*Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven*

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.003.0002
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter provides a background on commutative algebra and gives a self-contained proof of Johnson's Theorem 5.9.1 on regular solutions of systems of algebraic differential equations. It presents ...
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This chapter provides a background on commutative algebra and gives a self-contained proof of Johnson's Theorem 5.9.1 on regular solutions of systems of algebraic differential equations. It presents the facts on regular local rings and Kähler differentials needed for Theorem 5.9.1. It also recalls a common notational convention concerning a commutative ring R and an R-module M, with U and V as additive subgroups of R and M. Other topics include the Zariski topology, noetherian rings and spaces, rings and modules of finite length, integral extensions and integrally closed domains, Krull's Principal Ideal Theorem, differentials, and derivations on field extensions.Less

This chapter provides a background on commutative algebra and gives a self-contained proof of Johnson's Theorem 5.9.1 on regular solutions of systems of algebraic differential equations. It presents the facts on regular local rings and Kähler differentials needed for Theorem 5.9.1. It also recalls a common notational convention concerning a commutative ring *R* and an *R*-module *M*, with *U* and *V* as additive subgroups of *R* and *M*. Other topics include the Zariski topology, noetherian rings and spaces, rings and modules of finite length, integral extensions and integrally closed domains, Krull's Principal Ideal Theorem, differentials, and derivations on field extensions.

*Peter Scholze and Jared Weinstein*

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0014
- Subject:
- Mathematics, Geometry / Topology

This chapter presents a third lecture on one-legged shtukas. The goal is to complete the proof of Fargues' theorem. To complete the proof of Fargues' theorem, it remains to prove the following ...
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This chapter presents a third lecture on one-legged shtukas. The goal is to complete the proof of Fargues' theorem. To complete the proof of Fargues' theorem, it remains to prove the following result, where Y = Spa Ainf REVERSE SOLIDUS {xk}. Theorem 14.2.1 posits that there is an equivalence of categories between finite free Ainf-modules and vector bundles on Y. One should think of this as being an analogue of a classical result: If (R, m) is a 2-dimensional regular local ring, then finite free R-modules are equivalent to vector bundles on (Spec R)REVERSE SOLIDUS {m}. The chapter then provides a proof of Theorem 14.2.1.Less

This chapter presents a third lecture on one-legged shtukas. The goal is to complete the proof of Fargues' theorem. To complete the proof of Fargues' theorem, it remains to prove the following result, where *Y* = Spa A_{inf} REVERSE SOLIDUS {*x*_{k}}. Theorem 14.2.1 posits that there is an equivalence of categories between finite free A_{inf}-modules and vector bundles on *Y*. One should think of this as being an analogue of a classical result: If (*R*, m) is a 2-dimensional regular local ring, then finite free *R*-modules are equivalent to vector bundles on (Spec *R*)REVERSE SOLIDUS {m}. The chapter then provides a proof of Theorem 14.2.1.

*Christian Haesemeyer and Charles A. Weibel*

- Published in print:
- 2019
- Published Online:
- January 2020
- ISBN:
- 9780691191041
- eISBN:
- 9780691189635
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191041.003.0005
- Subject:
- Mathematics, Geometry / Topology

This chapter fixes a Rost variety 𝑋 for a sequence. It constructs a Rost motive 𝑀 = (𝑋, 𝑒) with coefficients ℤ(𝓁) under the inductive assumption that BL(n − 1) holds and discusses three important ...
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This chapter fixes a Rost variety 𝑋 for a sequence. It constructs a Rost motive 𝑀 = (𝑋, 𝑒) with coefficients ℤ(𝓁) under the inductive assumption that BL(n − 1) holds and discusses three important axioms. It introduces a candidate for the Rost motive and demonstrates how a motive satisfies two axioms. To further aid in the proof, the chapter argues that End(𝑀) is a local ring and then verifies an axiom proving that 𝑀 is a Rost motive whenever 𝑋 is a Rost variety. Finally, the chapter considers the historical background behind these equations. It reveals the eponymous Rost motive and considers Voevodsky's own construction of the Rost motive.Less

This chapter fixes a Rost variety 𝑋 for a sequence. It constructs a Rost motive 𝑀 = (𝑋, 𝑒) with coefficients ℤ_{(𝓁)} under the inductive assumption that BL(n − 1) holds and discusses three important axioms. It introduces a candidate for the Rost motive and demonstrates how a motive satisfies two axioms. To further aid in the proof, the chapter argues that End(𝑀) is a local ring and then verifies an axiom proving that 𝑀 is a Rost motive whenever 𝑋 is a Rost variety. Finally, the chapter considers the historical background behind these equations. It reveals the eponymous Rost motive and considers Voevodsky's own construction of the Rost motive.