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Type of short exact sequence in mathematics
In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible
Split_exact_sequence
Module components with flexibility in module theory
direct limit of projective modules, and a pure exact sequence is a direct limit of split exact sequences. Let R {\displaystyle R} be a ring (associative
Pure_submodule
Sequence of homomorphisms such that each kernel equals the preceding image
In mathematics, an exact sequence is a sequence of morphisms between objects (for example, groups, modules, and, more generally, objects of an abelian
Exact_sequence
About direct sums and exact sequences
holds, the sequence is called a split exact sequence, and the sequence is said to split. In the above short exact sequence, where the sequence splits, it
Splitting_lemma
Direct summand of a free module (mathematics)
short exact sequence of modules of the form 0 → A → B → P → 0 {\displaystyle 0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0} is a split exact sequence
Projective_module
properties of short exact sequences in an abelian category: E is closed under isomorphisms and contains the canonical ("split exact") sequences: M ′ → M ′ ⊕
Exact_category
Functor that preserves short exact sequences
mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations
Exact_functor
Topics referred to by the same term
destroyer Split, decommissioned in 1980 Yugoslav frigate Split, Koni-class Split (graph theory) Split (mathematics), a property of an exact sequence Split Lie
Split
Operation in group theory
words, there is a split exact sequence 1 → N → G → H → 1 {\displaystyle 1\to N\to G\to H\to 1} of groups (which is also known as a split extension of H {\displaystyle
Semidirect_product
Operation in abstract algebra
targets Krull–Schmidt theorem – Mathematical theorem Split exact sequence – Type of short exact sequence in mathematics Wedderburn, J. H. M. (1934). Lectures
Direct_sum_of_modules
Algebra term
valuation of the discriminant, taken mod 2). One will then have a split exact sequence 0 → Z → W ( Q ) → Z / 2 ⊕ ⨁ p ≠ 2 W ( F p ) → 0 {\displaystyle
Witt_group
Group of isotopy classes of a topological automorphism group
categories if n ≥ 5 {\displaystyle n\geq 5} , one has the following split-exact sequences: In the category of topological spaces 0 → Z 2 ∞ → MCG ( T n )
Mapping_class_group
Subject area in mathematics
relations for split exact sequences, not general exact sequences. In the category of projective modules over a ring, every short exact sequence splits, and
Algebraic_K-theory
𝒯=⟨𝒜,ℬ⟩ as closer to a split exact sequence, because the exact sequence 0→𝒜→𝒯→𝒯/𝒜→0 of triangulated categories is split by the subcategory ℬ⊂𝒯,
Semiorthogonal_decomposition
Tensoring the Euler sequence of P 1 {\displaystyle \mathbb {P} ^{1}} by O ( 1 ) {\displaystyle {\mathcal {O}}(1)} gives a non-split exact sequence 0 → O ( − 1
Stable_vector_bundle
Mathematical group of loops in a Lie group
inclusion of constant loops gives a splitting of ev1, so there is a split exact sequence 1 → Ω G → L G → ev 1 G → 1 , {\displaystyle 1\to \Omega G\to LG\xrightarrow
Loop_group
Concept in algebraic topology
{\displaystyle S^{15}} are contractible to a point. Further the short exact sequences split and there are families of isomorphisms: π i ( S 4 ) ≅ π i ( S 7
Fibration
The split gene theory offers an explanation for the origin of eukaryotic introns. It suggests that random primordial DNA sequences would only permit short
Split_gene_theory
Bioinformatics search algorithm
words to the database sequences. Repeat step 3 to 4 for each k-letter word in the query sequence. Scan the database sequences for exact matches with the remaining
BLAST_(biotechnology)
Group for which a given group is a normal subgroup
extension of Q {\displaystyle Q} by N {\displaystyle N} if there is a short exact sequence 1 → N → ι G → π Q → 1. {\displaystyle 1\to N\;{\overset {\iota }{\to
Group_extension
Homological construction in category theory
various quite different settings that a short exact sequence often gives rise to a "long exact sequence". Derived functors clarify and generalize many
Derived_functor
