SherbyNet - /page/wikibook/entry.mu

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Regular embedding

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Regular_scheme" title="Regular scheme">regular scheme</a>.</div>

In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, a <a href="Closed_immersion" title="Closed immersion">closed immersion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:X\hookrightarrow Y}">

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<annotation encoding="application/x-tex">{\displaystyle i:X\hookrightarrow Y}</annotation>

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</math><img src="./29f3f0662c620237a4fae4347bc9e48f13e34e47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.4ex; height:2.176ex;" alt="{\displaystyle i:X\hookrightarrow Y}" loading="lazy"> of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\cap U}">

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<annotation encoding="application/x-tex">{\displaystyle X\cap U}</annotation>

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</math><img src="./f371950b187adac5e4fe685d411a7fd09c0e082c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.345ex; height:2.176ex;" alt="{\displaystyle X\cap U}" loading="lazy"> is generated by a <a href="Regular_sequence" title="Regular sequence">regular sequence</a> of length r. A regular embedding of codimension one is precisely an <a href="Effective_Cartier_divisor" class="mw-redirect" title="Effective Cartier divisor">effective Cartier divisor</a>.

<div class="mw-heading mw-heading2"><h2 id="Examples_and_usage">Examples and usage</h2></div>

For example, if X and Y are <a href="Smooth_morphism" title="Smooth morphism">smooth</a> over a scheme S and if i is an S-morphism, then i is a regular embedding. In particular, every section of a smooth morphism is a regular embedding.<a href="#cite_note-1">[1]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} B}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} B}</annotation>

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</math><img src="./624cf3b2f5b4c7735d78ebd18c53cacb9a991dc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.801ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} B}" loading="lazy"> is regularly embedded into a <a href="Regular_scheme" title="Regular scheme">regular scheme</a>, then B is a <a href="Complete_intersection_ring" title="Complete intersection ring">complete intersection ring</a>.<a href="#cite_note-2">[2]</a>

The notion is used, for instance, in an essential way in Fulton's approach to <a href="Intersection_theory" title="Intersection theory">intersection theory</a>. The important fact is that when i is a regular embedding, if I is the ideal sheaf of X in Y, then the <a href="Normal_sheaf" class="mw-redirect" title="Normal sheaf">normal sheaf</a>, the dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I/I^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle I/I^{2}}</annotation>

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</math><img src="./19d4b3432c9980cba099ba225470ef439bd08262.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.605ex; height:3.176ex;" alt="{\displaystyle I/I^{2}}" loading="lazy">, is locally free (thus a vector bundle) and the natural map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sym} (I/I^{2})\to \oplus _{0}^{\infty }I^{n}/I^{n+1}}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} (I/I^{2})\to \oplus _{0}^{\infty }I^{n}/I^{n+1}}</annotation>

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</math><img src="./c8acd888ef3a40e7b670b22a0bbe0cb9f54c23c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.301ex; height:3.343ex;" alt="{\displaystyle \operatorname {Sym} (I/I^{2})\to \oplus _{0}^{\infty }I^{n}/I^{n+1}}" loading="lazy"> is an isomorphism: the <a href="Normal_cone_(algebraic_geometry)" class="mw-redirect" title="Normal cone (algebraic geometry)">normal cone</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} (\oplus _{0}^{\infty }I^{n}/I^{n+1})}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} (\oplus _{0}^{\infty }I^{n}/I^{n+1})}</annotation>

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</math><img src="./da639fb023c2f5c0165a4096f024c99e31e174e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.275ex; height:3.343ex;" alt="{\displaystyle \operatorname {Spec} (\oplus _{0}^{\infty }I^{n}/I^{n+1})}" loading="lazy"> coincides with the normal bundle.

