Proof.
Note that \mathcal{O}_{Y, y} is a Noetherian local ring. Consider the canonical morphism c : \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}) \to Y, see Schemes, Equation (26.13.1.1). This is a flat morphism as it identifies local rings. Denote momentarily f' : X' \to \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}) the base change of f to this local ring. We see that c^*R^ pf_*\mathcal{F} = R^ pf'_*\mathcal{F}' by Cohomology of Spaces, Lemma 69.11.2. Moreover, the fibres X_ y and X'_ y are identified. Hence we may assume that Y = \mathop{\mathrm{Spec}}(A) is the spectrum of a Noetherian local ring (A, \mathfrak m, \kappa ) and y \in Y corresponds to \mathfrak m. In this case R^ pf_*(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d})_ y = H^ p(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) for all p \geq 0. Denote f_ y : X_ y \to \mathop{\mathrm{Spec}}(\kappa ) the projection.
Let B = \text{Gr}_\mathfrak m(A) = \bigoplus _{n \geq 0} \mathfrak m^ n/\mathfrak m^{n + 1}. Consider the sheaf \mathcal{B} = f_ y^*\widetilde{B} of quasi-coherent graded \mathcal{O}_{X_ y}-algebras. We will use notation as in Cohomology of Spaces, Section 69.22 with I replaced by \mathfrak m. Since X_ y is the closed subspace of X cut out by \mathfrak m\mathcal{O}_ X we may think of \mathfrak m^ n\mathcal{F}/\mathfrak m^{n + 1}\mathcal{F} as a coherent \mathcal{O}_{X_ y}-module, see Cohomology of Spaces, Lemma 69.12.8. Then \bigoplus _{n \geq 0} \mathfrak m^ n\mathcal{F}/\mathfrak m^{n + 1}\mathcal{F} is a quasi-coherent graded \mathcal{B}-module of finite type because it is generated in degree zero over \mathcal{B} abd because the degree zero part is \mathcal{F}_ y = \mathcal{F}/\mathfrak m \mathcal{F} which is a coherent \mathcal{O}_{X_ y}-module. Hence by Cohomology of Schemes, Lemma 30.19.3 part (2) there exists a d_0 such that
H^ p(X_ y, \mathfrak m^ n \mathcal{F}/ \mathfrak m^{n + 1}\mathcal{F} \otimes _{\mathcal{O}_{X_ y}} \mathcal{L}_ y^{\otimes d}) = 0
for all p > 0, d \geq d_0, and n \geq 0. By Cohomology of Spaces, Lemma 69.8.3 this is the same as the statement that H^ p(X, \mathfrak m^ n \mathcal{F}/ \mathfrak m^{n + 1}\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) = 0 for all p > 0, d \geq d_0, and n \geq 0.
Consider the short exact sequences
0 \to \mathfrak m^ n\mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \to \mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \to \mathcal{F}/\mathfrak m^ n \mathcal{F} \to 0
of coherent \mathcal{O}_ X-modules. Tensoring with \mathcal{L}^{\otimes d} is an exact functor and we obtain short exact sequences
0 \to \mathfrak m^ n\mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d} \to \mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d} \to \mathcal{F}/\mathfrak m^ n \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d} \to 0
Using the long exact cohomology sequence and the vanishing above we conclude (using induction) that
H^ p(X, \mathcal{F}/\mathfrak m^ n \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) = 0 for all p > 0, d \geq d_0, and n \geq 0, and
H^0(X, \mathcal{F}/\mathfrak m^ n \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) \to H^0(X_ y, \mathcal{F}_ y \otimes _{\mathcal{O}_{X_ y}} \mathcal{L}_ y^{\otimes d}) is surjective for all d \geq d_0 and n \geq 1.
By the theorem on formal functions (Cohomology of Spaces, Theorem 69.22.5) we find that the \mathfrak m-adic completion of H^ p(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) is zero for all d \geq d_0 and p > 0. Since H^ p(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) is a finite A-module by Cohomology of Spaces, Lemma 69.20.3 it follows from Nakayama's lemma (Algebra, Lemma 10.20.1) that H^ p(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) is zero for all d \geq d_0 and p > 0. For p = 0 we deduce from Cohomology of Spaces, Lemma 69.22.4 part (3) that H^0(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) \to H^0(X_ y, \mathcal{F}_ y \otimes _{\mathcal{O}_{X_ y}} \mathcal{L}_ y^{\otimes d}) is surjective, which gives the final statement of the lemma.
\square
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