Lemma 107.4.4. Let $\mathcal{X}$ be an algebraic stack locally of finite type over a locally Noetherian scheme $S$. Let $x_0 : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ is a morphism where $k$ is a field of finite type over $S$ with image $s \in S$. If $\mathcal{O}_{S, s}$ is a G-ring, then the map of Lemma 107.4.2 preserves multiplicities.
Proof. By Lemma 107.2.8 we may assume there is a smooth morphism $U \to \mathcal{X}$ where $U$ is a scheme and a $k$-valued point $u_0$ of $U$ such that $\mathcal{O}_{U, u_0}^\wedge $ is a versal ring to $\mathcal{X}$ at $x_0$. By construction of our map in the proof of Lemma 107.4.2 (which simplifies greatly because $A = \mathcal{O}_{U, u_0}^\wedge $) we find that it suffices to show: the multiplicity of an irreducible component of $U$ passing through $u_0$ is the same as the multiplicity of any irreducible component of $\mathop{\mathrm{Spec}}(\mathcal{O}_{U, u_0}^\wedge )$ mapping into it.
Translated into commutative algebra we find the following: Let $C = \mathcal{O}_{U, u_0}$. This is essentially of finite type over $\mathcal{O}_{S, s}$ and hence is a G-ring (More on Algebra, Proposition 15.50.10). Then $A = C^\wedge $. Therefore $C \to A$ is a regular ring map. Let $\mathfrak q \subset C$ be a minimal prime and let $\mathfrak p \subset A$ be a minimal prime lying over $\mathfrak q$. Then
\[ R = C_\mathfrak p \longrightarrow A_\mathfrak p = R' \]
is a regular ring map of Artinian local rings. For such a ring map it is always the case that
\[ \text{length}_ R R = \text{length}_{R'} R' \]
This is what we have to show because the left hand side is the multiplicity of our component on $U$ and the right hand side is the multiplicity of our component on $\mathop{\mathrm{Spec}}(A)$. To see the equality, first we use that
\[ \text{length}_ R(R) \text{length}_{R'}(R'/\mathfrak m_ R R') = \text{length}_{R'}(R') \]
by Algebra, Lemma 10.52.13. Thus it suffices to show $\mathfrak m_ R R' = \mathfrak m_{R'}$, which is a consequence of being a regular homomorphism of zero dimensional local rings. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)