Theory of cohomology for commutative rings
A\longrightarrow B\longrightarrow C} is a sequence of homomorphisms of commutative rings. There is a natural right-exact sequence C ⊗ B Ω B / A ⟶ Ω C / A ⟶ Ω C /
André–Quillen_cohomology
Relates the homology of two objects to the homology of their product
_{1}^{R}(H_{i}(X;R),H_{j}(Y;R))\to 0.} Furthermore, these sequences split, but not canonically. The short exact sequences just described can easily be used to compute
Künneth_theorem
Complete set of nucleic acid sequences for humans
The human genome is a complete set of DNA sequences for each of the 22 autosomes and the two distinct sex chromosomes (X and Y) in humans. A small DNA
Human_genome
Malformation of the central digit(s) of the hand or foot
Ectrodactyly, split hand, or cleft hand (from Ancient Greek ἔκτρωμα (éktroma) 'miscarriage' and δάκτυλος (dáktulos) 'finger') involves the deficiency
Ectrodactyly
Unproven computational hardness assumption
current[when?] knowledge that there could be a sequence of 3-SAT algorithms, each with running time O(2δin) for a sequence of numbers δi tending towards zero, but
Exponential_time_hypothesis
{\mathfrak {g}}^{\mathrm {red} }} . However, the corresponding short exact sequence 0 → n i l ( g ) → g → g r e d → 0 {\displaystyle 0\to {\mathfrak {nil}}({\mathfrak
Nilradical_of_a_Lie_algebra
Question in abstract algebra
every short exact sequence 0 → Z → B → A → 0 {\displaystyle 0\rightarrow \mathbb {Z} \rightarrow B\rightarrow A\rightarrow 0} must split if B is also
Whitehead_problem
Substrings of length k contained in a biological sequence
k} contained within a biological sequence. Primarily used within the context of computational genomics and sequence analysis, in which k-mers are composed
K-mer
Abelian group extending a commutative monoid
+ [ N ] {\displaystyle [M\oplus N]=[M]+[N]} , because of the split short exact sequence 0 → M → M ⊕ N → N → 0. {\displaystyle 0\to M\to M\oplus N\to N\to
Grothendieck_group
Frame sequence for color composite video signals
editing and broadcast was known as color framing. The exact nature of the color frame sequence depends on the video standard being used. In the case of
Color_frame_sequence
Durable and flexible material created by tanning animal skins
A further subprocess, finishing, can be added into the leather process sequence, but not all leathers receive finishing. The preparatory stages are when
Leather
Sequence of DNA that determines traits in an organism
conversely a gene may be split across chromosomes but those transcripts are concatenated back together into a functional sequence by trans-splicing. It is
Gene
Formula One season
Formula One in 2021 whilst remaining partners of Red Bull Powertrains, split from the two Red Bull-owned teams and relaunched a fully-independent engine
2026 Formula One World Championship
2026_Formula_One_World_Championship
Discrete Fourier transform algorithm
computes the discrete Fourier transform (DFT), or its inverse (IDFT), of a sequence. A Fourier transform converts a signal from its original domain (often
Fast_Fourier_transform
Casino game of chance
variety of betting options. "Inside" bets involve selecting either the exact number on which the ball will land, or a small group of numbers adjacent
Roulette
Group of mathematical theorems
which complete the short exact sequence running from the lower left to the upper right of the diagram. The use of the exact sequence convention saves us from
Isomorphism_theorems
Most common system for writing numbers
from zero to nine, and allows representing any natural number by a unique sequence of these glyphs. The symbols (glyphs) used to represent the system are
Hindu–Arabic_numeral_system
Tools for studying groups based on techniques from algebraic topology
extent to which taking invariants doesn't respect exact sequences. This is expressed by a long exact sequence. The collection of all G {\displaystyle G} -modules
Group_cohomology
Instructions directly executable by a computer
its programmable interface. A computer program consists primarily of sequences of machine-code instructions. Machine code is classified as native with
Machine_code
Problem optimization method
