Seminar zur Motiven SS 2025

Quadratic intersection theory and motivic linking

This semester is a course on Rost's cycle modules and associated cycle complex, and their generalisation the Rost-Schmid complex for Milnor-Witt $K$-theory. We will apply this theory to a study of the intersection theory given by the Chow-Witt groups and finally give applications and concrete computations for quadratic linking degrees.

Schedule and program details

You can also find the program as pdf here.

Lecture 1 (April 7, Clémentine Lemarié --Rieusset) An overview
An overview of this seminar.

Lecture 2 (April 14, Linda Carnevale) Milnor $K$-theory and the Rost complex
The speaker will present Milnor $K$-theory $K^M_*$ and Rost complexes, with a particular focus on the Rost complex associated to $K^M$. This is the first lecture in a series of lectures about the Rost complex; our goal is to give intuition in this simpler setting to pave the way for the lectures on the Rost-Schmid complex (which is the main tool for motivic linking).
Main references: Sections 1-2 of Milnor's article [Mil70] and Sections 1-3 of Rost's article [Ros96].

Lecture 3 (April 23, Clémentine Lemarié --Rieusset) Rost groups and Chow groups
After recalling the classical definition of Chow groups, the speaker will present Rost groups. The Rost group $A^i(X;M,j)$, also denoted $A^i(X,j)$ when $M = K^M$, is the $i$-th cohomology group of the Rost complex $C^*(X;M,j)$. This generalises Chow groups as the Chow group $CH^i(X)$ is $A^i(X;K^M,i)$, or $A^i(X,i)$ for short. The speaker will end the talk by explicitly computing some Rost groups.
Main references: Sections 4-5 of Rost's article [Ros96]. Additional references: the books [Ful98] by Fulton and [EH16] by Eisenbud and Harris.

Lecture 4 (April 30, Linda Carnevale) An explicit proof of the homotopy property
The speaker will present the construction of a homotopy inverse of the morphism of Rost complexes induced by an affine bundle. This construction together with the homotopy axiom yields the homotopy property for affine bundles (see [Ros96, Proposition 8.6]). More importantly, it yields this property in an explicit manner: this construction takes place in the Rost complex before taking the cohomology that gives rise to the Rost groups, which is essential for computations.
Main reference: Section 9 of Rost's article [Ros96].

Lecture 5 (May 7, Marc Levine) Deformation to the normal cone and pull-backs
The speaker will present an important technique, deformation to the normal cone, which will then be used to define general pull-backs. These general pull-backs will themselves be used to define intersection products in the following talk.
Main references: Sections 10-12 of Rost's article [Ros96]. Additional references: the books [Ful98] by Fulton and [EH16] by Eisenbud and Harris.

Lecture 6 (May 14, Marc Levine) Intersection products
After recalling the classical definitions of the intersection product and of the Chow ring, the speaker will present the Rost ring, i.e. $\oplus_{i,j}A^i(X; M, j)$ endowed with Rost's intersection product. The Chow ring $CH^*(X)$ is the direct sum $\oplus_iCH^i(X)=\oplus_i A^i(X;K^M,i)$ endowed with Rost's intersection product, which coincides with the classical intersection product, so that it is a subring of the Rost ring associated to Milnor $K$-theory. If possible, the speaker will finish the talk by giving a formula to compute Rost's intersection product in nice cases and by applying this formula to explicit computations.
Main references: Sections 13-14 of Rost's article [Ros96]. Additional references: the books [Ful98] by Fulton and [EH16] by Eisenbud and Harris.

Lecture 7 (May 21, Linda Carnevale) From spheres to motivic spheres
After giving some background on the stable homotopy groups of (classical) spheres, the speaker will present motivic spheres and their stable homotopy groups. Thanks to Morel's theorem, which will be introduced in the following talk, this will give an idea of the importance of Milnor-Witt $K$-theory. The speaker will end this talk by presenting smooth models of motivic spheres, which will be used in the final two lectures.
Main reference: the article [ADF16] by Asok, Doran and Fasel. Additional references: Lecture 10, The Suspension Theorem and Homotopy Groups of Spheres in the book [FF16] by Fomenko and Fuchs, the article [IWX23] by Isaksen, Wang and Xu and Morel's book [Mor12].

Lecture 8 (May 28, N.N.) Milnor-Witt $K$-theory and the Rost-Schmid complex
After giving some background on the Witt ring and the Grothendieck-Witt ring, the speaker will present Milnor-Witt K-theory $K^{MW}_*$ and the Rost-Schmid complex associated to $K^{MW}$. This is the first lecture in a series of lectures about the Rost-Schmid complex, paving the way to the study of motivic linking. The speaker will end this talk by presenting the counterparts to the four basic maps discussed in Lecture 2, as well as the fifth basic map: multiplication by $\eta$.
Main references: Chapters 2-3 of Lemarié--Rieusset's PhD thesis [Lem23], Morel's book [Mor12] (especially Chapters 3-5) and Fasel's lecture notes [Fas20]. Additional references: Barge and Morel's article [BM00], Morel's lecture notes [Mor03], Schmid's PhD thesis [Sch98] and Feld's PhD thesis [Fel21].

