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.
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be a [[Scheme|scheme]] and let $ p $
cf. [[Group scheme|Group scheme]]) such that the sequences
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...braic variety|Complete algebraic variety]]; [[Projective scheme|Projective scheme]].
is a complete scheme over a field $ k $,
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be a commutative ring with unity. To an element $ a \in A $
function) given by the ring homomorphism $ \mathbf Z [ T] \rightarrow A $,
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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
...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.
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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
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over a ring of Witt vectors $ W (k) $ (
is a left module over the ring $ D _{k} $ (
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is an irreducible separable scheme of finite type over an algebraically closed field $ k $,
is a Noetherian scheme, then they are coherent. If $ X $
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The valuation ring $ R= \{ {a \in K } : {| a | \leq 1 } \} $
The ring $ \{ {f \in T _ {n} } : {\| f \| \leq 1 } \} $
8 KB (1,161 words) - 08:25, 6 June 2020
...$, then $K(C)$ is the Witt–Grothendieck group of $k$ (cf. [[Witt ring|Witt ring]]).
...is the [[Picard group|Picard group]] $\mathrm{Pic}(A)$ of a ring (or of a scheme). It is the Grothendieck group associated to the commutative monoid of isom
4 KB (701 words) - 06:11, 26 March 2023
be an associative, commutative ring with unit element. Witt vectors with components in $ A $
...Witt vectors with the operations introduced above form a ring, called the ring of Witt vectors and denoted by $ W(A) $.
17 KB (2,502 words) - 17:25, 22 December 2019
...[[Ring of polynomials|ring of polynomials]] is an associative-commutative ring without zero divisors (that is, a product of non-zero polynomials cannot be
...ivision with remainder can be carried out using the [[Horner scheme|Horner scheme]].
9 KB (1,497 words) - 10:44, 27 June 2015
...sm of (left) modules with common ring of scalars (in this, the role of the ring is played by the domain of definition of the functors, and the functors the
or a category of diagrams with scheme $ \mathfrak K $.
4 KB (489 words) - 16:56, 23 November 2023