# Fangzhou Jin

My work deals with foundational aspects of motivic homotopy theory. Here are some details of the papers published or posted while being supported by the ERC project.

The paper [DFJK21] (with F. Déglise, J. Fasel, and A. Khan) establishes a number of properties of the motivic stable homotopy category with rational coefficients: absolute purity, constructibility, local duality and SL-orientability. The \(\mathbb{A}^1\)-bivariant groups are identified with the Borel-Moore type Chow-Witt groups rationally.

The two preprints [JX20] (with H. Xie) and [Jin20a] study questions related to real étale cohomology. In the work [JX20] we discuss the relation between coherent duality and real étale duality via a Gersten-type complex; this is relevant for many questions arising in the ERC project due to the connection of coherent duality with the pushforward maps in hermitian \(K\)-theory.

In the work [Jin20a], we compute the Grothendieck group of the rational motivic stable homotopy category for reasonable schemes.

In the work [Jin20b] we define trace maps and local terms in the stable motivic homotopy category over a general base scheme, and prove an analogue of a theorem of Varshavsky: for a contracting correspondence, the local term agrees with an invariant of simpler form called the naive local term; we also show that some invariants from \(\mathbb{A}^1\)-enumerative geometry, such as the \(\mathbb{A}^1\)-Brouwer local degree and the local Euler class, can be interpreted as local terms.

**Publications**

[DFJK21] F. Déglise, J. Fasel, F. Jin, A. Khan, *On the rational motivic homotopy category*, J. Ec. Polytech. Math. 8 (2021), 533-583.

Publically available at arXiv:2005.10147 math.AG doi 10.5802/jep.153

**Preprints**

[JX20] F. Jin, H. Xie, *A Gersten complex on real schemes*, arXiv:2007.04625 math.AG

[Jin20a] F. Jin, *On some finiteness results in real étale cohomology*, arXiv:2009.00396 math.AG

[Jin20b] F. Jin, *Local terms of the motivic Verdier pairing*, arXiv:2010.09292 math.AG

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