Return to Answer

added 331 characters in body

Source Link

edited May 20, 2019 at 20:27

46
2

The usual crystalline comparison theorem involves a comparison of cohomology between the generic and special fibre of a proper smooth scheme over a $p$-adic discrete valuation ring. To formulate an analog where the generic fibre is merely a rigid space, one option is to use formal models, as in the following result.

Theorem: Fix a finite extension $K/\mathbf{Q}_p$. Say $\mathfrak{X}/\mathcal{O}_K$ is a proper smooth proper formal scheme whose generic fibre (in the sense of rigid analytic geometry) is the proper smooth rigid space $X/K$. Then there is a crystalline comparison isomorphism relating the crystalline cohomology of the special fibre $\mathfrak{X}_0$ of $\mathfrak{X}/\mathcal{O}_K$ with the 'etale cohomology of the geometric generic fibre $X_{\overline{K}}$.

One reference is the paper by Tan-Tong (https://arxiv.org/abs/1510.05543), settling the above result over unramified bases. They follow the Faltings-Scholze's method, build on previous work of Andreatta-Iovita (https://arxiv.org/abs/1212.3813), and also prove a result for non-trivial coefficient systems.

A reference for the general statement is the paper by Colmez-Niziol (https://arxiv.org/abs/1505.06471). This paper proves a general comparison result for semistable formal schemes, but uses a different method than Scholze's paper.

The paper by Bhatt-Morrow-Scholze (https://arxiv.org/abs/1602.03148) also includes a generalization of the above theorem for formal schemes that are merely defined over the valuation ring of algebraically closed field. This proof uses methods closer to the Faltings-Scholze approach, but with integral enhancements.

The usual crystalline comparison theorem involves a comparison of cohomology between the generic and special fibre of a proper smooth scheme over a $p$-adic discrete valuation ring. To formulate an analog where the generic fibre is merely a rigid space, one option is to use formal models, as in the following result.

Theorem: Fix a finite extension $K/\mathbf{Q}_p$. Say $\mathfrak{X}/\mathcal{O}_K$ is a proper smooth proper formal scheme whose generic fibre (in the sense of rigid analytic geometry) is the proper smooth rigid space $X/K$. Then there is a crystalline comparison isomorphism relating the crystalline cohomology of the special fibre $\mathfrak{X}_0$ of $\mathfrak{X}/\mathcal{O}_K$ with the 'etale cohomology of the geometric generic fibre $X_{\overline{K}}$.

One reference is the paper by Colmez-Niziol (https://arxiv.org/abs/1505.06471). This paper proves a general comparison result for semistable formal schemes, but uses a different method than Scholze's paper.

The paper by Bhatt-Morrow-Scholze (https://arxiv.org/abs/1602.03148) also includes a generalization of the above theorem for formal schemes that are merely defined over the valuation ring of algebraically closed field. This proof uses methods closer to the Faltings-Scholze approach, but with integral enhancements.

The usual crystalline comparison theorem involves a comparison of cohomology between the generic and special fibre of a proper smooth scheme over a $p$-adic discrete valuation ring. To formulate an analog where the generic fibre is merely a rigid space, one option is to use formal models, as in the following result.

Theorem: Fix a finite extension $K/\mathbf{Q}_p$. Say $\mathfrak{X}/\mathcal{O}_K$ is a proper smooth proper formal scheme whose generic fibre (in the sense of rigid analytic geometry) is the proper smooth rigid space $X/K$. Then there is a crystalline comparison isomorphism relating the crystalline cohomology of the special fibre $\mathfrak{X}_0$ of $\mathfrak{X}/\mathcal{O}_K$ with the 'etale cohomology of the geometric generic fibre $X_{\overline{K}}$.

One reference is the paper by Tan-Tong (https://arxiv.org/abs/1510.05543), settling the above result over unramified bases. They follow the Faltings-Scholze's method, build on previous work of Andreatta-Iovita (https://arxiv.org/abs/1212.3813), and also prove a result for non-trivial coefficient systems.

A reference for the general statement is the paper by Colmez-Niziol (https://arxiv.org/abs/1505.06471). This paper proves a general comparison result for semistable formal schemes, but uses a different method than Scholze's paper.

The paper by Bhatt-Morrow-Scholze (https://arxiv.org/abs/1602.03148) also includes a generalization of the above theorem for formal schemes that are merely defined over the valuation ring of algebraically closed field. This proof uses methods closer to the Faltings-Scholze approach, but with integral enhancements.

Source Link

created May 20, 2019 at 19:52

46
2

The usual crystalline comparison theorem involves a comparison of cohomology between the generic and special fibre of a proper smooth scheme over a $p$-adic discrete valuation ring. To formulate an analog where the generic fibre is merely a rigid space, one option is to use formal models, as in the following result.

Theorem: Fix a finite extension $K/\mathbf{Q}_p$. Say $\mathfrak{X}/\mathcal{O}_K$ is a proper smooth proper formal scheme whose generic fibre (in the sense of rigid analytic geometry) is the proper smooth rigid space $X/K$. Then there is a crystalline comparison isomorphism relating the crystalline cohomology of the special fibre $\mathfrak{X}_0$ of $\mathfrak{X}/\mathcal{O}_K$ with the 'etale cohomology of the geometric generic fibre $X_{\overline{K}}$.

One reference is the paper by Colmez-Niziol (https://arxiv.org/abs/1505.06471). This paper proves a general comparison result for semistable formal schemes, but uses a different method than Scholze's paper.

The paper by Bhatt-Morrow-Scholze (https://arxiv.org/abs/1602.03148) also includes a generalization of the above theorem for formal schemes that are merely defined over the valuation ring of algebraically closed field. This proof uses methods closer to the Faltings-Scholze approach, but with integral enhancements.