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Scooby-Doo

Scooby-Doo is an American media franchise owned by Warner Bros. Entertainment and created in 1969 by writers Joe Ruby and Ken Spears through their animated series, Scooby-Doo, Where Are You!, for Hanna-Barbera (which was absorbed into Warner Bros. Animation in 2001). The series features four teenagers: Fred Jones, Daphne Blake, Velma Dinkley, and Shaggy Rogers, and their talking Great Dane named Scooby-Doo, who solve mysteries involving supposedly supernatural creatures through a series of antics and missteps, while traveling using a brightly colored van called the "Mystery Machine". The franchise has several live-action films and shows.

Scooby-Doo was originally broadcast on CBS from 1969 to 1976, when it moved to ABC. ABC aired various versions of Scooby-Doo until canceling it in 1986, and presented a spin-off featuring the characters as children called A Pup Named Scooby-Doo from 1988 until 1991. Two Scooby-Doo reboots aired as part of Kids' WB on The WB and its successor The CW from 2002 until 2008. Further reboots were produced for Cartoon Network beginning in 2010 and continuing through 2018. Repeats of the various Scooby-Doo series are frequently broadcast on Cartoon Network's sister channel Boomerang in the United States and other countries. The most recent Scooby-Doo series, Scooby-Doo and Guess Who?, premiered on June 27, 2019, as an original series on Boomerang's streaming service and later HBO Max.

In 2013, TV Guide ranked Scooby-Doo the fifth-greatest TV cartoon of all time.

Development
In 1968, parent-run organizations, particularly Action for Children's Television (ACT), began protesting what they perceived as excessive violence in Saturday-morning cartoons. Most of these shows were Hanna-Barbera action cartoons such as Space Ghost, The Herculoids, and Birdman and the Galaxy Trio, and virtually all of them were canceled by 1969 because of pressure from the parent groups. Members of these watch groups served as advisers to Hanna-Barbera and other animation studios to ensure that new programs would be safe for children.

Fred Silverman, an executive for daytime programming at CBS, was then looking for a show that would both revitalize his Saturday-morning line and please the watch groups. The result was The Archie Show from Filmation, based on Bob Montana's teenage humor comic book Archie. Also successful were the musical numbers The Archies performed during each program (one of which, "Sugar, Sugar", was the most successful Billboard number-one hit of 1969). Eager to build upon this success, Silverman contacted producers William Hanna and Joseph Barbera about creating another show based on a teenage rock group, this time featuring teens who solved mysteries between gigs. Silverman envisioned the show as a cross between the popular I Love a Mystery radio serials of the 1940s and the Archie characters or the popular early 1960s television series The Many Loves of Dobie Gillis.

After attempting to develop his version of the show, called House of Mystery, Barbera, who developed and sold Hanna-Barbera shows while Hanna produced them, passed the task along to story writers Joe Ruby and Ken Spears, as well as artist/character designer Iwao Takamoto. Their treatment, based in part on The Archie Show, was titled Mysteries Five and featured five teenagers: Geoff, Mike, Kelly, Linda, and Linda's brother W.W., along with their bongo-playing dog, Too Much, who collectively formed the band Mysteries Five. When The Mysteries Five were not performing at gigs, they were out-solving spooky mysteries involving ghosts, zombies, and other supernatural creatures. Ruby and Spears were unable to decide whether Too Much would be a large cowardly dog or a small feisty one. When the former was chosen, Ruby and Spears wrote Too Much as a Great Dane but revised the dog character to a large sheepdog (similar to the Archies' sheepdog, Hot Dog) just before their presentation to Silverman, as Ruby feared the character would be too similar to the comic strip character Marmaduke. Silverman rejected their initial pitch, and after consulting with Barbera on next steps, got Barbera's permission to go ahead with Too Much being a Great Dane instead of a sheepdog.

During the design phase, lead character designer Takamoto consulted a studio colleague who was a breeder of Great Danes. After learning the characteristics of a prize-winning Great Dane from her, Takamoto proceeded to break most of the rules and designed Too Much with overly bowed legs, a double chin, and a sloped back, among other abnormalities.

Ruby and Spears' second pass at the show used Dobie Gillis as the template for the teenagers rather than Archie. The treatment retained the dog Too Much, while reducing the number of teenagers to four, removing the Mike character and retaining Geoff, Kelly, Linda, and W.W. As their personalities were modified, so were the characters' names: Geoff became "Ronnie" —later renamed "Fred" (at Silverman's behest), Kelly became "Daphne", Linda "Velma", and W.W. "Shaggy". The teens were now based on four teenage characters from The Many Loves of Dobie Gillis: Dobie Gillis, Thalia Menninger, Zelda Gilroy and Maynard G. Krebs, respectively.

The revised show was re-pitched to Silverman, who liked the material but, disliking the title Mysteries Five, decided to call the show Who's S-S-Scared? Silverman presented Who's S-S-Scared? to the CBS executives as the centerpiece for the upcoming 1969–70 season's Saturday-morning cartoon block. CBS president Frank Stanton felt that the presentation artwork was too scary for young viewers and, thinking the show would be the same, decided to pass on it.

Now without a centerpiece for the upcoming season's programming, Silverman had Ruby, Spears, and the Hanna-Barbera staff revise the treatments and presentation materials to tone down the show and better reflect its comedy elements. The rock band element was dropped, and more attention was focused on Shaggy and Too Much. According to Ruby and Spears, Silverman was inspired by Frank Sinatra's scat "doo-be-doo-be-doo" at the end of his recording of "Strangers in the Night" on a red-eye flight to one of the development meetings, and decided to rename the dog "Scooby-Doo" and retitled the show Scooby-Doo, Where Are You! The revised show was re-presented to CBS executives, who approved it for production.

Scooby-Doo, Where Are You!
The first episode of Scooby-Doo, Where Are You! "What a Night for a Knight" debuted on the CBS network Saturday, September 13, 1969, at 10:30 AM Eastern Time. The original voice cast featured Don Messick as Scooby-Doo, Casey Kasem as Shaggy, Frank Welker as Fred, actress Nicole Jaffe as Velma, and Indira Stefanianna as Daphne. Scooby's speech patterns closely resembled an earlier cartoon dog, Astro from The Jetsons (1962–63), also voiced by Messick. Seventeen episodes of Scooby-Doo Where Are You! were produced in 1969–70. The series theme song was written by David Mook and Ben Raleigh, and performed by Larry Marks.

Each of these episodes features Scooby and the four teenage members of Mystery, Inc.—Fred, Shaggy, Daphne, and Velma—arriving at a location in the Mystery Machine, a van painted with psychedelic colors and flower power imagery. Encountering a purportedly supernatural monster terrorizing the local populace, such as a ghost, they decide to investigate. The kids split up to look for clues and suspects, while being chased at turns by the monster. Eventually, the kids come to realize the paranormal activity is actually an elaborate hoax, and—often with the help of a Rube Goldberg-like trap designed by Fred—they capture the creature suit-wearing villain and unmask him or her. Revealed usually as a flesh and blood crook who used the costume to cover up their crimes, the villain is arrested and taken to jail, often with the catchphrase "if it weren't for those pesky/meddling kids". A few times though, the "villain" turns out to be innocent, such as a haywire robot or the owner disguised to scare away thieves.

Scheduled opposite another teenage mystery-solving show, ABC's The Hardy Boys, Scooby-Doo became a ratings success, with Nielsen ratings reporting that as many as 65% of Saturday-morning audiences were tuned in to CBS when Scooby-Doo was being broadcast. The show was renewed for a second season in 1970, for which eight episodes were produced. Seven of the second-season episodes featured chase sequences set to bubblegum pop songs recorded by Austin Roberts, who also re-recorded the theme song for this season. With Stefanianna Christopherson having married and retired from voice acting, Heather North assumed the role of Daphne, and she continued to voice the character until 1997.

The TV influences of I Love a Mystery and Dobie Gillis were apparent in the first episode. Of the similarities between the Scooby-Doo teens and the Dobie Gillis teens, the similarities between Shaggy and Maynard are the most noticeable; both characters share the same beatnik-style goatee, similar hairstyles, and demeanors. The core premise of Scooby-Doo, Where Are You! was also similar to Enid Blyton's Famous Five books. Both series featured four youths with a dog, and the Famous Five stories often revolved around a mystery which invariably turned out not to be supernaturally based, but simply a ruse to disguise the villain's true intent.

The role of each character was strongly defined in the series: Fred is the leader and the determined detective, Velma is the intelligent analyst, Daphne is danger-prone, Shaggy is a coward more motivated by hunger than any desire to solve mysteries, and Scooby is similar to Shaggy, save for a Bob Hope-inspired tendency towards temporary bravery. Later versions of the show made slight changes to the characters' established roles, such as showing the Daphne in 1990s and 2000s Scooby-Doo productions as knowing many forms of karate and having the ability to defend herself, and reducing her tendency towards being kidnapped.

Scooby-Doo itself influenced many other Saturday-morning cartoons of the 1970s. During that decade, Hanna-Barbera and its rivals produced several animated programs also featuring teenage detectives solving mysteries with a pet or mascot of some sort, including Josie and the Pussycats (1970–71), The Funky Phantom (1971–72), The Amazing Chan and the Chan Clan (1972–73), Speed Buggy (1973–74), Goober and the Ghost Chasers (1973–74), Jabberjaw (1976–78), and Captain Caveman and the Teen Angels (1977–80).

The New Scooby-Doo Movies
In the fall of 1972, new one-hour episodes under the title The New Scooby-Doo Movies were created; each episode featuring a real or fictitious guest star helping the gang solve mysteries, including characters from other Hanna-Barbera series such as Harlem Globetrotters, Josie and the Pussycats and Speed Buggy, the comic book characters Batman and Robin (adapted into their own Hanna-Barbera series, Super Friends, a year later), and celebrities such as Sandy Duncan, The Addams Family, Cass Elliot, Phyllis Diller, Don Knotts and The Three Stooges. Hanna-Barbera musical director Hoyt Curtin composed a new theme song for this series, and Curtin's theme remained in use for much of Scooby-Doo's original broadcast run. After two seasons and 24 episodes of the New Movies format from 1972 to 1973, CBS began airing reruns of the original Scooby-Doo, Where Are You! series until its option on the series expired in 1976.

