A transition from a commutative ring $ A $
to the ring of fractions (cf. [[Fractions, ring of|Fractions, ring of]]) $ A [ S ^ {-1} ] $,
4 KB (713 words) - 21:35, 4 January 2021
the local ring $ {\mathcal O} _ {X,x} $
is a flat module over the ring $ {\mathcal O} _ {Y,f( x) } $.
4 KB (646 words) - 19:39, 5 June 2020
in a commutative associative ring $ R $''
Consider the quotient ring $ R/A $
3 KB (505 words) - 08:09, 6 June 2020
[[Local ring|Local ring]]). In defining a morphism $(f,f^\sharp)$ between local ringed spaces $(X,\
[[Scheme|Scheme]]) constitute a full subcategory of the category of local ringed spaces.
3 KB (503 words) - 13:52, 24 November 2013
A mapping of modules over a commutative ring which is an analogue of the concept of a differential operator. Let $ R $
be a commutative ring, let $ S $
5 KB (785 words) - 19:35, 5 June 2020
be a [[Noetherian ring|Noetherian ring]] and let $ I $
as the graded ring $ S = \oplus _ {k \geq 0 } I ^ {k} $(
7 KB (941 words) - 10:59, 29 May 2020
...aximum of the injective (or projective) dimensions of the modules over the ring and is the analogue of the Lebesgue dimension of $ X $.
...homological dimension of a topological space for an algebraic variety or a scheme with a selected cohomology theory. Let $ X $
9 KB (1,321 words) - 17:45, 4 June 2020
...d let $\mathfrak{M}$ be its maximal ideal. Then the spectrum of the graded ring
...e, if the tangent cone is reduced, normal or regular, then so is the local ring $\mathcal{O}_{X,x}$. The dimension and multiplicity of $X$ at $x$ are the s
4 KB (653 words) - 13:02, 17 June 2016
...h $ X $ such as the [[Picard group|Picard group]]; the [[Chow ring|Chow ring]]; the [[K-functor| $ K $ -functor]], and the cohomology group are a tool
...or a smooth hypersurface, then the connected component of the unit of this scheme is a variety.
6 KB (923 words) - 18:31, 12 December 2019
be a [[Scheme|scheme]] and let $ p $
cf. [[Group scheme|Group scheme]]) such that the sequences
6 KB (885 words) - 08:04, 6 June 2020
...braic variety|Complete algebraic variety]]; [[Projective scheme|Projective scheme]].
is a complete scheme over a field $ k $,
9 KB (1,267 words) - 08:08, 6 June 2020
be a commutative ring with unity. To an element $ a \in A $
function) given by the ring homomorphism $ \mathbf Z [ T] \rightarrow A $,
3 KB (474 words) - 19:41, 5 June 2020
be a prime number; an $ \ell $-adic sheaf on a scheme $ X $
...e of the category of locally constant constructible sheaves on a connected scheme $ X $
6 KB (932 words) - 11:49, 8 April 2023
...d with $x$ such as the [[Picard group|Picard group]]; the [[Chow ring|Chow ring]]; the [[K-functor|$K$-functor]], and the cohomology group are a tool used
...or a smooth hypersurface, then the connected component of the unit of this scheme is a variety.
6 KB (971 words) - 13:36, 17 October 2019
...ideals (cf. [[Divisorial ideal|Divisorial ideal]]) of a [[Krull ring|Krull ring]] $ A $
into irreducible factors. Thus, a factorial ring has trivial divisor class group.
5 KB (820 words) - 19:36, 5 June 2020
The ''Chow ring'' of a non-singular quasi-projective algebraic variety
...Algebraic cycle|Algebraic cycle]]) on that variety. Multiplication in this ring is defined in terms of intersections of cycles (cf. [[Intersection theory|I
4 KB (714 words) - 21:54, 24 April 2012
''to an algebraic variety or scheme $ X $
is the maximal ideal of the [[Local ring|local ring]] $ {\mathcal O} _ {X ,x } $
2 KB (369 words) - 16:33, 10 February 2020
...e inclusion $\mathfrak{p}_x \subseteq \mathfrak{p}_y$ holds. When $A$ is a ring without zero divisors, the point $\{0\}$ is the generic one. The relation o
3 KB (403 words) - 08:08, 26 November 2023
over a ring of Witt vectors $ W (k) $ (
is a left module over the ring $ D _{k} $ (
3 KB (487 words) - 06:58, 21 March 2023
is an irreducible separable scheme of finite type over an algebraically closed field $ k $,
is a Noetherian scheme, then they are coherent. If $ X $
5 KB (799 words) - 08:03, 6 June 2020