# On the cohomology of reciprocity sheaves

@article{Binda2020OnTC,
title={On the cohomology of reciprocity sheaves},
author={Federico Binda and Kay R{\"u}lling and Shuji Saito},
journal={Forum of Mathematics, Sigma},
year={2020},
volume={10}
}
• Published 7 October 2020
• Mathematics
• Forum of Mathematics, Sigma
Abstract In this paper, we show the existence of an action of Chow correspondences on the cohomology of reciprocity sheaves. In order to do so, we prove a number of structural results, such as a projective bundle formula, a blow-up formula, a Gysin sequence and the existence of proper pushforward. In this way, we recover and generalise analogous statements for the cohomology of Hodge sheaves and Hodge-Witt sheaves. We give several applications of the general theory to problems which have been…
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## References

SHOWING 1-10 OF 82 REFERENCES

### The Brauer–Grothendieck Group

• Mathematics
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
• 2021

### Motives with modulus, III: The categories of motives

• Mathematics, Materials Science
Annals of K-Theory
• 2022
We construct and study a triangulated category of motives with modulus $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ over a field $k$ that extends Voevodsky's category

### Triangulated categories of logarithmic motives over a field

• Mathematics
• 2020
In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the

### Cancellation theorems for reciprocity sheaves

• Mathematics
• 2020
We prove cancellation theorems for reciprocity sheaves and cube-invariant modulus sheaves with transfers of Kahn--Saito--Yamazaki, generalizing Voevodsky's cancellation theorem for

### Tensor structures in the theory of modulus presheaves with transfers

• Mathematics
Mathematische Zeitschrift
• 2021
The tensor product of $${\mathbb {A}}^1$$ A 1 -invariant sheaves with transfers introduced by Voevodsky is generalized to reciprocity sheaves via the theory of modulus presheaves with transfers. We

### Motives with modulus, II: Modulus sheaves with transfers for proper modulus pairs

• Mathematics
Épijournal de Géométrie Algébrique
• 2021
We develop a theory of sheaves and cohomology on the category of proper modulus pairs. This complements [KMSY21], where a theory of sheaves and cohomology on the category of non-proper modulus pairs

### 3.

• Aulus Gellius
• Biology
Journal of the Royal Asiatic Society of Great Britain & Ireland
• 1893
It is suggested that polyploidization itself is not necessary in producing novel petal color patterns, and duplications of R2R3 - 36 MYB genes in the common ancestor of the two progenitors have apparently facilitated diversification of petal pigmentation patterns.

### Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairs

• Mathematics
Épijournal de Géométrie Algébrique
• 2021
We develop a theory of modulus sheaves with transfers, which generalizes Voevodsky's theory of sheaves with transfers. This paper and its sequel are foundational for the theory of motives with

### Smooth blowup square for motives with modulus

• Mathematics
Bulletin of the Polish Academy of Sciences Mathematics
• 2022
In this self-contained paper we prove that Voevodsky's smooth blowup triangle of motives generalises to a smooth blowup triangle of motives with modulus.