---- > [!definition] Definition. ([[regular local ring]]) > Let $(R, \mathfrak{m})$ be a [[local ring|local]] [[Noetherian ring|Noetherian]] [[ring]] with [[residue field]] $k=R / \mathfrak{m}$. We say $R$ is **regular** if $\text{dim}_{k} \frac{\mathfrak{m}}{\mathfrak{m}^{2}}=\text{dim }R$ Note the left quantity is a [[vector space]] [[dimension]] while the right quantity is a [[Krull dimension|Krull (ring) dimension]]. Often the [[quotient module|quotient]] $k$-[[module]] $\mathfrak{m / \mathfrak{m}}^{2}$ has a sort of '[[Zariski tangent space|Zariski cotangent space]]' connotation. ^definition > [!justification] What is the $k$-[[vector space]] $\mathfrak{m} / \mathfrak{m}^{2}$? > As [[ideal|ideals]] of $R$, $\mathfrak{m}$ and $\mathfrak{m}^{2}$ naturally carry $R$-([[submodule|sub]])[[module]] structure. [[quotient module|Quotienting]] gives a canonical $R$-[[module]] $\frac{\mathfrak{m}}{\mathfrak{m}^{2}}$. Call the structure map $\rho:R \to \text{End}_{\mathsf{Ab}} \frac{\mathfrak{m}}{\mathfrak{m}^{2}}$, $r \xmapsto{\rho} \big(r \cdot(-) + \mathfrak{m}^{2} \big)$. [[characterization of quotienting a ring|This action]] $\rho$ factors through $k=\frac{R}{\mathfrak{m}}$ since[^1] $\mathfrak{m} \subset \ker \rho$: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACUQBfU9TXfIRQBGclVqMWbADrScMAB45gAUTBQuAfVkBbOjgAWcAGbAAggCMuu-QeMAnOgGtgOrt14gM2PASJlhcXpmVkQQJwBedgB6G0MHZ1d3LnEYKABzeCJQBwgdJDIQHAgkACYeHPs8gupipFEJEJlpewMSipBc-MRS2pLEBoYsMFCQKDo4AzSQamCpMNlFLDgcOAACAEIZkAY6CxgGAAV+XyEQeyx0gxxuCi4gA > \begin{tikzcd} > R \arrow[d] \arrow[r, "\rho"] & \text{End}_\mathsf{Ab}\frac{\mathfrak{m}}{\mathfrak{m}^2} \\ > k=R/\mathfrak{m} \arrow[ru, "\exists !"', dashed] & > \end{tikzcd} > \end{document} > ``` > This naturally endows $\frac{\mathfrak{m}}{\mathfrak{m}^{2}}$ with $k$-[[module]] ($k$-[[vector space]]) structure. > [!intuition] > Maybe some intuition comes from [[smooth manifold]] theory (especially of the embedded Euclidean flavor a la Math 396 or Math 433). Viewing $\frac{\mathfrak{m}}{\mathfrak{m}^{2}}$ like a [[tangent space to a manifold|tangent space]], '[[differentiable Euclidean submanifold|regularity]]' is like saying that this tangent space should not be degenerate, i.e., its dimension should always match that of the manifold. ^intuition ---- #### [^1]: Indeed, if $m \in \mathfrak{m}$ then $\rho(m)=\big( m \cdot (-) + \mathfrak{m}^{2} \big)$, and $mm' \in \mathfrak{m}^{2}$ for all $m'$, so $\rho(m)$ is the zero map. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```