Milnor K-theory of complete discrete valuation rings with finite residue fields

Dahlhausen, Christian

doi:10.1016/j.jpaa.2017.07.002

Cited by 8 publications

(6 citation statements)

References 9 publications

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“…In this section we prove the Gesten conjecture for henselian DVRs with finite residue field. This follows from the complete case proved in [4] by an algebraization technique. The result is independent of the rest of the results of the article.…”

Section: Gersten Conjecture For Henselian Dvrs With Finite Residue Fieldmentioning

confidence: 79%

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On the relative Gersten conjecture for Milnor K-theory in the smooth case

Lüders¹

2020

Preprint

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Section: Gersten Conjecture For Henselian Dvrs With Finite Residue Fieldmentioning

confidence: 79%

“…The result is independent of the rest of the results of the article. and the fact that the upper horizontal map is injective by [4].…”

Section: Gersten Conjecture For Henselian Dvrs With Finite Residue Fieldmentioning

confidence: 99%

On the relative Gersten conjecture for Milnor K-theory in the smooth case

Lüders¹

2020

Preprint

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“…In this workflow, students culture microbes from almost any environmental sample, generate isolates from the sample(s), and use growth curves of the isolates to quantify aspects of the organism’s biology. The workflow was initially piloted as a less quantitatively oriented seminar course, using environmental samples ranging from nectar ( Dahlhausen, 2018 ) to abalone ( Vater et al, 2016 ) to koala feces ( Coil, 2017 ). These courses and other CUREs have been discussed by Vater et al (2019) .…”

Section: Methodsmentioning

confidence: 99%

Combining Microbial Culturing With Mathematical Modeling in an Introductory Course-Based Undergraduate Research Experience

et al. 2020

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Quantitative techniques are a critical part of contemporary biology research, but students interested in biology enter college with widely varying quantitative skills and attitudes toward mathematics. Course-based undergraduate research experiences (CUREs) may be an early way to build student competency and positive attitudes. Here we describe the design, implementation, and assessment of an introductory quantitative CURE focused on halophilic microbes. In this CURE, students culture and isolate halophilic microbes from environmental and food samples, perform growth assays, then use mathematical modeling to quantify the growth rate of strains in different salinities. To assess how the course may impact students' future academic plans and attitudes toward the use of math in biology, we used pre-and post-quarter surveys. Students who completed the course showed more positive attitudes toward science learning and an increased interest in pursuing additional quantitative biology experiences. We argue that the classroom application of microbiology methods, combined with mathematical modeling using student-generated data, provides a degree of student ownership, collaboration, iteration, and discovery that makes quantitative learning both relevant and exciting to students.

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“…y y 0 in which the top row is exact by (9) and the bottom row by Corollary 2.15(ii). The map ρ 2 j ⊕ σ 2 j is that of Corollary 2.15(ii).…”

Section: An Application To the Gersten Conjecturementioning

confidence: 97%

“…Remark 4.7. The focus of this subsection is in the incomplete case, but we mention that K M j (O) → K M j (F ) is known to be injective for all j ≥ 0 if O is compelte and has finite residue field [9].…”

Section: An Application To the Gersten Conjecturementioning

confidence: 99%

Milnor $K$-theory of $p$-adic rings

Lüders¹,

Morrow²

2021

Preprint

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We study the mod p r Milnor K-groups of p-adically complete and p-henselian rings, establishing in particular a Nesterenko-Suslin style description in terms of the Milnor range of syntomic cohomology. In the case of smooth schemes over complete discrete valuation rings we prove the mod p r Gersten conjecture for Milnor K-theory locally in the Nisnevich topology. In characteristic p we show that the Bloch-Kato-Gabber theorem remains true for valuation rings, and for regular formal schemes in a pro sense.1 Formulation of main theorems for p-henselian, ind-smooth algebrasIn this section we state our main theorems concerning ind-smooth algebras over complete discrete valuation rings, and establish some relations between them as well as the key reductions to the special case which will then be proved in Section 2. Main theoremsWe adopt the usual definition of Milnor K-theory, even for non-local rings:Definition 1.1 (Milnor K-theory). Let R be a (always commutative) ring. We define the j th Milnor K-group K M j (R) to be the quotient of (R × ) ⊗j by the Steinberg relations, i.e. the subgroup of (R × ) ⊗j generated by elements of the form a 1 ⊗ • • • ⊗ a j where a l + a k = 1 for some 1 ≤ l < k ≤ j. As usual, the image of a 1 ⊗ • • • ⊗ a j in K M j (R) is denoted by {a 1 , . . . , a j }.

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Milnor K-theory of complete discrete valuation rings with finite residue fields

Cited by 8 publications

References 9 publications

On the relative Gersten conjecture for Milnor K-theory in the smooth case

On the relative Gersten conjecture for Milnor K-theory in the smooth case

Combining Microbial Culturing With Mathematical Modeling in an Introductory Course-Based Undergraduate Research Experience

Milnor $K$-theory of $p$-adic rings

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