The Scooby-Doo Show and Scooby's All-Star Laff-A-Lympics
Now president of ABC, Fred Silverman made a deal with Hanna-Barbera to bring new episodes of Scooby-Doo to the ABC Saturday-morning lineup, where the show went through almost yearly lineup changes. For their 1976–77 season, 16 new episodes of Scooby-Doo were joined with a new Hanna-Barbera show, Dynomutt, Dog Wonder, to create The Scooby-Doo/Dynomutt Hour (the show became The Scooby-Doo/Dynomutt Show when a bonus Scooby-Doo, Where Are You! rerun was added to the package in November 1976). Joe Ruby and Ken Spears, now working for Silverman as supervisors of the ABC Saturday-morning programs, returned the program to its original Scooby-Doo, Where Are You! format, with the addition of Scooby's dim-witted country cousin Scooby-Dum, voiced by Daws Butler, as a recurring character. The voice cast was held over from The New Scooby-Doo Movies save for Nicole Jaffe, who retired from acting in 1973. Pat Stevens took over her role as the voice of Velma.

Then Joe Ruby and Ken Spears left again to start their own studio in 1977 as competition for Hanna-Barbera. They would remain away for the rest of the 1980s.

For the 1977–78 season, The Scooby-Doo/Dynomutt Show became the two-hour programming block Scooby's All-Star Laff-A-Lympics (1977–78) with the addition of Laff-a-Lympics and Captain Caveman and the Teen Angels. In addition to eight new episodes of Scooby-Doo and reruns of the 1969 show, Scooby-Doo also appeared during the All-Star block's Laff-a-Lympics series, which featured 45 Hanna-Barbera characters competing in Battle of the Network Stars-esque parodies of Olympic sporting events. Scooby was seen as the team captain of the Laff-a-Lympics "Scooby-Doobies" team, which also featured Shaggy and Scooby-Dum among its members.

Scooby's All-Star Laff-a-Lympics was retitled Scooby's All Stars for the 1978–79 season, reduced to 90 minutes when Dynomutt was spun off into its own half-hour and the 1969 reruns were dropped. Scooby's All-Stars continued broadcasting reruns of Scooby-Doo from 1976 and 1977, while new episodes of Scooby-Doo aired during a separate half-hour under the Scooby-Doo, Where Are You! banner. After nine weeks, the separate Where Are You! broadcast was cancelled, and the remainder of the 16 new 1978 episodes debuted during the Scooby's All-Stars block. The 40 total Scooby-Doo episodes produced from 1976 to 1978 were later packaged together for syndication as The Scooby-Doo Show, under which title they continue to air.

Scooby-Doo and Scrappy-Doo
The Scooby-Doo characters first appeared outside of their regular Saturday-morning format in Scooby Goes Hollywood, an hour-long ABC television special aired in prime time on December 13, 1979. The special revolved around Shaggy and Scooby attempting to convince the network to move Scooby out of Saturday morning and into a prime-time series, and featured spoofs of then-current television series and films such as Happy Days, Superman: The Movie, Laverne & Shirley and Charlie's Angels.

In 1979, Scooby's tiny nephew Scrappy-Doo was added to both the series and the billing, in an attempt to boost Scooby-Doo's slipping ratings. The 1979–80 episodes, aired under the new title Scooby-Doo and Scrappy-Doo as an independent half-hour show, succeeded in regenerating interest in the show. Lennie Weinrib voiced Scrappy in the 1979–80 episodes, with Don Messick assuming the role thereafter. Marla Frumkin replaced Pat Stevens as the voice of Velma mid-season.

Scooby-Doo and Scrappy-Doo shorts
As a result of Scooby-Doo and Scrappy-Doo success, the entire show was overhauled in 1980 to focus more upon Scrappy-Doo. At this time, Scooby-Doo started to walk and run anthropomorphically on two feet more often, rather than on four like a normal dog as he did previously. Fred, Daphne, and Velma were dropped from the series, and the new Scooby-Doo and Scrappy-Doo format now consisted of three seven-minute comedic adventures starring Scooby, Scrappy, and Shaggy instead of one half-hour mystery. Most of the supernatural villains in the seven-minute Scooby and Scrappy cartoons, who in previous Scooby series had been revealed to be human criminals in costume, were now real within the context of the series.

This version of Scooby-Doo and Scrappy-Doo first aired from 1980 to 1982 as part of The Richie Rich/Scooby-Doo Show, an hour-long program also featuring episodes of Hanna-Barbera's new Richie Rich cartoon, adapted from the Harvey Comics character. From 1982 to 1983, Scooby-Doo and Scrappy-Doo were part of The Scooby-Doo/Scrappy-Doo/Puppy Hour, a co-production with Ruby-Spears Productions which featured two Scooby and Scrappy shorts, a Scrappy and Yabba-Doo short featuring Scrappy-Doo and his Western deputy uncle Yabba-Doo, and The Puppy's New Adventures, based on characters from a 1977 Ruby-Spears TV special. Despite the popularity, this was negatively hated by fans for how it dropped the mystery format and other main characters like Fred, Daphne, and Velma.

Beginning in 1980, a half-hour of reruns from previous incarnations of Scooby-Doo were broadcast on ABC Saturday mornings in addition to first-run episodes. Airing under the titles Scooby-Doo Classics, Scary Scooby Funnies, The Best of Scooby-Doo, and Scooby's Mystery Funhouse, the rerun package remained on the air until the end of the 1986 season.

The New Scooby and Scrappy-Doo Show
Scooby-Doo was restored to a standalone half-hour in 1983 with The New Scooby and Scrappy-Doo Show in 1983, which comprised two 11-minute mysteries per episode in a format reminiscent of the original Scooby-Doo, Where Are You! mysteries. Heather North returned to the voice cast as Daphne, who in this incarnation solved mysteries with Shaggy, Scooby, and Scrappy while working undercover as a reporter for a teen magazine.

This version of the show lasted for two seasons, with the second season airing under the title The New Scooby-Doo Mysteries. The 1984–85 season episodes featured semi-regular appearances from Fred and Velma, with Frank Welker and Marla Frumkin resuming their respective roles for these episodes.

The 13 Ghosts of Scooby-Doo
1985 saw the debut of The 13 Ghosts of Scooby-Doo, which featured Daphne, Shaggy, Scooby, Scrappy, and new characters Flim-Flam (voiced by Susan Blu) and Vincent Van Ghoul (based upon and voiced by Vincent Price) traveling the globe to capture "thirteen of the most terrifying ghosts upon the face of the earth." The final first-run episode of The 13 Ghosts of Scooby-Doo aired in December 1985, and after its reruns were removed from the ABC lineup the following March, no new Scooby series aired on the network for the next two years.

A Pup Named Scooby-Doo
Hanna-Barbera reincarnated the original Scooby-Doo, Where Are You! cast as elementary school students (a common trope in 1980s children's TV) for a new series titled A Pup Named Scooby-Doo, which debuted on ABC in 1988. A Pup Named Scooby-Doo was an irreverent re-imagining of the series, heavily inspired by the classic cartoons of Tex Avery and Bob Clampett, and eschewed the realistic aesthetic of the original Scooby series for a more Looney Tunes-like style, including an episode where Scooby-Doo's parents show up and reveal his real name to be "Scoobert". At the same time, the series returned to its original formula in that the group unmasked human villains in costume, as opposed to the supernatural monsters of the early to mid-1980s. The series also established "Coolsville" as the name of the gang's hometown; this setting was retained for several of the later Scooby productions. The retooled show was a success, remaining in production for four seasons and on ABC's lineup until 1991.

A Pup Named Scooby-Doo was developed and produced by Tom Ruegger, who had been the head story editor on Scooby-Doo since 1983. Following the first season of A Pup Named Scooby-Doo, Ruegger and much of his unit defected from Hanna-Barbera to Warner Bros. Animation to develop Steven Spielberg Presents Tiny Toon Adventures and later Animaniacs, Pinky and the Brain, and Freakazoid!.

What's New, Scooby-Doo?
In 2002, following the successes of the Cartoon Network reruns, the direct to video franchise, and the first feature film, Scooby-Doo returned to Saturday morning for the first time in a decade with What's New, Scooby-Doo?, which aired on Kids' WB from 2002 until 2006. Produced by Warner Bros. Animation, the show follows the format of the original series but places it in the 21st century, featuring a heavy promotion of modern technology (computers, DVD, the Internet, cell phones) and culture.

Beginning with this series, Frank Welker took over as Scooby's voice actor, while continuing to provide the voice of Fred as well. Casey Kasem returned as Shaggy, on the condition that the character be depicted as a vegetarian like Kasem himself. Grey DeLisle continued to voice Daphne, and former Facts of Life star Mindy Cohn voiced Velma. The series was produced by Chuck Sheetz, who had worked on The Simpsons.

Shaggy & Scooby-Doo Get a Clue!
In September 2006 a new show entitled, Shaggy & Scooby-Doo Get a Clue!, debuted on The CW's Kids' WB Saturday-morning programming block. In the new premise, Shaggy inherits money and a mansion from an uncle, an inventor who has gone into hiding from villains trying to steal his secret invention. The villains, led by "Dr. Phibes" (based primarily upon Dr. Evil from the Austin Powers series, and named after Vincent Price's character from The Abominable Dr. Phibes), then use different schemes to try to get the invention from Shaggy and Scooby, who handle the plots alone. Fred, Daphne, and Velma are normally absent, but do make appearances at times to help. The characters were redesigned and the art style revised for the new series. Scott Menville voiced Shaggy in the series, with Casey Kasem appearing as the voice of Shaggy's Uncle Albert. Shaggy & Scooby-Doo Get a Clue! ran for two seasons on The CW.