a decision by breaking it down into a sequence of decision steps over time. This is done by defining a sequence of value functions V1, V2, ..., Vn taking
Dynamic_programming
Form of alternating current
other indicator lights to show the sequence of voltages at the terminals for the given direction of shaft rotation. Split-phase electric power Used when three-phase
Three-phase_electric_power
Type of machine learning model
development of word embeddings (e.g., Word2Vec by Mikolov in 2013) and sequence-to-sequence (seq2seq) models using LSTM. In 2016, Google transitioned its translation
Large_language_model
Species of small cat
term tilcayo with Leopardus tigrinus. It was originally reported that the exact etymology of the term is unknown, as it did not appear to be of Spanish
Tilcayo
Paraphyletic group of mammals
larynx is attached to the hyoid bone that is hanging from a sequence of bones. This sequence of bones the hyoid hangs from are tympanohyal, stylohyal, epihyal
Big_cat
South Korean character set
known as Johab. This represents a Hangul syllable as the sequence of three five-bit values, split across two 8-bit bytes, most significant bit first. The
KS_X_1001
Sequence of operations for a task
mathematics and computer science, an algorithm (/ˈælɡərɪðəm/ ) is a finite sequence of mathematically rigorous instructions, typically used to solve a class
Algorithm
On polynomial rings over fields
i . {\displaystyle G_{i}\mapsto g_{i}.} In other words, one has an exact sequence 0 → R 1 → L 0 → M → 0. {\displaystyle 0\to R_{1}\to L_{0}\to M\to 0
Hilbert's_syzygy_theorem
2005 Indian film by Shankar
festival and the extensive use of time slice photography in an action sequence. Cinematographer V. Manikandan discontinued the project halfway through
Anniyan
Free online crowdsourced encyclopedia
have far more topics than can be covered by any printed encyclopedia. The exact degree and manner of coverage on Wikipedia is under constant review by its
Wikipedia
2022 anime series
Jaki, Rafal [@GwentBro] (September 24, 2022). "We did not specify the exact timeline—the first time skip is a few months but less than a year. If I
Cyberpunk:_Edgerunners
Star at the centre of the Solar System
initiated nuclear fusion in its core. It is now classified as a G-type main-sequence star (G2V). Every second, the Sun's core fuses about 600 billion kilograms
Sun
Descendants of Genghis Khan
Dayan Khan's sons. And when the Y-chromosomes were sequenced, all seven matched, sharing the exact same downstream mutations, and determined Dayan Khan
Chinggisids
list of sequence alignment software is a compilation of software tools and web portals used in pairwise sequence alignment and multiple sequence alignment
List of sequence alignment software
List_of_sequence_alignment_software
Surjective ring homomorphism with a given codomain
ring homomorphism ϕ {\displaystyle \phi } that fits into the short exact sequence of abelian groups: 0 → I → E → ϕ R → 0. {\displaystyle 0\to I\to E{\overset
Algebra_extension
imagery, cockpit voice recordings, and official reports to reconstruct the sequences of events. As of 2 June 2026,[update] 297 episodes of Mayday have aired
List_of_Mayday_episodes
its fossil incompleteness and the proximity to the human-chimpanzee split, the exact position of Ardipithecus in the fossil record is unclear. However,
Chimpanzee–human last common ancestor
Chimpanzee–human_last_common_ancestor
Superorder of mammals
treeshrews, primates, and colugos. The Euarchontoglires clade is based on DNA sequence analyses and retrotransposon markers that combine the clades Glires (Rodentia
Euarchontoglires
Clade of birds
earliest split in Neoaves. A reanalysis with an extended dataset by Reddy and colleagues in 2017 suggested this was due to the type of sequence data, with
Neoaves
by finitary means. Alternatively, those modules which leave pure-exact sequence exact after applying Hom. annihilator 1. The annihilator of a left R {\displaystyle
Glossary_of_module_theory
Gene editing method
a DNA repair template allows for the insertion of a specific DNA sequence at an exact location within the genome. The repair template should extend 40
CRISPR_gene_editing
