The conormal and normal sheaves, and adjunction

(Conormal and normal)

Let $i:Z\hookrightarrow X$ be a closed immersion of schemes with ideal sheaf $\mci$. The conormal sheaf is $\mci/\mci^2$, regarded as an $\OO_Z$-module. Its dual is the normal sheaf \[ \mcn_{Z/X}\coloneqq\sheafhom_{\OO_Z}(\mci/\mci^2,\OO_Z). \]

(Determinant)

For a locally free sheaf $\mcf$ of finite rank $r$ on a scheme $X$, its determinant is the invertible sheaf $\det\mcf\coloneqq\bigwedge^r\mcf$. For a smooth variety $X$ over a field $k$, the canonical invertible sheaf is $\omega_X=\det\Omega_{X/k}$ [@Har10a, Chapter II, §8].

(Determinants of exact sequences)

Let $X$ be a scheme and let $0\to\mcf_1\to\cdots\to\mcf_n\to0$ be an exact sequence of locally free sheaves of finite rank. With exponent $-1$ denoting the dual of an invertible sheaf, there is a canonical isomorphism \[ \bigotimes_{i=1}^n(\det\mcf_i)^{\otimes(-1)^{i+1}}\cong\OO_X. \] For $n=3$ it is $\det\mcf_2\cong\det\mcf_1\otimes\det\mcf_3$. The short-exact-sequence map is proved in P-AGH2516TENSOROPS, part (d); applying it to the successive image sheaves gives the displayed cancellation for a longer exact sequence [@Har10a, Exercise II.5.16].

(Adjunction)

Let $i:Z\hookrightarrow X$ be a closed immersion of smooth varieties over a field $k$, of dimensions $m$ and $n$, with ideal sheaf $\mci$. Then the conormal sequence is short exact, \[ 0\longrightarrow\mci/\mci^2\longrightarrow i^*\Omega_{X/k}\longrightarrow\Omega_{Z/k}\longrightarrow0, \] and $\mci/\mci^2$ is locally free of rank $n-m$. Taking determinants gives \[ \omega_Z\cong i^*\omega_X\otimes_{\OO_Z}\bigwedge^{n-m}\mcn_{Z/X}. \] [@Har10a, Theorem II.8.17 and Proposition II.8.20]

(Adjunction for a divisor)

Let $X$ be a smooth variety over a field $k$, and let $i:Z\hookrightarrow X$ be a smooth effective Cartier divisor. Its ideal sheaf is $\OO_X(-Z)$, and \[ \omega_Z\cong i^*\bigl(\omega_X\otimes_{\OO_X}\OO_X(Z)\bigr). \] [@Har10a, Proposition II.8.20]

(A free conormal module on a nonsmooth subscheme)

Let $k$ be a field, let $X=\Spec k[x]$, and let $Z$ be the closed subscheme defined by $I=(x^2)$. Put $B=k[x]/(x^2)$. The map \[ B\longrightarrow I/I^2,\qquad a\bmod x^2\longmapsto ax^2\bmod x^4 \] is an isomorphism: it is surjective, and $ax^2\in(x^4)$ holds exactly when $a\in(x^2)$. Thus the conormal sheaf is free of rank one on $Z$. The element $x\bmod x^2$ is nonzero and nilpotent, so $Z$ is not reduced and is not smooth over $k$ [@Har10a, Chapter III, §10]. Consequently local freeness of the conormal sheaf does not imply smoothness of the closed subscheme.

(Canonical sheaf and genus of a smooth plane curve)

Let $k$ be algebraically closed and let $C\subseteq\PP_k^2$ be a smooth integral plane curve of degree $d\ge1$. The identities $\omega_{\PP_k^2}\cong\OO(-3)$ and $\OO(C)\cong\OO(d)$ give \[ \omega_C\cong\OO_C(d-3),\qquad\deg\omega_C=d(d-3). \] Since $\deg\OO_C(1)=d$ and $\deg\omega_C=2g(C)-2$, it follows that \[ g(C)=\frac{(d-1)(d-2)}2. \] [@Har10a, Example II.8.20.3 and Chapter IV, §1]

What is the conormal sheaf of a closed subscheme?

State the adjunction formula.

What is the canonical sheaf of a smooth degree-$d$ plane curve?

Dependencies

D-MODIDEALThe ideal sheaf of a closed subschemeuses
D-4GCH6The sheaf of differentials, and its quasicoherenceuses

What links to this

Differentials