Lecture 9 (June 4, N.N.) Rost-Schmid groups and Chow- Witt groups
The speaker will present Rost-Schmid groups, and in particular the Chow-Witt groups. The Rost-Schmid group $H^i(X, \underline{K}^{MW}_j\{L\})$, where $L$ is an invertible $\mathcal{O}_X$-module, is the $i$-th cohomology group of the Rost-Schmid complex $C^*(X, K^{MW}\{L\}, j)$. The Rost-Schmid group $H^i(X, \underline{K}^{MW}_i\{L\})$ is also denoted $\widetilde{CH}^i(X, L)$ and called the $i$-th Chow-Witt group. The speaker will end the talk by explicitly computing some Rost-Schmid groups and, if possible, presenting the construction of a homotopy inverse of the morphism of Rost-Schmid complexes induced by an affine space bundle.
Main references: Chapters 2-3 of Lemarié--Rieusset's PhD thesis [Lem23], Morel's book [Mor12] (especially Chapters 3-5) and Fasel's lecture notes [Fas20]. Additional references: The article [BM00] by Barge and Morel, Morel's lecture notes [Mor03], Schmid's PhD thesis [Sch98] and Feld's PhD thesis [Fel21].

Lecture 10 (June 11, Marc Levine) Pull-backs
The speaker will define general pull-backs. These general pull-backs will themselves be used to define intersection products in the following talk. Main references: Chapters 2-3 of Lemarié--Rieusset's PhD thesis [Lem23], Morel's book [Mor12] (especially Chapters 3-5) and Fasel's lecture notes [Fas20]. Additional references: The article [BM00] by Barge and Morel, Morel's lecture notes [Mor03], Schmid's PhD thesis [Sch98] and Feld's PhD thesis [Fel21].

Lecture 11 (June 18, N.N.) Intersection products
The speaker will present the Rost-Schmid ring, i.e. $\oplus_{i,j} H^i(X,\underline{K}^{MW}_j\{L\})$, together with the quadratic intersection product. The subring of the Rost-Schmid ring whose underlying group is the direct sum of the Chow-Witt groups $\oplus_iH^i(X,\underline{K}^{MW}_i\{L\})=\oplus_i\widetilde{CH}^i(X,L)$ is called the Chow-Witt ring. If possible, the speaker will finish the talk by giving a formula to compute the quadratic intersection product in nice cases and by applying this formula to explicit computations.
Main references: Chapters 2-3 of Lemarié--Rieusset's PhD thesis [Lem23], Morel's book [Mor12] (especially Chapters 3-5) and Fasel's lecture notes [Fas20]. Additional references: The article [BM00] by Barge and Morel, Morel's lecture notes [Mor03], Schmid's PhD thesis [Sch98] and Feld's PhD thesis [Fel21].

Lecture 12 (June 23, N.N.) Knot theory and projective knot theory
After giving some general background on the linking number, the speaker will present linking in (classical) knot theory and in projective knot theory. Specifically, the speaker will define oriented (classical) knots and links before studying the linking number of an oriented link with two components, first in the most classical case, $S^1\amalg S^1\to S^3$, then in the more general case $S^p\amalg S^q\to S^{p+q+1}$. The speaker will then define oriented projective knots and links before studying the linking number of an oriented projective link with two components, first in the most classical case, $\mathbb{R}\mathbb{P}^1\amalg\mathbb{R}\mathbb{P}^1\to \mathbb{R}\mathbb{P}^3$, then in the more general case $\mathbb{R}\mathbb{P}^p\amalg\mathbb{R}\mathbb{P}^q\to \mathbb{R}\mathbb{P}^{p+q+1}$.
Main references: Chapter 10 of the book [ST80] by Seifert and Threlfall, Chapter 1 of Lemarié--Rieusset's PhD thesis [Lem23] and Julia Viro's article [Dro91]. Additional references: The articles [Dro94] and [Vir07] by Julia Viro and the chapter [VV21] by Julia Viro and Oleg Viro.

Lecture 13 (July 2, Clémentine Lemarié --Rieusset) The linking number and the quadratic linking degree.
The speaker will present motivic linking, a theory in algebraic geometry which is a counterpart to classical linking. Specifically, the speaker will define quadratic linking degrees which are counterparts to the linking number and describe how two disjoint closed $F$-subschemes of an $F$-scheme are linked (i.e. intertwined in some sense), for $F$ a perfect field. The quadratic linking degrees are thus called because their definition uses quadratic intersection theory and because, in some interesting settings, they take values in either the Witt group $W(F)$, the Grothendieck-Witt group $GW(F)$ or the first Milnor-Witt $K$-theory group $K^{MW}_1(F)$, depending on the setting.
Main references: Chapters 4-5 of Lemarié--Rieusset's PhD thesis [Lem23], her article [Lemb] and her preprint [Lema].