American film production and distribution company
retained by Disney via 20th Century Animation. Although Disney did not give an exact date as to when the studio would be closing down initially, former animator
20th_Century_Studios
Stimulating neuropeptide
residues in 6 and 12 and 7 and 14 positions.[citation needed] The amino acid sequence is: Pyroglu-Pro-Leu-Pro-Asp-Cys-Cys-Arg-Gln-Lys-Thr-Cys-Ser-Cys-Arg-Le
Orexin-A
Strategy board game, variant of Pachisi
track to the home area safely without being captured by another player. The exact location of the starting point may vary according to the version, but it
Aggravation_(board_game)
2026 American superhero series by Oren Uziel
September 12, 2020. Travis, Ben (March 9, 2026). "Spider-Noir Won't Be The Exact Version From Spider-Verse: 'Same Character, Different Universe'". Empire
Spider-Noir
Large self-illuminated object in space
Such stars are said to be on the main sequence and are called dwarf stars. Starting at zero-age main sequence, the proportion of helium in a star's core
Star
Number expressed in the base-2 numeral system
In this method, multiplying one number by a second is performed by a sequence of steps in which a value (initially the first of the two numbers) is either
Binary_number
3.2-million-year-old fossilized hominid
pumice clasts at Hadar. (The Lucy skeleton occurs in the part of the Hadar sequence that accumulated with the fastest rate of deposition, which partly accounts
Lucy_(hominid)
Concept in mathematics
section with trivial normal bundle (a section corresponds to a short exact sequence 0 → J → A + I → K → 0 where J, K are ideal classes and the normal bundle
Reductive_group
How often each letter appears in written language
variations in letter compartment sizes in typographer's type cases. No exact letter frequency distribution underlies a given language, since all writers
Letter_frequency
How spheres of various dimensions can wrap around each other
π3(S7) and π7(S15) are zero. Thus the long exact sequences again break into families of split short exact sequences, implying two families of relations. π
Homotopy_groups_of_spheres
Construction in homological algebra
a short exact sequence 0 → K → L → M → 0 {\displaystyle 0\rightarrow K\rightarrow L\rightarrow M\rightarrow 0} induces a long exact sequence of the form
Ext_functor
American television series (2004–2010)
primary storyline set on the island, augmented by flashback or flashforward sequences which provide additional insight into the involved characters. Lindelof
Lost_(TV_series)
Type of defect in crystalline materials
materials form repeating patterns of layers of atoms. Errors can occur in the sequence of these layers and are known as stacking faults. Stacking faults are in
Stacking_fault
2010 film by Edgar Wright
The opening title sequence was designed by Richard Kenworthy of Shynola, and was inspired by drawn-on-film animation. The sequence also begins with an
Scott_Pilgrim_vs._the_World
Natural disaster in the US
they knew severe storms were in the area, although they did not know the exact location, extent, and movement of the storms. Fujita, T. T.; Bradbury, D
1965 Palm Sunday tornado outbreak
1965_Palm_Sunday_tornado_outbreak
there is now a gap in the numbering sequence. Benedict: Antipope Benedict X (1058–1059) was kept in the numbering sequence. Boniface: Antipope Boniface VII
List_of_popes
Electric motor driven by an AC electrical input
seconds. These motors are relatively costly, and are used where exact speed (assuming an exact-frequency AC source) and rotation with low flutter (high-frequency
AC_motor
Communication aspects of Universal Serial Bus
maximum packet size for that endpoint, called a "short" packet. When the exact amount of data specified by the wLength field of the setup packet has been
USB_communications
System for identifying vehicles
section or fourth to eighth segment of the VIN. In order to identify the exact year in passenger cars and multipurpose passenger vehicles with a GVWR of
Vehicle_identification_number
Heterogeneous class of stars with unusual spectra
and WC respectively. The two classes WN and WC were further split into temperature sequences WN5–WN8 and WC6–WC8 based on the relative strengths of the
