and $ \mathop{\rm Ext} $.
The derived functors (cf. [[Derived functor|Derived functor]]) $ \mathop{\rm Tor} $
5 KB (636 words) - 22:11, 5 June 2020
...strategy is given by the [[Tilting theory|tilting theory]] and the tilting functors, as now described.
...)$, which are called tilting functors. The importance of considering such functors is that they give equivalences between subcategories of the module categori
3 KB (507 words) - 16:45, 1 July 2020
...e set of equivalence classes, which thus becomes an Abelian group $\mathrm{Ext}^1_R(A,B)$, where $R$ is the ring over which $A$ is a module. This construc
corresponding to the group $\mathrm{Ext}^n_R(A,B)$. The groups $\mathrm{Ext}^n_R(A,B)$, $n=1,2,\ldots$, are the [[derived functor]]s of the functor $\m
2 KB (354 words) - 21:55, 30 October 2016
is denoted by $ \mathop{\rm Ext} _ {R} ^ {n} $.
The group $ \mathop{\rm Ext} _ {R} ^ {1} ( A , C ) $
3 KB (470 words) - 07:09, 10 May 2022
ii) $\operatorname { Ext } _ { A } ^ { 1 } ( T , T ) = 0$; and
...through the tilting functors $\text{Hom}_A( T , - )$ and $\operatorname { Ext } _ { A } ^ { 1 } ( T , - )$ (cf. also [[Tilting functor|Tilting functor]])
3 KB (459 words) - 16:55, 1 July 2020
The branch of algebra whose main study is derived functors on various categories of algebraic objects (modules over a given ring, shea
...preserve the mappings. These correspondences became known as [[Functor|'''functors''']]. The principal advantages of this language — the amount of informati
12 KB (1,885 words) - 23:48, 23 April 2017
be the chain [[Complex|complex]] of functors from $ O _ {G} $
denotes the set of natural transformations of functors. The functors $ c _ {n} ( X ) $
5 KB (732 words) - 08:45, 26 March 2023
ii) $\operatorname{Ext}_{H} ^ {1} (T, T) = 0$; and
...lso says that $T$ is maximal with respect to the property $\operatorname { Ext } _ { H } ^ { 1 } ( T , T ) = 0$. Note further, that a tilting module $T$ o
8 KB (1,215 words) - 19:50, 24 December 2023
...cal{C} , - ) : n \geq 0 \}$ form a universal connected (exact) sequence of functors and that
...rightarrow A$ (here, $\mathcal{A } ^ { \text{C} }$ denotes the category of functors from $\mathcal{C}$ to $\mathcal{A}$, and $\underline{\operatorname{lim}} \l
9 KB (1,283 words) - 20:55, 8 February 2024
...ule if $\operatorname{p.dim } _ { \Lambda } T \leq 1$ and $\operatorname { Ext } _ { \Lambda } ^ { 1 } ( T , T ) = 0$ and there is a short [[Exact sequenc
\begin{equation*} \operatorname { Ext } _ { \Lambda } ^ { 1 } ( T , . ) : \mathcal F \rightarrow \mathcal X . \en
16 KB (2,221 words) - 09:47, 11 November 2023
...hown that on a suitable category of $C ^ { * }$-algebras, $\operatorname { Ext } ( A )$ fits into a short [[Exact sequence|exact sequence]]
...{Z} } ^ { 1 } ( K _ { 0 } ( A ) , \mathbf{Z} ) \rightarrow \operatorname { Ext } ( A ) \rightarrow \end{equation*}
10 KB (1,450 words) - 17:44, 1 July 2020
\mathop{\rm Ext} _ { {\mathcal O} _ {X} } ^ {i}
modules into itself. Its derived functors are the local cohomology functors $ {\mathcal H} _ {A} ^ {i} ( M) $(
10 KB (1,477 words) - 22:17, 5 June 2020
...ticular case of Poincaré-type duality relations which are true for derived functors in homological algebra (another particular case is Poincaré-type duality f
...> E.G. Sklyarenko, "Poincaré duality and relations between the functors Ext and Tor" ''Math. Notes'' , '''28''' : 5 (1980) pp. 841–845 ''Mat. Za
8 KB (1,257 words) - 05:06, 7 March 2022
\mathrm{Ext} _ {R} ^ {n}
(see [[Functor|Functor]] $ \mathop{\rm Ext} $),
19 KB (2,870 words) - 09:48, 26 March 2023
...d } _ { Q } ( \mathbf{X} ) - \operatorname { dim } _ { K } \operatorname { Ext } _ { Q } ^ { 1 } ( \mathbf{X} , \mathbf{X} )$, along the isomorphism $\und
...^ { \infty } ( - 1 ) ^ { m } \operatorname { dim } _ { K } \operatorname { Ext } _ { R } ^ { m } ( X , X )$, under a group isomorphism $\underline{\operat
18 KB (2,636 words) - 06:50, 15 February 2024
...scription given by Grothendieck of the spectral sequence for the functor $\Ext$ is essential in algebraic geometry. A much simpler method of constructing
...groups. The higher direct images $f_*^q\cF$ of $\cF$ are the right derived functors of $f_*$. The sheaf $f_*^q\cF$ is the sheaf associated to the pre-sheaf $U\
26 KB (4,342 words) - 15:06, 15 July 2014
is hereditary and $ H ^ {*} ( \mathop{\rm Ext} ^ {1} ( A, C) ) = 0 $(
0 \rightarrow \mathop{\rm Ext} ^ {1} ( H _ {*} ( A),\
16 KB (2,248 words) - 22:15, 5 June 2020
...rcwise connected space|arcwise connected]] if and only if $ \mathop{\rm Ext} ( X( G), \mathbf Z ) = 0 $.
are the homology functors with closed supports contained in $ \Phi $(
10 KB (1,483 words) - 17:06, 13 June 2020
A class of special functors from the category of pairs of spaces into the category of graded Abelian gr
...[6]]]) there exists a spectral sequence with initial term $ \mathop{\rm Ext} _ {A _ {h} } ^ {**} ( h ^ {*} ( Y) , h ^ {*} ( X) ) $,
23 KB (3,297 words) - 19:41, 5 June 2020
...ng is factorial. The study was then begun of the functors $ \mathop{\rm Ext} $
16 KB (2,400 words) - 17:45, 4 June 2020