# Motives Seminar

## Motivic Steenrod Operations

The motives seminar this semester will accompany the Research Seminar on the Milnor Conjecture. The main goal is to give Voevodsky's construction of the motivic reduced power operations and the motivic Steenrod algebra, following [V]. For some additionall details on some of the lectures, see the Program Notes

We will meet on Tuesdays, 16:15-18 Uhr, in WSC-S-U-3.01.

### Previous Motives Seminars

Follow the ** Teaching** link at the top of this page to find links to previous motives seminars.

### Program

Lecture 1. April 4: Marc Levine. * The Steenrod algebra in topology*

Lecture 1 Video.

Recall the axioms and construction of the classical Steenrod algebra, following [SE]. See the Program Notes for details.

Lecture 2. April 11: Herman Rohrbach * Motivic Background*.

Lecture 2 Video.

Give an overview of Voevodsky's category of (effective) geometric motives, his category of effective motives, defined using presheaves with transfer, and the Morel-Voevodsky unstable homotopy category. See the Program Notes for details.

Lecture 3. April 18: Jan Hennig * [V], Chapters 1 and 2. *

Lecture 3 Video.

Introduce the main goal: the construction of the reduced power operations and their properties. Introduce the motivic Eilenberg-MacLane spaces and their properties, as expressed by the results of Chapter 2.

Lecture 4. April 25: Andrei Konovalov * [V] Chapters 3 and 4*.

Lecture 4 Video.

Introduce the sheaf model for the Eilenberg-MacLane spaces and discuss their properties, following [V, Chap. 3]. Define Euler classes and the Thom isomorphism and their properties, as in [V, Chap. 4]

Lecture 5. May 2: Pietro Gigli * [V] Chapter 5*

Lecture 5 Video.

Define the reduced power operation and discuss its properties, following [V, Chap. 5].

Lecture 6. May 9: Alessandro D'Angelo * [V] Chapter 6, part 1*

Recall the construction of \(BG:=B_{gm}G\cong B_{et}G\) from [MV, sec. 4.2] and note the independence of its motivic homotopy from the various choices involved in the construction. Make the computation of the motivic cohomology of \(B\mu_\ell\), and cover the material in [V, Chap. 6] including up to Theorem 6.10 and its proof.

Lecture 7. May 16: Herman Rohrbach *[V] Chapter 6, part 2*

Present the remainder of [V, Chap. 6]

Lecture 8. May 23: Chirantan Chowdhury *[V] Chapters 7, 8*

Present the results of [V, Chap. 7], leading the the main result Theorem 7.8. Present the results of [V, Chap. 8], leading to the main result Theorem 8.4.

Lecture 9. May 30: Sabrina Pauli *[V] Chapter 9*

Derive all the formulas for the reduced power operations in this chapter.

Lecture 10. June 6: Chirantan Chowdhury *[V] Chapter 10*

Give the proofs for Theorem 10.2 and Theorem 10.3.

Lecture 11. June 13: Dhyan Aranha *[V] Chapter 11*

Prove the main results of this Chapter: that the admissible monomials form a basis for the operations, Cor. 11.5. Present the paragraph following Cor. 11.5 about the topological realisation. Discuss as well the properties of the comultiplication \(\psi\) and the subring of operator-like elements.

Lecture 12. June 20: N.N. * [V] Chapter 12, part 1*

Prove theorem 12.4, corollary 12.5 and Theorem 12.6.

Lecture 13. June 27: N.N. * [V] Chapter 12, part 2*

Present the computations on [V, pg. 47-49] that lead to a description of the dual algebra, following Remark 12.12, especially the case \(\ell=2\), and the description of \(\mathcal{H}(k, \mathbb{Z}/2)\) as the twisted product of \(\mathcal{H}(2)\) and \(H^{**}\).

Lecture 14. July 4: N.N. *[V] Chapter 13*

This chapter discusses the motivic Milnor operations \(Q(E)\) and their properties, as well as the power operations \(\mathcal{P}^R\) and the general operations \(\rho(E,R)\). The Milnor operations are essential for the proof of the Milnor conjecture (\(\ell=2\)) and the Bloch-Kato conjectures (\(\ell>2\)).

Lecture 15. July 11: N.N. *[V] Chapter 14*

This gives the connection between the Steenrod operations and the characteristic classes \(s_*\) used in the proof of the Milnor conjecture: Theorem 14.2, Cor. 14.3, 14.4, as well as the remaining computations in this chapter.

### Bibliography

[LNotes] M. Levine, * 3 Lectures on motivic cohomology*

[MV] F. Morel, V. Voevodsky, * \(\mathbb{A}^1\)-homotopy theory of schemes. * Inst. Hautes Études Sci. Publ. Math. No. 90 (1999), 45–143 (2001).

[SE] Steenrod, N. E., **Cohomology operations**. Lectures by N. E. Steenrod written and revised by D. B. A. Epstein. Annals of Mathematics Studies, No. 50 Princeton University Press, Princeton, N.J. 1962

[V] Voevodsky, Vladimir, * Reduced power operations in motivic cohomology*. Publ. Math. Inst. Hautes Études Sci. No. 98 (2003), 1–57.