On some 𝑝-adic and mod 𝑝 representations of quaternion algebra over ℚ_𝑝

We say 𝑀 is Cohen–Macaulay if Ext Λ i ⁡ ( M , Λ ) = 0 for any i ≠ j Λ ⁢ ( M ) , and 𝑀 is pure if any nonzero submodule has the same grade as 𝑀.

Cohen–Macaulay Λ-modules are pure. Conversely, a pure Λ-module of grade d − 1 is Cohen–Macaulay.

Proof

The first statement is well known (for example, by combining [38, Proposition 3.5 (v) (a), Proposition 3.9 (i)]). For the second, we first prove that Ext Λ d ⁡ ( M , Λ ) = 0 if 𝑀 is a pure finitely generated Λ-module 𝑀 with j Λ ⁢ ( M ) ≠ d . If not, then j Λ ⁢ ( Ext Λ d ⁡ ( M , Λ ) ) = d by Auslander’s condition. But, since 𝑀 is pure and j Λ ⁢ ( M ) ≠ d , it follows from [26, Chapter III, Theorem 4.2.6] that Ext Λ d ⁡ ( Ext Λ d ⁡ ( M , Λ ) , Λ ) = 0 , a contradiction. This clearly implies the second statement by definition. ∎

Let 𝑀 be a finitely generated Λ-module of grade d − 1 . By [38, §3.1], 𝑀 has a unique submodule M d whose grade is 𝑑 (equivalently M d is finite-dimensional over 𝔽). Then M / M d is pure of grade d − 1 , hence Cohen–Macaulay by Lemma A.2.

Let 𝑀 be a pure finitely generated Λ-module. Two good filtrations 𝐹 and F ′ of 𝑀 are called tamely close if j R Λ ⁢ ( M F + F ′ / M F ∩ F ′ ) ≥ j Λ ⁢ ( M ) + 2 .

Let 𝑀 be a pure finitely generated Λ-module. Then every equivalence class of good filtrations on 𝑀 contains a unique 𝐹 such that gr F ⁡ ( M ) is a pure gr ⁡ ( Λ ) -module of grade equal to j Λ ⁢ ( M ) . Moreover, if F ′ is another good filtration in the equivalence class of 𝐹, then F ′ < F in the sense that F ′ ⁣ n ⁢ M ⊆ F n ⁢ M for all n ∈ Z , and the kernel of the induced morphism gr F ′ ⁡ ( M ) → gr F ⁡ ( M ) is identified with the largest submodule of gr F ′ ⁡ ( M ) of grade greater than j Λ ⁢ ( M ) . In particular, the morphism gr F ′ ⁡ ( M ) → gr F ⁡ ( M ) is nonzero.

Proof

The first statement and the fact F ′ < F are proved in [2, Theorem 5.23]. For the convenience of the reader, we recall the proof. Below, we write

here the second equality is a standard fact; see [26, Chapter III, Theorem 2.5.2].

First, the construction of 𝐹 in a given equivalence class of good filtrations such that gr F ⁡ ( M ) is pure is well known; see [2, Lemma 5.19, Corollary 5.21], or [26, Chapter III, Theorem 4.2.13 (4)] for a more detailed presentation. For the uniqueness of 𝐹, we need to show that M F = M F + F ′ for any good filtration F ′ which is tamely close to 𝐹. Put Q = M F + F ′ / M F . Then j R Λ ⁢ ( Q ) ≥ n + 2 by Definition A.4. Using the 𝑇-torsion-freeness of M F + F ′ and (A.1), multiplication by 𝑇 induces an exact sequence

where Ker T ⁡ ( Q ) denotes the kernel of T : Q → Q . Note that, since R Λ / T ≅ gr ⁡ ( Λ ) , Ker T ⁡ ( Q ) has a gr ⁡ ( Λ ) -module structure. Since 𝑇 is a central nonzero-divisor in R Λ , it is easy to see that

where the first inequality holds as Ker T ⁡ ( Q ) ⊆ Q . However, gr F ⁡ ( M ) is pure of grade 𝑛 by construction; any nonzero submodule of gr F ⁡ ( M ) also has grade 𝑛. This forces Ker T ⁡ ( Q ) = 0 . On the other hand, since 𝐹 and F + F ′ are both good filtrations on 𝑀, we have

(here [ T − 1 ] means taking localization at the multiplicative subset { T k : k ≥ 0 } ). This implies that 𝑄 is a 𝑇-torsion R Λ -module. Combined with the fact Ker T ⁡ ( Q ) = 0 , we get Q = 0 , as required.

The last assertion essentially follows from the above argument. More precisely, since F ′ < F , we may consider the following short exact sequence:

As above, multiplication by 𝑇 induces an exact sequence

Since j gr ⁡ ( Λ ) ⁢ ( Ker T ⁡ ( M F / M F ′ ) ) ≥ n + 1 and gr F ⁡ ( M ) is pure of grade 𝑛, the result easily follows. ∎