categories with k perfect [2, Sections 5.13-5.17].
Talk 5: Proof of the main theorem when the base field is perfect [2, Section 6].
Present the proof of [2, Theorem 1.12] as given in [2, Sections 6.1-6.15].
The tricky point is to show that the Hopf algebroid L reconstructed from the Tannakian category C is faithfully flat over the base ring B ⊗ B. To show this, Deligne expresses L as the image under the fiber functor to B ⊗ B-mod of an object Λ in C ⊗ C. As the latter category is a tensor category the conclusion follows.
Talk 6: The case k is not perfect
This is essential to present the proof of [2, Lemma 6.9] for the case k is not perfect. The materials are given in Sections 5.18-5.21 and 6.16-6.20 of [2].
Topic 3: Group schemes of prime order (12 hours).
The main goal of this workshop is to obtain a thorough understanding of the paper [2] of John Tate and Frans Oort on group schemes of prime order. More specifically, we will try to understand the classification of finite group schemes of prime order over rather general base schemes S, including (the spectrum of) a complete local noetherian ring of residue characteristic p. Time permitting, we will learn some generalizations, for instance some results of Eremichev’s thesis [1] which studied finite group schemes of order p2, p3 over a perfect field.
References
[1] Eremichev, Vladimir. Families of group schemes of prime power order. PhD thesis, University of Warwick (2020). http://wrap.warwick.ac.uk/145366/
[2] Tate, John and Oort, Frans. Group schemes of prime order, Ann. Sci. Ecole Norm. Sup. (4) 3, 1–21 (1970).
[3] Tate, John. Finite flat group schemes, in Modular forms and Fermat’s last theorem, eds. Gary Cornell, Joseph H. Silverman, and Glenn Stevens, New York: Springer-Verlag, 1997, pp. 121– 154.
[4] Waterhouse, William C. Introduction to affine group schemes, Graduate texts in mathematics, vol. 66, New York-Berlin: Springer-Verlag, 1979.
Talk 1: Introduction to finite group schemes: we will cover Section 1 of [2] and [3].
Some examples of finite group scheme of prime order and a classification in the case p = 2 will be presented. Cartier duality and a theorem of Deligne (see Section 1 of [2]) will be discussed.
Talk 2: A classification of finite group schemes of prime order over a complete noetherian local ring of residue characteristic p. This is Section 2 of [2].
Talk 3: A classification of finite group schemes of prime order over a ring of integers in a number field. This is Section 3 of [2].
Talk 4: Canonical decomposition of finite commutative group schemes over a perfect field (see [4] and Section 1.4 of [1]).
The result is as follows: any finite commutative group scheme G over a perfect field has a unique and functorial decomposition:
G Grr ⊕ Grl ⊕ Glr ⊕ Gll, where Grr, Grl, Glr, and Gll are of the reduced-reduced, reduced-local, local-reduced, and local-local types respectively. Some properties of Gll will be needed for the next talk.
Talk 5: The Dieudonné correspondence and finite group schemes of order p2 (see Sections 2.6, 3.1, and 3.2 of [1]).
By using the Dieudonné correspondence between local-local commutative group schemes and Dieudonné modules, we will classify group schemes of order p2 over Fp.
Talk 6: Magma codes, examples for low rank group schemes and Tate-Oort group schemes.