Scooby-Doo! Mystery Incorporated
The next Scooby series, Scooby-Doo! Mystery Incorporated, premiered on Cartoon Network on April 5, 2010. The first Scooby series produced for cable television, Mystery Incorporated is a reboot of the franchise, re-establishing the characters' relationships, personalities, and locations, and expanding their world to feature their parents, high school, and neighbors. The series also borrowed pieces from many parts of Scooby-Doo's long history, as well as characters and elements of other Hanna-Barbera shows to form its back story and the bases of some of its episodes. Matthew Lillard was brought over from the live-action theatrical series as the new voice of Shaggy, while Welker, Cohn, and DeLisle continued in their respective roles. Patrick Warburton, Linda Cardellini, Lewis Black, Vivica A. Fox, Gary Cole, Udo Kier, Tim Matheson, Tia Carrere, and Kate Higgins were added as new semi-regular cast members. Casey Kasem appeared in a recurring role as Shaggy's father, one of his last roles before retiring due to declining health.

The series, while still following the basic mystery-solving format of its predecessors, was broadcast as a 52-chapter animated televised novel and included elements similar to live-action mystery/adventure shows such as Buffy the Vampire Slayer and Lost. An overarching mystery surrounding the gang's hometown of Crystal Cove, California became the series' main story arc, with pieces to the mystery unfolding episode by episode. Also featured were romantic entanglements and interpersonal conflict between the lead characters. The series ran for 52 episodes over two seasons, with a three-part finale airing across April 4 and 5, 2013—exactly three years from the debut.

Be Cool, Scooby-Doo!
On March 10, 2014, Cartoon Network announced several new series based on classic cartoons, including a new Scooby-Doo animated series titled Be Cool, Scooby-Doo!. The show features the gang "living it up" the summer after the gang's senior year of high school. Along the way, they run into monsters and mayhem. The series premiered October 5, 2015 on Cartoon Network and concluded on March 18, 2018.

Scooby-Doo and Guess Who?
The Scooby-Doo series Scooby-Doo and Guess Who? premiered on the Boomerang streaming service and app on June 27, 2019. It ran for two seasons, with the second half of the second season airing on HBO Max. The series features the Mystery Inc. gang teaming up with a variety of guest stars to solve mysteries. Guest stars included Halsey, Sia, Bill Nye, Mark Hamill, Neil deGrasse Tyson, Ricky Gervais, Kenan Thompson, and Chris Paul. The series also includes fictional guest stars, including Steve Urkel (played by Jaleel White), Batman (played by Kevin Conroy), Wonder Woman (played by Rachel Kimsey), the Flash, and Sherlock Holmes.

Scooby-Doo and Guess Who?
The remaining eleven episodes of the second season were released through the streaming service on October 1, 2021.

Velma
Velma is an adult-oriented animated series which premiered on HBO Max on January 12, 2023. It ran for two seasons, and marked the first full original Scooby Doo related show on HBO Max since the previous series Guess Who? was picked up by it at the end of its run (originating on Boomerang). The series is an alternate reality prequel and spinoff to the main franchise, taking place before the formation of Mystery Inc., and does not include Scooby-Doo himself. Unlike in the previous series and films, the main characters (and main voice cast) in Velma are multi-racial.

Scooby-Doo! The Live-Action Series
On April 29, 2024, Deadline reported that a live action Scooby-Doo! series is in development by Berlanti Productions on Netflix titled ''Scooby-Doo! The Live-Action Series'', with Josh Appelbaum and Scott Rosenberg writing; and Greg Berlanti, Sarah Schechter, Leigh London Redman, André Nemec, Jeff Pinkner, Jonathan Gabay and Midnight Radio's Adrienne Erickson executive producing.

Television films, reruns, and direct-to-video films
From 1987 to 1988, Hanna-Barbera Productions produced Hanna-Barbera Superstars 10, a series of syndicated television films featuring their most popular characters, including Yogi Bear, Huckleberry Hound, The Flintstones, and The Jetsons. Scooby-Doo, Scrappy-Doo and Shaggy starred in three of these films: Scooby-Doo Meets the Boo Brothers (1987), Scooby-Doo and the Ghoul School (1988), and Scooby-Doo and the Reluctant Werewolf (1988). These three films took their tone from the early-1980s Scooby-Doo and Scrappy-Doo entries, and featured the characters encountering actual monsters and ghosts rather than masqueraded people. Scooby-Doo and Shaggy later appeared as the narrators of the television film Arabian Nights, originally broadcast by TBS in 1994, Don Messick's final outing as the original voice of Scooby-Doo.

Reruns of Scooby-Doo have been in syndication since 1980, and have also been shown on cable television networks such as TBS Superstation (until 1989) and USA Network (as part of the USA Cartoon Express from 1990 to 1994). In 1993, A Pup Named Scooby-Doo, having just recently ended its network run on ABC, began reruns on Cartoon Network. With Turner Broadcasting purchasing Hanna-Barbera in 1991, in 1994 the Scooby-Doo franchise became exclusive to the Turner networks: Cartoon Network, TBS Superstation, and TNT. Canadian network Teletoon began airing Scooby-Doo, Where Are You! in 1997, with the other Scooby series soon following. When TBS and TNT ended their broadcasts of H-B cartoons in 1998, Scooby-Doo became the exclusive property of both Cartoon Network and sister station Boomerang.

With Scooby-Doo's restored popularity in reruns on Cartoon Network, Warner Bros. Animation and Hanna-Barbera (by then a subsidiary of Warner Bros. following the merger of Time Warner and Turner Entertainment in 1996) began producing one new Scooby-Doo direct-to-video film a year, beginning in 1998. These films featured a slightly older version of the original five-character cast from the Scooby-Doo, Where Are You! days. The first four DTV entries were Scooby-Doo on Zombie Island (1998), Scooby-Doo! and the Witch's Ghost (1999), Scooby-Doo and the Alien Invaders (2000), and Scooby-Doo and the Cyber Chase (2001). Frank Welker was the only original voice cast member to return for these productions. Don Messick had died in 1997 and Casey Kasem, a strict vegetarian, relinquished the role of Shaggy after having to provide the voice for a 1995 Burger King commercial. Therefore, Scott Innes took over as both Scooby-Doo and Shaggy (Billy West voiced Shaggy in Scooby-Doo on Zombie Island). B.J. Ward took over as Velma, and Mary Kay Bergman voiced Daphne until her death in November 1999, and was replaced by Grey DeLisle.

These first four direct-to-video films differed from the original series format by placing the characters in plots with a darker tone and pitting them against actual supernatural forces. Scooby-Doo on Zombie Island, featured the original 1969 gang, reunited after years of being apart, fighting voodoo-worshiping cat creatures in the Louisiana bayou. ''Scooby-Doo! and the Witch's Ghost'' featured an author (voice of Tim Curry) returning to his Massachusetts hometown with the gang, to find out that an event is being haunted by the author's dead ancestor Sarah, who was an actual witch. The Witch's Ghost introduced a goth rock band known as The Hex Girls, who became recurring characters in the Scooby-Doo franchise.

Scooby-Doo and the Cyber Chase was the final production made by the Hanna-Barbera studio, which was absorbed into parent company Warner Bros. Animation following William Hanna's death in 2001. Warner Animation continued production of the direct-to-video series while also producing new Scooby-Doo series for television.

The direct-to-video productions continued to be produced concurrently with at least one entry per year. Two of these entries, Scooby-Doo! and the Legend of the Vampire and Scooby-Doo! and the Monster of Mexico (both 2003) were produced in a retro-style reminiscent of the original series, and featured Heather North and Nicole Jaffe as the voices of Daphne and Velma, respectively. Later entries produced between 2004 and 2009 were done in the style of What's New, Scooby-Doo, using that show's voice cast. Entries from 2010 on use the original 1969 designs and feature Matthew Lillard as the voice of Shaggy, the character Lillard portrayed in the live-action theatrical Scooby-Doo films. Two Scooby-Doo! movies were released in 2016, named Lego Scooby-Doo! Haunted Hollywood and Scooby-Doo! and WWE: Curse of the Speed Demon.

Scooby-Doo! direct-to-video specials
Beginning in 2012, Warner Bros. Animation began producing direct-to-video special episodes in the style of the concurrently produced films for inclusion on Scooby-Doo compilation DVD sets otherwise including episodes from previous Scooby series. These include Scooby-Doo! Spooky Games, included on the July 2012 release ''Scooby-Doo! Laff-A-Lympics: Spooky Games, Scooby-Doo! Haunted Holidays, from the October 2012 release Scooby-Doo! 13 Spooky Tales: Holiday Chills and Thrills, and Scooby-Doo! and the Spooky Scarecrow and Scooby-Doo! Mecha Mutt Menace, from the September 2013 DVD releases Scooby-Doo! 13 Spooky Tales: Run for Your 'Rife! and Scooby-Doo! 13 Spooky Tales: Ruh-Roh Robot!''. On May 13, 2014, another episode, Scooby-Doo! Ghastly Goals was released on the ''Scooby-Doo! 13 Spooky Tales: Field of Screams'' DVD. On May 5, 2015, Scooby-Doo! and the Beach Beastie, the sixth direct-to-video special, was released on the ''Scooby-Doo! 13 Spooky Tales: Surf's Up Scooby-Doo'' DVD.

The direct-to-video series' 34th installment, Trick or Treat Scooby-Doo! (2022), made headlines for portraying Velma as a lesbian (by showing her "crushing big time" on a female guest character), which was in accordance with long-held fan speculation but had never previously been depicted.

Live-action films
A feature-length live-action film version of Scooby-Doo was released by Warner Bros. Pictures on June 14, 2002. Directed by Raja Gosnell, the film starred Freddie Prinze Jr. as Fred, Sarah Michelle Gellar as Daphne, Matthew Lillard as Shaggy, and Linda Cardellini as Velma. Scooby-Doo, voiced by Neil Fanning, was created on-screen by computer-generated special effects. Scooby-Doo was a financially successful release, with a domestic box office gross of over US$130 million.