<div class="mw-heading mw-heading3"><h3 id="Non-examples">Non-examples</h3></div>

One non-example is a scheme which isn't equidimensional. For example, the scheme

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<annotation encoding="application/x-tex">{\displaystyle X={\text{Spec}}\left({\frac {\mathbb {C} [x,y,z]}{(xz,yz)}}\right)}</annotation>

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</math><img src="./9d0617982c750b846a0b670e576b76d6ca7afa41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.986ex; height:6.509ex;" alt="{\displaystyle X={\text{Spec}}\left({\frac {\mathbb {C} [x,y,z]}{(xz,yz)}}\right)}" loading="lazy"></dd></dl>

is the union of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{2}}</annotation>

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</math><img src="./34b8964bc8a01b0cc19f98a398c2e2cfcdf8c0f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} ^{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{1}}</annotation>

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</math><img src="./67530a5bd8c23c0be226ac63ddf5f6b2619e682d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} ^{1}}" loading="lazy">. Then, the embedding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\hookrightarrow \mathbb {A} ^{3}}">

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<annotation encoding="application/x-tex">{\displaystyle X\hookrightarrow \mathbb {A} ^{3}}</annotation>

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</math><img src="./709c97874bcdb86ab889c36570eaf559af457428.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.619ex; height:2.676ex;" alt="{\displaystyle X\hookrightarrow \mathbb {A} ^{3}}" loading="lazy"> isn't regular since taking any non-origin point on the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">-axis is of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> while any non-origin point on the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}">

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<annotation encoding="application/x-tex">{\displaystyle xy}</annotation>

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</math><img src="./c72eb345e496513fb8b2fa4aa8c4d89b855f9a01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}" loading="lazy">-plane is of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}">

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<annotation encoding="application/x-tex">{\displaystyle 2}</annotation>

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</math><img src="./901fc910c19990d0dbaaefe4726ceb1a4e217a0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Local_complete_intersection_morphisms_and_virtual_tangent_bundles">Local complete intersection morphisms and virtual tangent bundles</h2></div>

A morphism of finite type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

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<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

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</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is called a (local) complete intersection morphism if each point x in X has an open affine neighborhood U so that f |U factors as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U{\overset {j}{\to }}V{\overset {g}{\to }}Y}">

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<annotation encoding="application/x-tex">{\displaystyle U{\overset {j}{\to }}V{\overset {g}{\to }}Y}</annotation>

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</math><img src="./01c095f316b67aa02efd24a35f073becc0872d60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.991ex; height:3.676ex;" alt="{\displaystyle U{\overset {j}{\to }}V{\overset {g}{\to }}Y}" loading="lazy"> where j is a regular embedding and g is <a href="Smooth_morphism" title="Smooth morphism">smooth</a>.

For example, if f is a morphism between <a href="Smooth_variety" class="mw-redirect" title="Smooth variety">smooth varieties</a>, then f factors as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to X\times Y\to Y}">

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<annotation encoding="application/x-tex">{\displaystyle X\to X\times Y\to Y}</annotation>

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</math><img src="./903a23d38192714499567879c0e8b3bc4a4914cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.575ex; height:2.176ex;" alt="{\displaystyle X\to X\times Y\to Y}" loading="lazy"> where the first map is the <a href="Graph_morphism_(algebraic_geometry)" class="mw-redirect" title="Graph morphism (algebraic geometry)">graph morphism</a> and so is a complete intersection morphism. Notice that this definition is compatible with the one in EGA IV for the special case of <a href="Flat_morphism" title="Flat morphism">flat morphisms</a>.<a href="#cite_note-4">[4]</a>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

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<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

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</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> be a local-complete-intersection morphism that admits a global factorization: it is a composition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X{\overset {i}{\hookrightarrow }}P{\overset {p}{\to }}Y}">

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<annotation encoding="application/x-tex">{\displaystyle X{\overset {i}{\hookrightarrow }}P{\overset {p}{\to }}Y}</annotation>

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</math><img src="./c47b4a9db5ede63806dc778a1bffa06e90ae4d12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.439ex; height:3.509ex;" alt="{\displaystyle X{\overset {i}{\hookrightarrow }}P{\overset {p}{\to }}Y}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> is a regular embedding and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> a smooth morphism. Then the virtual tangent bundle is an element of the <a href="Grothendieck_group" title="Grothendieck group">Grothendieck group</a> of vector bundles on X given as:<a href="#cite_note-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle T_{f}=[i^{*}T_{P/Y}]-[N_{X/P}]}</annotation>

