...ps, Abelian groups, etc.) on $X_{\text{et}}$ is defined as a contravariant functor $\mathcal{F}$ from the category $X_{\text{et}}$ into that of sets (groups,
...The functor $f^*$ adjoint to $f*$ on the left is called the inverse-image functor. In particular, the stalk of $\mathcal{F}$ at a geometric point $\eta : \ma
3 KB (538 words) - 20:07, 3 June 2017
...e limit is the notion of an inductive limit (direct limit or colimit) of a functor. An object $ A $
is called an inductive limit of the covariant functor $ F : \mathfrak D \rightarrow \mathfrak K $
7 KB (1,089 words) - 22:12, 5 June 2020
...w.encyclopediaofmath.org/legacyimages/b/b015/b015310/b0153104.png" /> is a
functor from the category of <img align="absmiddle" border="0" src="https://www.enc
...clopediaofmath.org/legacyimages/b/b015/b015310/b01531052.png" />-th
direct image of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopedi
11 KB (1,513 words) - 17:08, 7 February 2011
...(Y) \to \CH(X)$ is a homomorphism of rings, and for $f$ proper, the direct-image homomorphism $f_*: \CH(X)\to \CH(Y)$ is a homomorphism of $\CH(Y)$-modules.
...g that preserves the degree and commutes with the inverse-image and direct-image homomorphisms.
4 KB (714 words) - 21:54, 24 April 2012
In an additive category the direct sum $X\oplus Y$ of any two objects exists. It is isomorphic to their produc
A functor $F\colon\mathfrak C\to\mathfrak C_1$ from an additive category $\mathfrak C
3 KB (490 words) - 23:53, 10 December 2018
the image of $ \sigma ^ {n} $
which is isomorphic to the direct sum (over all $ \sigma ^ {n} $)
8 KB (1,130 words) - 08:14, 6 June 2020
called the translation functor. A triangle in $ {\mathcal C} $
...$ \delta $-functor (or exact functor) if it commutes with the translation functor and preserves distinguished triangles.
16 KB (2,338 words) - 06:56, 10 May 2022
as well as the image of $ G $
is the direct product of a finite Abelian group and an algebraic torus defined and split
4 KB (534 words) - 11:05, 17 December 2019
$ \mathop{\rm Im} f $ (the image of $ f $),
and $ \mathop{\rm Coim} f $ (the co-image of $ f $),
11 KB (1,695 words) - 20:37, 23 December 2023
right $ f ^ { - 1 } \mathcal D _ {Y} $-bimodule. The inverse image functor $ L f ^ { * } $
The direct image functor $ f _ {+} $
24 KB (3,511 words) - 07:03, 10 May 2022
...es) it does not form a cohomology functor (see [[Homology functor|Homology functor]]). In the case when $ {\mathcal F} $
Grothendieck cohomology. One considers the functor $ {\mathcal F} \rightarrow \Gamma ( X , {\mathcal F} ) $
16 KB (2,386 words) - 16:47, 20 January 2024
...usions, is exact, i.e. the kernel of every homomorphism coincides with the image of the previous one;
...joint subspaces be naturally isomorphic to the direct sum of the homology (direct product of the cohomology) of the subspaces (Milnor's additivity axiom). An
5 KB (779 words) - 19:51, 16 January 2024
...y spaces are replaced by the sheaves $R^n f_*$ of the derived direct image functor $f_*$; here an important role is played by the behaviour of these sheaves u
4 KB (573 words) - 18:20, 17 April 2017
...$, $gm = \overline g m \overline g\inv$, where $\overline g$ is an inverse image of $g$ in $F$.
...$f : M \to N$ the submodules $\Ker f$ (the kernel of $f$) and $\Im f$ (the image of $f$), and also the quotient modules $\Coker f = N / \Im f$ (the cokernel
23 KB (3,918 words) - 04:31, 23 July 2018
...ncrete case, is given explicitly. A functor of families is a contravariant functor $ {\mathcal M} $
be a functor of the points in this category, that is, $ h _{M} = \mathop{\rm Hom}\noli
16 KB (2,402 words) - 11:49, 16 December 2019
inverse image); and
direct image).
12 KB (1,730 words) - 22:13, 5 June 2020
is exact, that is, the image of each incoming homomorphism equals the kernel of the outgoing one.
is a covariant [[Functor|functor]] from some category of pairs of spaces into the category of groups. Axiom
23 KB (3,393 words) - 08:51, 25 April 2022
...e discrete, the stalk-wise algebraic operations defined on $\cF$ by taking direct limits are continuous and the natural projection $p:\cF\to X$, where $\cF =
...defined in the standard way. For any sheaf homomorphism $f:\cF'\to\cF$ the image $f(\cF')$ is an open subset of $\cF$, closed with respect to the stalk-wise
26 KB (4,342 words) - 15:06, 15 July 2014
and the image of the element $ [ f] $
onto the direct summand $ \pi _ {m} ( S ^ {2n - 1 } ) $
12 KB (1,812 words) - 22:11, 5 June 2020
...osable group of finite rank. Equivalently, a Butler group is an epimorphic image of a completely decomposable group of finite rank [[#References|[a5]]].
be a direct sum, $ A = A _ {1} \oplus A _ {2} \oplus A _ {3} $,
9 KB (1,361 words) - 16:10, 13 January 2021