A sequel, Scooby-Doo 2: Monsters Unleashed, followed in March 2004 with the same cast and director. Scooby-Doo 2 earned US$84 (€55.98) million at the U.S. box office. A third film was planned, but later scrapped following Warner Bros.' disappointment at the returns from Scooby-Doo 2.

In addition, a live-action television film, Scooby-Doo! The Mystery Begins, was released on DVD and simultaneously aired on Cartoon Network on September 13, 2009, the 40th anniversary of the series' debut. The film starred Nick Palatas as Shaggy, Robbie Amell as Fred, Kate Melton as Daphne, Hayley Kiyoko as Velma, and Frank Welker as the voice of Scooby-Doo. A second live-action TV movie, Scooby-Doo! Curse of the Lake Monster, retained the same cast and aired on October 16, 2010, and a direct-to-video spin-off Daphne & Velma in 2018. The Mystery Begins and Curse of the Lake Monster serve as reboots to the 2002 and 2004 films while Daphne and Velma serves as a spin-off/prequel to them.

Theatrical animated film
In 2013, Warner Bros. Pictures was developing a fully animated Scooby-Doo feature film with Atlas Entertainment. Charles Roven and Richard Suckle, who produced the first two live-action films, were producing the animated film, and Matt Lieberman was writing the film. In 2014, Warner Bros. was restarting the film series with Randall Green writing a new movie. In 2015, Warner Bros. had Tony Cervone lined up to direct an animated film, with Allison Abbate as producer and Dan Povenmire as executive producer. Originally planned for a September 21, 2018 release, it was later pushed back to May 15, 2020, with Dax Shepard co-directing and co-writing. The Hollywood Reporter announced that Frank Welker will be reprising his voice role as Scooby, and that he will be joined by Will Forte and Gina Rodriguez voicing Shaggy and Velma, while Tracy Morgan will be voicing Captain Caveman, from the Hanna-Barbera series Captain Caveman and the Teen Angels and Deadline reported that Zac Efron and Amanda Seyfried will voice Fred and Daphne. In addition, Ken Jeong will be voicing Dynomutt, Dog Wonder from Hanna-Barbera series of the same name and Kiersey Clemons will voice Dee Dee Sykes, a character from Captain Caveman and the Teen Angels. Dick Dastardly, from Hanna-Barbera's Wacky Races, will be the film's main antagonist, voiced by Jason Isaacs. In March 2020, the film's theatrical release was delayed indefinitely due to the COVID-19 pandemic. On April 22, 2020, Warner Bros. announced that due to movie theater closures the theatrical release for Scoob! had been cancelled, with the film released instead on Premium video on demand in the United States and Canada on May 15, 2020, the original date of release. In July 2020, Warner Bros. confirmed the film would still play in theaters in select countries with relaxed COVID-19 restrictions. The film subsequently received a secondary theatrical release in the United States beginning on May 21, 2021, in selected markets.

Cast

 * Scooby-Doo: Don Messick was the original voice of Scooby-Doo from 1969 until 1995. Hadley Kay performed the voice for the Johnny Bravo episodes "Bravo Dooby-Doo" and "'Twas the Night", as well as in commercials, in 1997. Scott Innes was the voice of Scooby-Doo from 1998 to 2002. Neil Fanning voiced Scooby-Doo in the live-action Warner Bros. theatrical films produced in 2002 and 2004. Frank Welker is the current voice of Scooby-Doo, having taken over the role from Innes in 2002, although Innes voiced the character in video game projects (including PC, DVD and board games), commercials and some toys until 2008. Dave Coulier (2005) and Seth Green (2007, 2012, 2018) voiced Scooby in the Robot Chicken parodies.
 * Norville "Shaggy" Rogers: Casey Kasem was the original voice of Shaggy from 1969 until 1997. Billy West voiced Shaggy in Scooby-Doo on Zombie Island and Scooby-Doo: Behind the Scenes in 1998. Scott Innes voiced the character from 1999 to 2002 and he continued to voice Shaggy in video game projects (including PC, DVD and board games), commercials and some toys until 2009. Casey Kasem returned to the voice role in 2002 and continued as Shaggy until 2009. In 2006, Kasem continued to voice Shaggy only in the direct-to-video film series until 2009, while Scott Menville performed the voice of Shaggy in the 2006–08 CW series Shaggy & Scooby-Doo Get a Clue!. Matthew Lillard appeared as Shaggy in the live action 2002 and 2004 theatrical films, and took over as the voice of the animated character in 2010. He also voiced Shaggy in four stop-motion parody sketches for the Adult Swim show Robot Chicken. Nick Palatas appeared as Shaggy in the 2009 and 2010 live-action TV movies.
 * Fred Jones: Frank Welker has always performed the voice of the animated versions of Fred since 1969, with the exception of the 1988–91 ABC series A Pup Named Scooby-Doo, where Carl Steven performed the voice of preteen Fred. Freddie Prinze Jr. appears as Fred in the live-action theatrical films and voiced the character in the Robot Chicken parodies. Robbie Amell played Fred in the live-action TV movies.
 * Daphne Blake: Stefanianna Christopherson was the voice of Daphne in the first season of Scooby-Doo, Where Are You! in 1969–70. Heather North assumed the role for season two in 1970, and continued as Daphne through 1997, save for Kellie Martin's turn as preteen Daphne in A Pup Named Scooby-Doo. Mary Kay Bergman performed the voice of Daphne from 1998 to 2000, with Grey DeLisle assuming the role. She continues to perform the role to this day. North reprised her voice role for two 2003 direct-to-video films, Scooby-Doo! and the Legend of the Vampire and Scooby-Doo! and the Monster of Mexico. Sarah Michelle Gellar appears as Daphne in the live-action theatrical films and as Daphne's voice in the Robot Chicken parodies. Kate Melton played Daphne in the live-action TV movies.
 * Velma Dinkley: Nicole Jaffe was the original voice of Velma from 1969 to 1973. Pat Stevens assumed the role from 1976 to 1979, with Marla Frumkin taking over midseason on Scooby-Doo and Scrappy-Doo in the latter year. Frumkin returned to voice Velma on a recurring basis for The New Scooby-Doo Mysteries in 1984, and Christina Lange voiced preteen Velma in A Pup Named Scooby-Doo. B. J. Ward voiced Velma from 1997 to 2002, with Mindy Cohn assuming the role in 2002. As with North, Jaffe reprised her voice role for Scooby-Doo! and the Legend of the Vampire and Scooby-Doo! and the Monster of Mexico in 2003. Stephanie D'Abruzzo voiced Velma for the 2013 puppet film Scooby-Doo! Adventures: The Mystery Map. In 2015, Kate Micucci took on the role for the series Be Cool, Scooby-Doo! and Lego Scooby-Doo shorts and specials; in 2016 she took over the role from Cohn completely. Linda Cardellini appears as Velma in the live-action theatrical films and as the voice of Velma in the Robot Chicken parodies. Hayley Kiyoko played Velma in the live-action TV movies.
 * Scrappy-Doo: Lennie Weinrib voiced Scrappy-Doo during the first version of Scooby-Doo and Scrappy-Doo in 1979–80. Don Messick assumed the role in 1980 for the Scooby-Doo and Scrappy-Doo segments of The Richie Rich/Scooby-Doo Show and continued as Scrappy through 1988. Scrappy has only appeared sporadically since 1988, with Scott Innes performing the voice in the 2002 live-action film, which portrays Scrappy as the main villain, as well as in Cartoon Network bumpers, video games and toys since 1999. Dan Milano voiced Scrappy in a 2007 Robot Chicken sketch.

Comic books

Gold Key Comics began publication of Scooby-Doo, Where Are You! comic books in December 1969. The comics initially contained adaptations of episodes of the television show drawn by Phil DeLara, Jack Manning and Warren Tufts. The comic books later moved to all-original stories until ending with issue #30 in 1974. Several of these issues were written by Mark Evanier and drawn by Dan Spiegle. Charlton published Scooby comics, many drawn by Bill Williams, for 11 issues in 1975. From 1977 to 1979, Marvel Comics published nine issues of Scooby-Doo, all written by Evanier and drawn by Spiegel. Harvey Comics published reprints of the Charlton comics, as well as a handful of special issues, between 1993 and 1994.

In 1995, Archie Comics began publishing a monthly Scooby-Doo comic book, the first year of which featured Scrappy-Doo among its cast. Evanier and Spiegel worked on three issues of the series, which ended after 21 issues in 1997 when Warner Bros.' DC Comics acquired the rights to publish comics based on Hanna-Barbera characters. DC's Scooby-Doo series continues publication to this day. In 2013, DC began a digital bi-monthly comic book titled Scooby-Doo Team-Up, crossing over Mystery Inc. with other DC and Hanna-Barbera characters. Since then, the series has become a monthly comic book available in print.

In 2004, a limited series of a 100 comic books called ''Scooby-Doo! World of Mystery'' was released. In each issue, Mystery Inc. go from country to country solving mysteries. Each issue came with a pack of exclusive cards, with 350 in total able to be collected.

In 2016, DC launched a new monthly comic book entitled Scooby Apocalypse, with the characters being reinvented in a story set in a post-apocalyptic world, where monsters roam the streets and Scooby and the gang must find a way to survive at all costs, while also trying to find a way to reverse the apocalypse.

Merchandising
Early Scooby-Doo merchandise included a 1973 Milton Bradley board game, decorated lunch boxes, iron-on transfers, coloring books, story books, records, underwear, and other such goods. When Scrappy-Doo was introduced to the series in 1979, he, Scooby, and Shaggy became the foci of much of the merchandising, including a 1983 Milton-Bradley Scooby-Doo and Scrappy-Doo board game. The first Scooby-Doo video game appeared in arcades in 1986, and has been followed by a number of games for both home consoles and personal computers. Scooby-Doo multivitamins also debuted at this time, and have been manufactured by Bayer since 2001.