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</math><img src="./504ce0d6ddf24f2633f5e4afa7847e71217a900e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.331ex; height:3.176ex;" alt="{\displaystyle T_{f}=[i^{*}T_{P/Y}]-[N_{X/P}]}" loading="lazy">,</dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{P/Y}=\Omega _{P/Y}^{\vee }}">

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<annotation encoding="application/x-tex">{\displaystyle T_{P/Y}=\Omega _{P/Y}^{\vee }}</annotation>

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</math><img src="./7ccdaa9db45f86e5041dedaf2abcbe35aa661541.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:13.219ex; height:3.509ex;" alt="{\displaystyle T_{P/Y}=\Omega _{P/Y}^{\vee }}" loading="lazy"> is the relative tangent sheaf of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">

(which is <a href="Locally_free_sheaf" class="mw-redirect" title="Locally free sheaf">locally free</a> since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is smooth)

and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> is the normal sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {I}}/{\mathcal {I}}^{2})^{\vee }}">

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {I}}/{\mathcal {I}}^{2})^{\vee }}</annotation>

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</math><img src="./1ee69d6ca692b68cc811185e48af79d96d1d5b0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.407ex; height:3.176ex;" alt="{\displaystyle ({\mathcal {I}}/{\mathcal {I}}^{2})^{\vee }}" loading="lazy">

(where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}}">

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</math><img src="./0e9730a0ada0426927ff64141eb9f505eca132d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.069ex; width:1.561ex; height:2.176ex;" alt="{\displaystyle {\mathcal {I}}}" loading="lazy"> is the ideal sheaf of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">), which is locally free since

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> is a regular embedding.

More generally,

if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\rightarrow Y}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon X\rightarrow Y}</annotation>

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</math><img src="./6f986e95e93b70de25a0084daf075cb02c3ccae8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\rightarrow Y}" loading="lazy"> is a any local complete intersection morphism of schemes, its

<a href="Cotangent_complex" title="Cotangent complex">cotangent complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{X/Y}}">

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<annotation encoding="application/x-tex">{\displaystyle L_{X/Y}}</annotation>

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</math><img src="./a637bc6665c20b6a6d9e8060d17e4e4565626613.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.291ex; height:3.009ex;" alt="{\displaystyle L_{X/Y}}" loading="lazy"> is <a href="Perfect_complex" title="Perfect complex">perfect</a> of Tor-amplitude [-1,0].

If moreover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is locally of finite type and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> locally Noetherian, then the converse is also true.<a href="#cite_note-6">[6]</a>

These notions are used for instance in the <a href="Grothendieck%E2%80%93Riemann%E2%80%93Roch_theorem" title="Grothendieck–Riemann–Roch theorem">Grothendieck–Riemann–Roch theorem</a>.

<div class="mw-heading mw-heading2"><h2 id="Non-Noetherian_case">Non-Noetherian case</h2></div>

<a href="S%C3%A9minaire_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique_du_Bois_Marie" class="mw-redirect" title="Séminaire de géométrie algébrique du Bois Marie">SGA 6 Exposé VII</a> uses the following slightly weaker form of the notion of a regular embedding, which agrees with the one presented above for Noetherian schemes:

First, given a <a href="Projective_module" title="Projective module">projective module</a> E over a commutative ring A, an A-linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u:E\to A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u:E\to A}</annotation>

</semantics>

</math><img src="./664a38dcfbd1da976abc507729ba49a45e237c75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.4ex; height:2.176ex;" alt="{\displaystyle u:E\to A}" loading="lazy"> is called Koszul-regular if the <a href="Koszul_complex" title="Koszul complex">Koszul complex</a> determined by it is <a href="Acyclic_complex" class="mw-redirect" title="Acyclic complex">acyclic</a> in dimension > 0 (consequently, it is a resolution of the cokernel of u).<a href="#cite_note-7">[7]</a>

Then a closed immersion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\hookrightarrow Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\hookrightarrow Y}</annotation>