Wolf–Rayet_star
Irish English dialects both use sequences of a vowel then /r/ rather than r-colored vowels, and both lack the foot–strut split; which result in comparable
English-language vowel changes before historical /r/
English-language_vowel_changes_before_historical_/r/
Series of language models developed by DeepSeek
into a sequence of embedding vectors in the same way of a standard LLM. The images are converted by a vision transformer DeepSeek-ViT into a sequence of image
DeepSeek_(AI)
"faithfully flat", the original sequence is exact. ◻ {\displaystyle \square } Bhatt and Scholze show that the Amitsur complex is exact if R {\displaystyle R} and
Amitsur_complex
Sequence of data points over time
In mathematics, a time series is a sequence of data points indexed, listed, or graphed in chronological order. Most commonly, a time series consists of
Time_series
British TV sitcom (1992–2005)
the BBC. Dench's real-life daughter Finty Williams appeared in the title sequence as the young Jean in one of the two pictures. The other was Palmer's son
As_Time_Goes_By_(TV_series)
Sound changes
Stentoften Runestone). This was the period after the East Germanic languages had split off. Changes during this time were shared with the North Germanic dialects
Phonological history of English
Phonological_history_of_English
Stellar classification
black holes or neutron stars. Most of these stars are young massive main sequence, giant, or supergiant stars, but also some central stars of planetary nebulae
O-type_star
River in Italy
maritime and alluvial sediments occurs as far west as Piacenza. The exact sequences at various locations have been studied extensively. Apparently, the
Po_(river)
Canadian writer (1879–1961)
the Whiteoak family from 1854 to 1954. The sixteen novels do not form a sequence, however, and each can be read as an independent story. There are similarities
Mazo_de_la_Roche
Mathematical group
image is the inner automorphism group). This can be summarized by the exact sequence Z ( G ) ↪ G ⟶ σ A u t ( G ) ↠ O u t ( G ) {\displaystyle Z(G)\hookrightarrow
Outer_automorphism_group
Creative and technical part of the post-production process of filmmaking
post-production process of filmmaking in which shots are combined into sequences that create a finished motion picture. The term is derived from the traditional
Film_editing
Literary element
various story structures and components that might be considered. Story is a sequence of events, which can be true or fictitious, that appear in prose, verse
Story_structure
Software design modeling notation
between components Interaction overview diagram Sequence diagram – shows interactions arranged in time sequence; can be drawn via tools such as Lucidchart
Unified_Modeling_Language
British crime TV series (1962–1969)
switch to colour, this was replaced by simple narration. The pre-credit sequence usually ended with someone referring to (and/or addressing) the Saint by
The_Saint_(TV_series)
2004 video game
place one or multiple precise shots on an enemy's body. When the targeting sequence ends, the player character automatically fires to all marked locations
Red_Dead_Revolver
Proof assistant and programming language
goal is split into two subgoals, one is q and the other is p · exact h.right -- the first subgoal is exactly the right part of h : p ∧ q · exact h.left
Lean_(proof_assistant)
1966 film directed by Sergio Leone
Cemetery. When Tuco returns with water, Carson has died, having revealed the exact grave in which the gold is hidden to Blondie. The two reluctantly agree
The Good, the Bad and the Ugly
The_Good,_the_Bad_and_the_Ugly
metric for the amount of difference between two sequences Trigram search: search for text when the exact syntax or spelling of the target object is not
List_of_algorithms
Unsolved murders of teenage girl and domestic worker
were conflicting media reports about the exact sequence of events, which was somewhat like this: Possible sequence of events according to CBI Joint Director
2008_Noida_double_murder_case
travel, tourism, insurance
SPLIT EXACT-SEQUENCE
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travel, tourism, insurance