Scooby-Doo merchandising tapered off during the late 1980s and early 1990s, but increased after the series' revival on Cartoon Network in 1995. Today, all manner of Scooby-Doo-branded products are available for purchase, including Scooby-Doo breakfast cereal, plush toys, action figures, car decorations, Barbie dolls from Mattel and much more. Real "Scooby Snacks" dog treats are produced by Del Monte Pet Products. Hasbro has created a number of Scooby board games, including a Scooby-themed edition of the popular mystery board game Clue. In 2007, the Pressman Toy Corporation released the board game ''Scooby-Doo! Haunted House''. Beginning in 2001, a Scooby-Doo children's book series was authorized and published by Scholastic. These books, written by Suzanne Weyn, include original stories and adaptations of Scooby theatrical and direct-to-video features.

From 1990 to 2002, Shaggy and Scooby-Doo appeared as characters in the Funtastic World of Hanna-Barbera simulator ride at Universal Studios Florida. The ride was replaced in the early 2000s with a Jimmy Neutron attraction, and The Funtastic World of Hanna-Barbera instead became an attraction at several properties operated by Paramount Parks. Shaggy and Scooby-Doo are currently costumed characters at Universal Studios Florida, and can be seen driving the Mystery Machine around the park.

In 2001, Scooby-Doo in Stagefright, a live stage play based upon the series, began touring across the world. A follow-up, Scooby-Doo and the Pirate Ghost, followed in 2009.

The Mystery Machine has been used as the basis for many die-cast models and toys, such as from Hot Wheels.

The brand made $800 million in retail sales in 1999. In 2004, Scooby-Doo merchandise had generated US$1 billion in retail sales Licensed merchandise also sold US$496 million in 2015, US$501 million in 2016, and US$353 million in 2017.

Reception and legacy
During its five-decade broadcast history, Scooby-Doo has received two Emmy nominations: a 1989 Daytime Emmy nomination for A Pup Named Scooby-Doo, and a 2003 Daytime Emmy nomination for What's New, Scooby-Doo's Mindy Cohn in the "Outstanding Performer in an Animated Program" category. Science advocate Carl Sagan favorably compared the predominantly skeptic oriented formula to that of most television dealing with paranormal themes, and considered that an adult analogue to Scooby-Doo would be a great public service.

Scooby-Doo has maintained a significant fan base, which has grown steadily since the 1990s due to the show's popularity among both young children and nostalgic adults who grew up with the series. Several television critics have stated that the show's mix of the comedy-adventure and horror genres was the reason for its widespread success. As Fred Silverman and the Hanna-Barbera staff had planned when they first began producing the series, Scooby-Doo's ghosts, monsters and spooky locales tend more towards humor than horror, making them easily accessible to younger children. "Overall, [Scooby-Doo is] just not a show that is going to overstimulate kids' emotions and tensions," offered American Center for Children and Media executive director David Kleeman in a 2002 interview. "It creates just enough fun to make it fun without getting them worried or giving them nightmares.

Older teenagers and adults have admitted to enjoying Scooby-Doo because of presumed subversive themes which involve theories of drug use and sexuality, in particular that Shaggy is assumed to be a user of cannabis and Velma is assumed to be a lesbian. Such themes were pervasive enough in popular culture to find their way into Warner Bros.' initial Scooby-Doo feature film in 2002, though several of the scenes were edited before release to secure a family-friendly "PG" rating. Series creators Joe Ruby and Ken Spears reported that they "took umbrage" to the inclusion of such themes in the Scooby-Doo feature and other places, and denied intending their characters to be drug users in any way.

Like many Hanna-Barbera shows, the early Scooby-Doo series have been criticized at times for their production values and storytelling. In 2002, Jamie Malanowski of The New York Times commented that "[Scooby-Doo's] mysteries are not very mysterious, and the humor is hardly humorous. As for the animation—well, the drawings on your refrigerator may give it competition."

By the 2000s, Scooby-Doo had received recognition for its popularity by placing in a number of top cartoon or top cartoon character polls. The August 3, 2002, issue of TV Guide featured its list of the 50 Greatest Cartoon Characters of All Time, in which Scooby-Doo placed twenty-second. Scooby also ranked thirteenth in Animal Planet's list of the 50 Greatest TV Animals. For one year from 2004 to 2005, Scooby-Doo held the Guinness World Record for having the most episodes of any animated television series ever produced, a record previously held by and later returned to The Simpsons. Scooby-Doo was published as holding this record in the 2006 edition of the Guinness Book of Records.

In January 2009, entertainment website IGN named Scooby-Doo #24 on its list of the Top 100 Best Animated TV Shows. Writing in 2020, Christopher Orr of The Atlantic queried why the franchise had remained popular for several decades, concluding that it was primarily due to the many differing ways in which the relationship between the main characters could be interpreted or used as a metaphor.

Five College folklore
A popular urban legend among Five College students holds that the characters on Scooby-Doo represent the five colleges. The legend has Velma representing Smith College and Daphne as Mount Holyoke College (or vice-versa), Fred as Amherst College, Shaggy as Hampshire College, and Scooby as UMass Amherst. Hanna-Barbera Productions, CBS executive Fred Silverman, and Mark Evanier, one of the show's writers, have stated that the legend is false. Moreover, Scooby-Doo creators Joe Ruby and Ken Spears have been explicit in the cartoon show being based on the radio program I Love a Mystery and the TV sitcom The Many Loves of Dobie Gillis, with the four teenagers being based directly on characters from Dobie Gillis. In addition, Scooby-Doo made its television debut in 1969, one year before Hampshire College opened.

In popular culture
As with most popular franchises, Scooby-Doo has been parodied and has done parodies.
 * The cult television and comic book series Buffy the Vampire Slayer features a group of characters that refer to themselves as the "Scooby Gang", who similarly battle supernatural forces and solve supernatural monster mysteries. The show contains obvious influences of Scooby-Doo, where "The Scoobies" use books to look up monsters. Sarah Michelle Gellar, the actress who plays Buffy Summers on the series, later went on to appear as Daphne Blake in the live-action films Scooby-Doo and Scooby-Doo 2: Monsters Unleashed.
 * Scooby-Doo and the Mystery Inc. gang (based on their classic 1972 incarnation as opposed to their more recent incarnations) appear in the second part of the Batman: The Brave and the Bold episode "Bat-Mite Presents: Batman's Strangest Cases" in which they team up with Batman and Robin to rescue Weird Al who was kidnapped by the Joker and the Penguin.
 * The song Scooby-Doo and the Snowmen Mystery was released in 1972 in the United Kingdom by the label Music for Pleasure.
 * The film Wayne's World includes an alternate ending called the "Scooby-Doo Ending" in which a character in the film is revealed to have been wearing a mask. It also includes a reference to the iconic line "Let's see who this really is" before removing the mask. When the culprit is revealed to be Old Man Withers, owner of the local haunted amusement park, Withers mutters "And I would have gotten away with it, too, if it wasn't for those meddling kids!"
 * Jay and Silent Bob Strike Back has a brief scene where the title characters hitch a ride in the Mystery Machine with Scooby and the gang.
 * The filk band Ookla the Mok open their 2003 album Oh Okay LA with the song "W.W.S.D.?" ("What Would Scooby Do?"), which proposes a deontological system of moral philosophy based on the actions of Scooby-Doo.
 * In October 1999, Cartoon Network made a Scooby-Doo spoof of The Blair Witch Project called The Scooby-Doo Project.
 * A Scooby-Doo parody appeared in the Mad episode "Kitchen Nightmares Before Christmas / How I Met Your Mummy".
 * Scooby-Doo was parodied on Futurama episode "Saturday Morning Fun Pit", where the characters from Planet Express take on the roles of the gang (Bender as Scooby, Hermes as Fred, Leela as Daphne, Amy as Velma and Fry as Shaggy).
 * The Venture Bros. episode "¡Viva los Muertos!" features a thinly parodied version of the gang as aging, gone-to-seed miscreants with the characters matched to corresponding serial killers and radical figures, e.g. Fred being mixed with Ted Bundy into the composite character "Ted".
 * The series is parodied in the animated music video for the song "Ghost" by Mystery Skulls.
 * The animated series Arthur has a parody of Scooby-Doo called "Spooky-Poo".
 * In the South Park episode "Korn's Groovy Pirate Ghost Mystery", the nu metal band Korn, parodying Scooby and the gang, tackle an invasion of mysterious "Pirate Ghosts". They enlist the help of Stan Marsh, Kyle Broflovski, Eric Cartman and Kenny McCormick, and after they solve the mystery they perform "Falling Away from Me" from their album Issues.
 * The gang was featured in Harvey Birdman, Attorney at Law where the title character defends Shaggy and Scooby against possession charges in the 2002 episode "Shaggy Busted".
 * After defeating and capturing a pirate crew in the role playing video game Golden Sun: The Lost Age, one of the imprisoned pirates declares that, "Everything would have been fine if it hadn't been for you meddling kids!"
 * In the Teen Titans Go! episode "The Cruel Giggling Ghoul", each Titan assumes the role of a Scooby Gang member (with Beast Boy as Scooby) to investigate a mystery at a spooky amusement park, with the help of LeBron James. The Scooby Gang later appears in the crossover episode "Cartoon Feud", where Control Freak forces them to compete in Family Feud. The Scooby Gang later appears in the episodes "Intro" and "Warner Bros. 100th Anniversary".
 * The novel Meddling Kids (2017) by Edgar Cantero parodies not only Scooby-Doo, but also teen-detective dramas (such as the Hardy Boys, Nancy Drew, and the Famous Five) in general.
 * The CW's television series Supernatural crossed over with the Scooby-Doo franchise in the episode Scoobynatural, which aired March 29, 2018. The animated collaboration featured the three main characters of Supernatural (Sam, Dean, and Castiel) along with Scooby and the gang as they team up to solve a supernatural mystery.
 * Velma made a cameo appearance in The Lego Movie 2: The Second Part, voiced by Trisha Gum.
 * The Harvey Street Kids episode "Crush 4U, Where RU?" fully references the Scooby-Doo series, especially the title.
 * Scooby-Doo and the gang appear in the 2021 film Space Jam: A New Legacy. Their design is the same from Scoob! They appear among the other Warner Bros. characters in the film.
 * Scooby-Doo and Shaggy both appeared in "The Official BBC Children in Need Medley" in 2009.
 * In the Black Mirror episode "Loch Henry", a character sings a line from the opening theme of Scooby-Doo, Where Are You! when investigating a cellar where murders had taken place.
 * The term "Don't have a Scooby" is recognised rhyming slang for "clue".
 * The Scooby Gang appeared in the Jellystone! third season episode "Frankenhooky". They tie up the Ghost Chasers with a rope before unmasking them to reveal Sooey Pig, Orful Octopus, and Magic Rabbit of the Really Rottens. The Scooby Gang are revealed to actually be Cindy Bear, Ranger Smith, Hardy Har Har, Tubb, and a cotton ball in a jar.

title: Scooby-Doo


text:
User:Stokerm

Chemical engineer employed in the semiconductor industry.