</semantics>

</math><img src="./a5d77a675aa10c404233db9c24ab7b9dfbcde726.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.66ex; height:2.176ex;" alt="{\displaystyle X\hookrightarrow Y}" loading="lazy"> is called Koszul-regular if the ideal sheaf determined by it is such that, locally, there are a finite free A-module E and a Koszul-regular surjection from E to the ideal sheaf.<a href="#cite_note-8">[8]</a>

It is this Koszul regularity that was used in SGA 6

<a href="#cite_note-9">[9]</a> for the definition of local complete intersection morphisms; it is indicated there that Koszul-regularity was intended to replace the definition given earlier in this article and that had appeared originally in the already published EGA IV.<a href="#cite_note-10">[10]</a>

(This questions arises because the discussion of zero-divisors is tricky for non-Noetherian rings in that one cannot use the theory of associated primes.)

<ul><li><a href="Regular_submanifold" class="mw-redirect" title="Regular submanifold">Regular submanifold</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFSernesi2006">Sernesi 2006</a>, D. Notes 2.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFSernesi2006">Sernesi 2006</a>, D.1.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFSGA_61971">SGA 6 1971</a>, Exposé VIII, Definition 1.1.; <a href="#CITEREFSernesi2006">Sernesi 2006</a>, D.2.1.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFEGA_IV1967">EGA IV 1967</a>, Definition 19.3.6, p. 196

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFFulton1998">Fulton 1998</a>, Appendix B.7.5.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFIllusie1971">Illusie 1971</a>, Proposition 3.2.6 , p. 209

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFSGA_61971">SGA 6 1971</a>, Exposé VII. Definition 1.1. NB: We follow the terminology of the <a href="Stacks_project" class="mw-redirect" title="Stacks project">Stacks project</a>.<a rel="nofollow" class="external autonumber" href="https://stacks.math.columbia.edu/tag/061T">[1]</a>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFSGA_61971">SGA 6 1971</a>, Exposé VII, Definition 1.4.

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFSGA_61971">SGA 6 1971</a>, Exposé VIII, Definition 1.1.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFEGA_IV1967">EGA IV 1967</a>, § 16 no 9, p. 45

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFSGA_61971" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="Pierre_Berthelot_(mathematician)" class="mw-redirect" title="Pierre Berthelot (mathematician)">Berthelot, Pierre</a>; <a href="Alexandre_Grothendieck" class="mw-redirect" title="Alexandre Grothendieck">Alexandre Grothendieck</a>; <a href="Luc_Illusie" title="Luc Illusie">Luc Illusie</a>, eds. (1971). Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225) (in French). Berlin; New York: <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. xii+700. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0066283">10.1007/BFb0066283</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-05647-8</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0354655">0354655</a>.</cite></li>

<li><cite id="CITEREFFulton1998" class="citation cs2"><a href="William_Fulton_(mathematician)" title="William Fulton (mathematician)">Fulton, William</a> (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-62046-4</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1644323">1644323</a></cite>, section B.7</li>

<li><cite id="CITEREFEGA_IV1967" class="citation journal cs1"><a href="Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck, Alexandre</a>; <a href="Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean</a> (1967). <a rel="nofollow" class="external text" href="http://www.numdam.org/articles/PMIHES_1967__32__5_0">"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie"</a>. <a href="Publications_Math%C3%A9matiques_de_l'IH%C3%89S" title="Publications Mathématiques de l'IHÉS">Publications Mathématiques de l'IHÉS</a>. 32: 5–361. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02732123">10.1007/bf02732123</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0238860">0238860</a>.</cite>, section 16.9, p. 46</li>

<li><cite id="CITEREFIllusie1971" class="citation cs2 cs1-prop-foreign-lang-source"><a href="Luc_Illusie" title="Luc Illusie">Illusie, Luc</a> (1971), Complexe Cotangent et Déformations I, Lecture Notes in Mathematics 239 (in French), Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-05686-7</bdi></cite></li>

<li><cite id="CITEREFSernesi2006" class="citation book cs1">Sernesi, Edoardo (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xkcpQo9tBN8C">Deformations of Algebraic Schemes</a>. Physica-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783540306153</bdi>.</cite></li></ul></div><div><div class="zim-footer">

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