In algebraic geometry, a closed immersion
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
is generated by a regular sequence of length r. A regular embedding of codimension one is precisely an effective Cartier divisor.
For example, if X and Y are smooth 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.
is regularly embedded into a regular scheme, then B is a complete intersection ring.
[2] The notion is used, for instance, in an essential way in Fulton's approach to intersection theory.
The important fact is that when i is a regular embedding, if I is the ideal sheaf of X in Y, then the normal sheaf, the dual of
, is locally free (thus a vector bundle) and the natural map
is an isomorphism: the normal cone
coincides with the normal bundle.
One non-example is a scheme which isn't equidimensional.
isn't regular since taking any non-origin point on the
A morphism of finite type
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
where j is a regular embedding and g is smooth.
[3] For example, if f is a morphism between smooth varieties, then f factors as
Notice that this definition is compatible with the one in EGA IV for the special case of flat morphisms.
be a local-complete-intersection morphism that admits a global factorization: it is a composition
Then the virtual tangent bundle is an element of the Grothendieck group of vector bundles on X given as:[5] where
is the relative tangent sheaf of
is a any local complete intersection morphism of schemes, its cotangent complex
is locally of finite type and
locally Noetherian, then the converse is also true.
[6] These notions are used for instance in the Grothendieck–Riemann–Roch theorem.
SGA 6 Exposé VII 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 projective module E over a commutative ring A, an A-linear map
is called Koszul-regular if the Koszul complex determined by it is acyclic in dimension > 0 (consequently, it is a resolution of the cokernel of u).
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.
[8] It is this Koszul regularity that was used in SGA 6 [9] 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.
[10] (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.)