I was an early contributor to the following articles, many of which have evolved significantly (or become extinct!):

 * Allotropy
 * Absolute zero
 * Adiabatic
 * Anneal
 * Argentine cuisine
 * Atom
 * Auger electrons
 * Avogadros Number
 * Binary
 * Boiling point
 * Carnot heat engine
 * Catalyst
 * Catalysis
 * Chemical engineering - Not much here, yet!
 * Chemical Vapor Deposition
 * Chromatography
 * Classical Mechanics
 * Columbia River
 * Covalent bond
 * Critical properties
 * Density
 * Derivation of the partition function - Please read & comment on this.
 * Depth of field
 * Deuterium
 * Diamagnetism
 * Diffusion pump
 * Dipole
 * Dispersion
 * Electromagnetic radiation
 * Empanadas
 * Equations of State
 * Equations of state/History
 * Ferromagnetism
 * Free radical
 * Gas Laws - should probably just be combined with Equations of State
 * Gibbs phase rule
 * Haber process
 * Heat
 * Isotope
 * Lithium
 * Magnetism
 * Maxwell-Boltzmann distribution
 * Metastability
 * Mole fraction
 * Mormon
 * Mössbauer Effect
 * Osmosis
 * Paramagnetism
 * The particle in a box
 * Phases of matter
 * Phenols
 * Photolithography
 * Photoresist
 * Polymorphism
 * Quartz
 * Semiconductor
 * Semiconductor device fabrication
 * Scanning electron microscope
 * Schrodinger wave equation
 * Sputtering
 * Statistical Mechanics
 * Superparamagnetism
 * Temperature
 * Thermodynamic reversibility
 * Thermodynamics
 * Titanium
 * Triple point
 * Turbomolecular pump
 * White trash

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Stokerm


text:
Saint Casimir

Casimir Jagiellon (Casimirus; Kazimieras; Kazimierz; 3 October 1458 – 4 March 1484) was a prince of the Kingdom of Poland and of the Grand Duchy of Lithuania. The second son of King Casimir IV Jagiellon, he was tutored by Johannes Longinus, a Polish chronicler and diplomat. After his elder brother Vladislaus was elected as King of Bohemia in 1471, Casimir became the heir apparent. At the age of 13, Casimir participated in the failed military campaign to install him as King of Hungary. He became known for his piety, devotion to God, and generosity towards the sick and poor. He became ill (most likely with tuberculosis) and died at the age of 25. He was buried in Vilnius Cathedral. His canonization was initiated by his brother King Sigismund I the Old in 1514 and the tradition holds that he was canonized in 1521.

Veneration of Casimir saw a resurgence in the 17th century when his feast day was confirmed by the pope in 1602 and the dedicated Chapel of Saint Casimir was completed in 1636. Casimir became a patron saint of Lithuania and Lithuanian youth. In Vilnius, his feast day is marked annually with Kaziuko mugė (a trade fair) held on the Sunday nearest to 4 March, the anniversary of his death. There are more than 50 churches named after Casimir in Lithuania and Poland, including Church of St. Casimir, Vilnius and St. Kazimierz Church, Warsaw, and more than 50 churches in Lithuanian and Polish diaspora communities in America. Women's congregation Sisters of Saint Casimir was established in 1908 and remains active in the United States.

Early life and education
A member of the Jagiellon dynasty, Casimir was born in Wawel Castle in Kraków. Casimir was the third child and the second son of the King of Poland and Grand Duke of Lithuania Casimir IV and Queen Elisabeth Habsburg of Austria. Elisabeth was a loving mother and took active interest in her children's upbringing. The Queen and the children often accompanied the King in his annual trips to the Grand Duchy of Lithuania.

Casimir was a polyglot and knew Lithuanian, Polish, German and Latin languages. From the age of nine, Casimir and his brother Vladislaus were educated by the Polish priest Fr. Jan Długosz. The boys were taught Latin and German, law, history, rhetoric, and classical literature. Długosz was a strict and conservative teacher who emphasized ethics, morality, and religious devotion. According to Stanisław Orzechowski (1513–1566), the princes were subject to corporal punishment which was approved by their father. Długosz noted Casimir's skills in oratory when he delivered speeches to greet his father returning to Poland in 1469 and Jakub Sienienski, the Bishop of Kujawy, in 1470.

Hungarian campaign
Prince Casimir's uncle Ladislaus the Posthumous, King of Hungary and Bohemia, died in 1457 at the age of 17, without leaving an heir. Casimir's father, King Casimir IV, subsequently advanced his claims to Hungary and Bohemia, but could not enforce them due to the Thirteen Years' War (1454–66). Instead, Hungarian nobles elected Matthias Corvinus and Bohemian nobles selected George of Poděbrady as their kings. George of Poděbrady died in March 1471. In May 1471, Vladislaus, eldest son of Casimir IV, was elected to the throne of Bohemia. However, a group of Catholic Bohemian nobles supported Matthias Corvinus instead of Vladislaus II. In turn, a group of Hungarian nobles conspired against Matthias Corvinus and invited the Polish king to overthrow him. King Casimir IV decided to install his son, Casimir, in Hungary.

Poland amassed an army of 12,000 men, commanded by Piotr Dunin and Dziersław of Rytwiany. Both King Casimir and Prince Casimir participated in the campaign. In October 1471, the Polish army crossed the Hungarian border and slowly marched towards Buda. Matthias Corvinus managed to win over the majority of the Hungarian nobles, including the main conspirator Archbishop János Vitéz, and the Polish army did not receive the expected reinforcements. Only Deák, Perény and Rozgonyi families sent troops. Upon hearing that Corvinus' army of 16,000 men camped outside of Pest, the Polish army decided to retreat from Hatvan to Nitra. There the soldiers battled food shortages, spreading infectious diseases, and the upcoming winter. The Polish King also lacked funds to pay the mercenaries. As a result, the Polish army decreased by about a third. In December 1471, Prince Casimir, fearing for his safety, was sent to Jihlava closer to the Polish border and that further eroded their soldiers' morale. Corvinus took Nitra and a one-year truce was completed in March 1472 in Buda. Prince Casimir returned to Kraków to resume his studies with Długosz.

Długosz remarked that Prince Casimir felt "great sorrow and shame" regarding the failure in Hungary. Polish propaganda, however, portrayed him as a savior, sent by divine providence, to protect the people from a godless tyrant (i.e. Matthias Corvinus) and marauding pagans (i.e. Muslim Ottoman Turks). Prince Casimir was also exposed to the cult of his uncle King Władysław III of Poland who died in the 1444 Battle of Varna against the Ottomans. This led some researchers, including Jacob Caro, to conclude that the Hungarian campaign pushed Prince Casimir into religious life.

Later life and death

As his elder brother, Vladislaus II, ruled Bohemia, Prince Casimir became the heir apparent to the throne of Poland and Lithuania. Italian humanist writer Filippo Buonaccorsi (also known as Filip Callimachus) was hired to become Casimir's tutor in political matters, but his Renaissance views had less influence on Casimir than Długosz. In 1474, the Italian merchant and traveler Ambrogio Contarini met with Prince Casimir and was impressed by his wisdom. Prince Casimir completed his formal education at age 16 and spent most of his time with his father. In 1476, Prince Casimir accompanied his father to Royal Prussia where he tried to resolve the conflict with the Prince-Bishopric of Warmia (see War of the Priests). In 1478 Seimas of the Grand Duchy of Lithuania demanded that King Casimir IV leave either Prince Casimir or Prince John I Albert in Lithuania as a regent. King Casimir IV feared separatist moods and refused, but after settling the conflict in Prussia, moved to Vilnius.

Between 1479 and 1484 his father spent most of his time in Vilnius attending to the affairs of Lithuania. In 1481, Mikhailo Olelkovich and his relatives planned to murder King Casimir and Prince Casimir during a hunt at a wedding of Feodor Ivanovich Belsky. The plan was discovered and Prince Casimir, perhaps fearing for his safety, was sent to Poland to act as vice-regent. Around the same time his father tried to arrange his marriage to Kunigunde of Austria, daughter of Emperor Frederick III. It is often said that Prince Casimir refused the match, preferring to remain celibate and sensing his approaching death. According to Maciej Miechowita, Prince Casimir developed tuberculosis. In May 1483, Prince Casimir joined his father in Vilnius. There, after the death of Andrzej Oporowski, Bishop and Vice-Chancellor of the Crown, Prince Casimir took over some of his duties in the chancellery. However, his health deteriorated while rumors about his piety and good deeds spread further. In February 1484, the Polish sejm in Lublin was aborted as King Casimir IV rushed back to Lithuania to be with his ill son. Prince Casimir died on 4 March 1484, in Grodno. His remains were interred in Vilnius Cathedral, where the dedicated Saint Casimir's Chapel was built in 1636.