Lecture 14 (July 9, Clémentine Lemarié --Rieusset) Computations of the quadratic linking degree
The speaker will present two particularly interesting settings for motivic linking: the setting $\mathbb{A}^2_F\setminus\{0\}\amalg\mathbb{A}^2_F\setminus\{0\}\to \mathbb{A}^4_F\setminus\{0\}$ and the setting $\mathbb{P}^1_F\amalg\mathbb{P}^1_F\to \mathbb{P}^3_F$, with $F$ a perfect field. The speaker will then compute quadratic linking degrees on examples taking place in these settings, highlighting what information is gained on how these closed subschemes of $\mathbb{A}^4_F\setminus\{0\}$ or of $\mathbb{P}^3_F$ are linked. The speaker will end the talk by a discussion of other settings and examples which are of particular interest for motivic linking.
Main references: Chapters 6-7 of Lemarié--Rieusset's PhD thesis [Lem23], her article [Lemb] and her preprint [Lema].

References

[ADF16] Aravind Asok, Brent Doran, and Jean Fasel. Smooth Models of Motivic Spheres and the Clutching Construction. International Mathematics Research Notices, 2017(6):1890--1925, 06 2016.

[BM00] Jean Barge and Fabien Morel. Groupe de Chow des cycles orientés et classe d'Euler des fibrés vectoriels. Comptes Rendus De L'Académie Des Sciences Serie I-mathematique - C R ACAD SCI SER I MATH, 330:287--290, 02 2000.

[Dro91] Yu. V. Drobotukhina. An analogue of the Jones polynomial for links in $\mathbb{R}\mathbb{P}^3$ and a generalization of the Kauffman-Murasugi theorem. Leningrad Math. J., 2(3):613--630, 1991.

[Dro94] Julia Drobotukhina. Classification of links in $\mathbb{R}\mathbb{P}^3$ with at most six crossings. Advances in Soviet Mathematics, 18(1):87--121, 1994.

[EH16] David Eisenbud and Joe Harris. 3264 and All That: A Second Course in Algebraic Geometry . Cambridge University Press, 2016.

[Fas20] Jean Fasel. Lectures on Chow-Witt groups. In Federico Binda, Marc Levine, Manh Toan Nguyen, and Oliver Röndigs, editors, Motivic homotopy theory and refined enumerative geometry, volume 745, pages 83-122. American Mathematical Soc., 2020.

[Fel21] Niels Feld. Milnor-Witt sheaves and modules. PhD thesis, Université Grenoble- Alpes, 2021. theses.hal.science/tel-03225375 .

[FF16] Anatoly Fomenko and Dmitry Fuchs. Homotopical Topology. Springer, Cham, second edition, 2016.

[Ful98] William Fulton. Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin, second edition, 1998.

[IWX23] Daniel C. Isaksen, Guozhen Wang, and Zhouli Xu. Stable homotopy groups of spheres: from dimension 0 to 90 . Publ. math. IHES, 137:107-243, 2023. link.springer.com/article/10.1007/s10240-023-00139-1.

[Lema] Clémentine Lemarié --Rieusset. Motivic linking and projective spaces. It will be on the arXiv by summer 2025.

[Lemb] Clémentine Lemarié --Rieusset. The quadratic linking degree. arxiv.org/abs/2210.11048.

[Lem23] Clémentine Lemarié --Rieusset. Motivic knot theory. PhD thesis, Université Bourgogne Franche-Comté, 2023. theses.hal.science/tel-04427361.

[Mil70] John Milnor. Algebraic $K$-theory and quadratic forms. Invent. Math., 9:318--344, 1970.

[Mor03] Fabien Morel. An introduction to $\mathbb{A}^1$-homotopy theory. ICTP Lecture Notes, 15:357--441, 01 2003.

[Mor12] Fabien Morel. $\mathbb{A}^1$-algebraic topology over a field, volume 2052. Springer, 2012.

[Ros96] Markus Rost. Chow groups with coefficients. Doc. Math., 1(16):319--393, 1996. www.emis.de/journals/DMJDMV/vol-01/16.pdf.

[Sch98] Manfred Schmid. Wittringhomologie. PhD thesis, Universit&auml:t Regensburg, 1998. www.math.uni-bielefeld.de/~rost/schmid.html.

[ST80] Herbert Seifert and William Threlfall. Seifert and Threlfall: A Textbook of Topology, volume 89 of Pure and Applied Mathematics. Academic Press, Inc. New York-London, 1980. Translated from the German edition of 1934 by Michael A. Goldman, With a preface by Joan S. Birman, With Topology of 3-dimensional fibered spaces by Seifert, Translated from the German by Wolfgang Heil.

[Vir07] Julia Viro. Linking number in a projective space as the degree of a map. Journal of Knot Theory and Its Ramifications, 16(4):489--497, 2007.

[VV21] Julia Viro and Oleg Viro. Fundamental groups in projective knot theory. In Topology and Geometry, A Collection of Essays Dedicated to Vladimir G. Turaev, Athanase Papadopoulos, editor, pages 75--92. EMS Press, Berlin, 2021.