Pious life and attributed miracles

Surviving contemporary accounts described Prince Casimir as a young man of exceptional intellect and education, humility and politeness, who strove for justice and fairness. Early sources do not attest to his piety or devotion to God, but his inclination to religious life increased towards the end of his life. Later sources provide some stories of Casimir's religious life. Marcin Kromer (1512–1589) said Casimir refused his physician's advice to have sexual relations with women in hopes to cure his illness. Other accounts say Casimir contracted his lung disease after a particularly hard fast or that he could be found pre-dawn, kneeling by the church gates, waiting for a priest to open them. Zacharias Ferreri (1479–1524) wrote that Casimir composed a prayer in hexameter on Christ's incarnation but this text has not survived. Later, a copy of Omni die dic Mariae ("Daily, Daily Sing to Mary") was found in Casimir's coffin. The hymn became so strongly associated with Casimir that sometimes it is known as "Hymn of St. Casimir" and he is credited as its author. The lengthy hymn has an intricate meter and rhyme scheme (alternate acatalectic and catalectic trochaic dimeter with internal rhyme in the first and third verses (aa/b, cc/b)) and was most likely written by Bernard of Cluny.

One of the first miracles attributed to Casimir was his appearance before the Lithuanian army during the Siege of Polotsk in 1518. Casimir showed where Lithuanian troops could safely cross the Daugava River and relieve the city, besieged by the army of the Grand Duchy of Moscow. Ferreri's hagiography of 1521 mentions many miracles of Casimir are known but describes only one – a Lithuanian victory against the Russians. The description lacks specifics, such as date or location, but most likely refers to the Lithuanian victory in 1519 against Russian troops that raided the environs of Vilnius, and not the more popular story of the Siege of Polotsk.

Canonization and official veneration
Casimir's official cult started spreading soon after his death. In 1501, Pope Alexander VI, citing Casimir's miracles as well as the splendor of the chapel where he was buried, granted a special indulgence to those who would pray in the chapel from one vespers to another during certain Catholic festivals, and would contribute to the upkeep of the chapel. In 1513, Andrzej Krzycki wrote a poem mentioning numerous wax votive offerings on Casimir's grave. In 1514, during the Fifth Council of the Lateran, Casimir's brother Sigismund I the Old petitioned the pope to canonize Casimir. After repeated requests, in November 1517, Pope Leo X appointed a three-bishop commission and later sent his legate Zacharias Ferreri to investigate. He arrived at Vilnius in September 1520 and completed his work in about two months. His findings, the first short hagiography of Casimir, was published in 1521 in Kraków as Vita Beati Casimiri Confessoris. The canonization was all but certain but Pope Leo X died in December 1521. Research of Zenonas Ivinskis and Paulius Rabikauskas showed that there is no documentary proof that he issued a papal bull canonizing Casimir but many important documents were lost during the Sack of Rome (1527). The Protestant Reformation attacked the cult of saints and there were no new canonizations between 1523 and 1588. However, Casimir was included in the first Roman Martyrology, published in 1583.

The cause of Casimir's cult was taken up by the new Bishop of Vilnius Benedykt Woyna (appointed in 1600). He sent canon Gregorius Swiecicki to Rome with a letter from King Sigismund III Vasa requesting to add the feast of Casimir to the Roman Breviary and Roman Missal. The Sacred Congregation of Rites refused the request but on 7 November 1602 Pope Clement VIII issued a papal brief Quae ad sanctorum which authorized his feast sub duplici ritu on 4 March but only in Poland and Lithuania. The brief also mentioned that Casimir was added to the ranks of saints by Pope Leo X. In the absence of any earlier known papal document explicitly mentioning Casimir as saint, the brief is often cited as Casimir's canonization. Swiecicki returned to Vilnius with the papal brief and red velvet labarum with the image of Saint Casimir. The city organized a large three-day festival on 10–12 May 1604 to properly accept the papal flag. On the third day, the cornerstone was laid for the new Church of St. Casimir. The coffin of Casimir was taken out of the crypt and elevated to the altar. Swiecicki testified that when the coffin was opened in August 1604 a wonderful smell filed the cathedral for three days.

In 1607 and 1613, Bishop Woyna declared Casimir patron saint of Lithuania (Patronus principalis Lithuaniae). The issue of a universal Casimir's feast was not forgotten and in 1620 Bishop Eustachy Wołłowicz petitioned Pope Paul V to add Casimir to the Roman Breviary and Roman Missal. This time the Sacred Congregation of Rites granted the request in March 1621 and added his feast sub ritu semiduplici. In March 1636, Pope Urban VIII allowed the celebration of the feast of Casimir with an octave (duplex cum octava) in the Diocese of Vilnius and the Grand Duchy of Lithuania. That is equivalent of proclaiming Casimir as the patron saint of Lithuania. On 28 September 1652, Pope Innocent X allowed a fest of transfer of relics of Casimir on a Sunday following the Assumption of Mary. On 11 June 1948, when many Lithuanians were displaced war refugees, Pope Pius XII named Casimir the special patron of Lithuanian youth.

Iconography
Saint Casimir's painting in Vilnius Cathedral is considered to be miraculous. The painting, probably completed around 1520, depicts the saint with two right hands. According to a legend, the painter attempted to redraw the hand in a different place and paint over the old hand, but the old hand miraculously reappeared. More conventional explanations claim that three-handed Casimir was the original intent of the painter to emphasize the exceptional generosity of Casimir ("But when you give to someone in need, don't let your left hand know what your right hand is doing." Matthew 6:3) or that the old hand bled through a coat of new paint (similar to a palimpsest). Around 1636 the painting was covered in gilded silver clothing (riza).

Casimir's iconography usually follows the three-handed painting. He is usually depicted as a young man in long red robe lined with stoat fur. Sometimes he wears a red cap of the Grand Duke of Lithuania, but other times, to emphasize his devotion to spiritual life, the cap is placed near Casimir. Almost always he holds a lily, a symbol of virginity, innocence, and purity. He might also hold a cross, a rosary, or a book with words from Omni die dic Mariae (Daily, Daily Sing to Mary). The towns of Kvėdarna and Nemunaitis in Lithuania have Saint Casimir depicted on their coat of arms.

Physical remains and relics
Casimir was buried in the crypt under the Royal Chapel of Vilnius Cathedral (present-day Wołłowicz Chapel to the left from the main entrance), constructed by his father in the Gothic style in 1474. In 1604, the coffin was elevated from the crypt to the altar and in 1636 moved to the dedicated Chapel of Saint Casimir. The present-day sarcophagus was made in 1747 under the last will of Bishop of Warmia Christopher Johan Szembek (1680–1740). It is made of linden wood and covered with silver plates; its corners are decorated with gilded eagles. The sarcophagus was removed from the cathedral on three occasions. In 1655, before the Battle of Vilnius during the Deluge, the relics were removed most likely by Jerzy Białłozor and hid by Cyprian Paweł Brzostowski and later by the Sapiehas in the Ruzhany Palace. They were returned to the devastated chapel in 1663. The relics were removed for a short time in 1702 during the Battle of Vilnius of the Great Northern War. In October 1952, the relics were quietly moved to the Church of St. Peter and St. Paul by the order of Soviet authorities. The cathedral was turned into an art gallery. The relics returned to their place in 1989 when the cathedral was reconsecrated.

After the rediscovery of the Catacombs of Rome in 1578, the cult of relics spread throughout Europe (see also catacomb saints) and the trend did not skip Casimir. The coffin of Casimir was opened in early 1602 and in August 1604. At the time, canon Gregorius Swiecicki testified that despite humidity the body was intact. But in 1667 there were only bones left; they were inventoried and placed into six cloth bags. Surviving written records indicate that the coffin was opened in 1664, 1667, 1677, 1690, 1736, 1838, 1878 (twice), and 1922. There are several recorded instances when Casimir's relics were gifted to prominent figures and societies: to musicians' confraternity at San Giorgio Maggiore, Naples in early 1650s, to King John III Sobieski and Cosimo III de' Medici, Grand Duke of Tuscany in October 1677, to the Sodality of Our Lady of the Jesuit academy in Mechelen and the Order of Malta in October 1690, to Queen Maria Josepha of Austria in February 1736, to Cistercian abbot Sztárek Lajos of in 1860. Many more relics of Casimir can be found in local churches. In particular, in 1838, two teeth and ten unspecified bones were removed from the coffin; the bones were cut into small pieces and distributed among various churches. In 1922, the bones were wrapped in a new cloth and the old cloth was distributed as a relic. It was the last time that a relic of Casimir (one tooth for the Church of St. Casimir) was taken.

Dedications
In his 1970 monograph priest Florijonas Neviera (Florian Niewiero, 1896–1976) counted churches named after Casimir. He found 12 churches in Lithuania (as of 1940), 48 churches and 5 chapels in Poland, 23 Lithuanian and 36 Polish churches in United States (as of 1964), five churches in Canada (Montreal, Winnipeg, Toronto, Portneuf, and Ripon), two churches in United Kingdom (London and Manchester), and two churches in Belarus (Vselyub and Lepiel). The women's congregation Sisters of Saint Casimir was established in 1908 by Maria Kaupas and is active in the United States. In 1945, the College of Saint Casimir was established in Rome to educate Lithuanian priests who fled west after World War II.

While the devotion to Casimir is most prevalent in Lithuania and Poland and their diaspora communities, his cult can be found in other countries as well. In the 17th century, at least two societies of Saint Casimir were active in Mechelen and Antwerp (now Belgium). In the 17th century, Casimir's cult also spread in Italy, particularly Florence, Palermo, Naples; his cult in Rome was more associated with Polish dignitaries and émigrés. Musical performances were organized in Rome in 1675 (words by Sebastiano Lazzarini, music by Francesco Beretta, performed at Santo Spirito in Sassia) and in 1678 (words by Ottavio Santacroce, music probably by Giovanni Bicilli, performed at Santa Maria in Vallicella on the occasion of the visit by Michał Kazimierz Radziwiłł and his wife Katarzyna Sobieska), and in Florence in 1706 (words possibly by Cardinal Pietro Ottoboni, music by Alessandro Scarlatti). In Palermo, Pietro Novelli was commissioned painting Coronation of Saint Casimir (l’Incoronazione di s. Casimiro) for the altar of (now held at the Galleria Regionale della Sicilia).

The settlements of Saint-Casimir in Canada (founded 1836) and San Casimiro in Venezuela (founded 1785) are named after him. Sculptures of Casimir, among other canonized royals, can be found in San Ferdinando, Livorno, Italy and Metropolitan Cathedral, Mexico City. Stained glass windows with Casimir can be found at the Cathedral Basilica of St. Joseph in San Jose, California, and at the Church of St. Peter in Chevaigné, France. Since 1846, there is a nursing home in Paris named Maison Saint-Casimir. It was created by the Polish community of France and is run since its opening by Polish nuns Daughters of Charity of Saint Vincent de Paul.

title: Saint Casimir


text:
Spectrum of a ring

In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted by $$\operatorname{Spec}{R}$$; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings $$\mathcal{O}$$.

Zariski topology
For any ideal I of R, define $$V_I$$ to be the set of prime ideals containing I. We can put a topology on $$\operatorname{Spec}(R)$$ by defining the collection of closed sets to be
 * $$\{ V_I \colon I \text{ is an ideal of } R \}.$$

This topology is called the Zariski topology.

A basis for the Zariski topology can be constructed as follows. For f ∈ R, define Df to be the set of prime ideals of R not containing f. Then each Df is an open subset of $$\operatorname{Spec}(R)$$, and $$\{D_f:f\in R\}$$ is a basis for the Zariski topology.

$$\operatorname{Spec}(R)$$ is a compact space, but almost never Hausdorff: in fact, the maximal ideals in R are precisely the closed points in this topology. By the same reasoning, $$\operatorname{Spec}(R)$$ is not, in general, a T1 space. However, $$\operatorname{Spec}(R)$$ is always a Kolmogorov space (satisfies the T0 axiom); it is also a spectral space.

Sheaves and schemes
Given the space $$X = \operatorname{Spec}(R)$$ with the Zariski topology, the structure sheaf $$\mathcal O_X$$ is defined on the distinguished open subsets $$D_f$$ by setting $$\Gamma(D_f,\mathcal O_X) = R_f,$$ the localization of $R$ by the powers of $f$. It can be shown that this defines a B-sheaf and therefore that it defines a sheaf. In more detail, the distinguished open subsets are a basis of the Zariski topology, so for an arbitrary open set $U$, written as the union of $U=\bigcup_{i\in I}D_{f_i}$, we set $\Gamma(U,\mathcal O_X)=\varprojlim_{i\in I}R_{f_i},$ where $$\varprojlim$$ denotes the inverse limit with respect to the natural ring homomorphisms $$R_f\to R_{fg}.$$ One may check that this presheaf is a sheaf, so $$\operatorname{Spec}(R)$$ is a ringed space. Any ringed space isomorphic to one of this form is called an affine scheme. General schemes are obtained by gluing affine schemes together.

Similarly, for a module $M$ over the ring $R$, we may define a sheaf $$\tilde{M}$$ on $$\operatorname{Spec}(R)$$. On the distinguished open subsets set $$\Gamma(D_f,\tilde M)=M_f,$$ using the localization of a module. As above, this construction extends to a presheaf on all open subsets of $$\operatorname{Spec}(R)$$ and satisfies the gluing axiom. A sheaf of this form is called a quasicoherent sheaf.

If $P$ is a point in $$\operatorname{Spec}(R)$$, that is, a prime ideal, then the stalk of the structure sheaf at $P$ equals the localization of $R$ at the ideal $P$, and this is a local ring. Consequently, $$\operatorname{Spec}(R)$$ is a locally ringed space.

If $R$ is an integral domain, with field of fractions $K$, then we can describe the ring $$\Gamma(U, \mathcal O_X)$$ more concretely as follows. We say that an element $f$ in $K$ is regular at a point $P$ in $X$ if it can be represented as a fraction $f = a/b$ with $b$ not in $P$. Note that this agrees with the notion of a regular function in algebraic geometry. Using this definition, we can describe $$\Gamma(U, \mathcal O_X)$$ as precisely the set of elements of $K$ that are regular at every point $P$ in $U$.

Functorial perspective
It is useful to use the language of category theory and observe that $$\operatorname{Spec}$$ is a functor. Every ring homomorphism $$f: R \to S$$ induces a continuous map $$\operatorname{Spec}(f): \operatorname{Spec}(S) \to \operatorname{Spec}(R)$$ (since the preimage of any prime ideal in $$S$$ is a prime ideal in $$R$$). In this way, $$\operatorname{Spec}$$ can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces. Moreover, for every prime $$\mathfrak{p}$$ the homomorphism $$f$$ descends to homomorphisms
 * $$\mathcal{O}_{f^{-1}(\mathfrak{p})} \to \mathcal{O}_\mathfrak{p} $$

of local rings. Thus $$\operatorname{Spec}$$ even defines a contravariant functor from the category of commutative rings to the category of locally ringed spaces. In fact it is the universal such functor, and hence can be used to define the functor $$\operatorname{Spec}$$ up to natural isomorphism.

The functor $$\operatorname{Spec}$$ yields a contravariant equivalence between the category of commutative rings and the category of affine schemes; each of these categories is often thought of as the opposite category of the other.

Motivation from algebraic geometry
Following on from the example, in algebraic geometry one studies algebraic sets, i.e. subsets of Kn (where K is an algebraically closed field) that are defined as the common zeros of a set of polynomials in n variables. If A is such an algebraic set, one considers the commutative ring R of all polynomial functions A → K. The maximal ideals of R correspond to the points of A (because K is algebraically closed), and the prime ideals of R correspond to the subvarieties of A (an algebraic set is called irreducible or a variety if it cannot be written as the union of two proper algebraic subsets).

The spectrum of R therefore consists of the points of A together with elements for all subvarieties of A. The points of A are closed in the spectrum, while the elements corresponding to subvarieties have a closure consisting of all their points and subvarieties. If one only considers the points of A, i.e. the maximal ideals in R, then the Zariski topology defined above coincides with the Zariski topology defined on algebraic sets (which has precisely the algebraic subsets as closed sets). Specifically, the maximal ideals in R, i.e. $$\operatorname{MaxSpec}(R)$$, together with the Zariski topology, is homeomorphic to A also with the Zariski topology.

One can thus view the topological space $$\operatorname{Spec}(R)$$ as an "enrichment" of the topological space A (with Zariski topology): for every subvariety of A, one additional non-closed point has been introduced, and this point "keeps track" of the corresponding subvariety. One thinks of this point as the generic point for the subvariety. Furthermore, the sheaf on $$\operatorname{Spec}(R)$$ and the sheaf of polynomial functions on A are essentially identical. By studying spectra of polynomial rings instead of algebraic sets with Zariski topology, one can generalize the concepts of algebraic geometry to non-algebraically closed fields and beyond, eventually arriving at the language of schemes.

Examples

 * The spectrum of integers: The affine scheme $$\operatorname{Spec}(\mathbb{Z})$$ is the final object in the category of affine schemes since $$\mathbb{Z}$$ is the initial object in the category of commutative rings.
 * The scheme-theoretic analogue of $$\mathbb{C}^n$$: The affine scheme $$\mathbb{A}^n_\mathbb{C} = \operatorname{Spec}(\mathbb{C}[x_1,\ldots, x_n])$$. From the functor of points perspective, a point $$(\alpha_1,\ldots,\alpha_n) \in \mathbb{C}^n$$ can be identified with the evaluation morphism $$\mathbb{C}[x_1,\ldots, x_n] \xrightarrow{ev_{(\alpha_1,\ldots, \alpha_n)}} \mathbb{C}$$. This fundamental observation allows us to give meaning to other affine schemes.
 * The cross: $$\operatorname{Spec}(\mathbb{C}[x,y]/(xy))$$ looks topologically like the transverse intersection of two complex planes at a point, although typically this is depicted as a $$+$$, since the only well defined morphisms to $$\mathbb{C}$$ are the evaluation morphisms associated with the points $$\{(\alpha_1,0), (0,\alpha_2) : \alpha_1,\alpha_2 \in \mathbb{C} \}$$.
 * The prime spectrum of a Boolean ring (e.g., a power set ring) is a compact totally disconnected Hausdorff space (that is, a Stone space).
 * (M. Hochster) A topological space is homeomorphic to the prime spectrum of a commutative ring (i.e., a spectral space) if and only if it is compact, quasi-separated and sober.

Non-affine examples
Here are some examples of schemes that are not affine schemes. They are constructed from gluing affine schemes together.
 * The projective $n$-space $$\mathbb{P}^n_k = \operatorname{Proj}k[x_0,\ldots, x_n]$$ over a field $$k$$. This can be easily generalized to any base ring, see Proj construction (in fact, we can define projective space for any base scheme). The projective $$n$$-space for $$ n \geq 1 $$ is not affine as the global section of $$\mathbb{P}^n_k$$ is $$k$$.
 * Affine plane minus the origin. Inside $$\mathbb{A}^2_k = \operatorname{Spec} k[x,y]$$ are distinguished open affine subschemes $$ D_x, D_y $$. Their union $$ D_x \cup D_y = U$$ is the affine plane with the origin taken out. The global sections of $$U$$ are pairs of polynomials on $$D_x,D_y $$ that restrict to the same polynomial on $$ D_{xy} $$, which can be shown to be $$ k[x,y] $$, the global section of $$\mathbb{A}^2_k $$. $$U$$ is not affine as $$ V_{(x)} \cap V_{(y)} = \varnothing $$ in $$ U$$.

Non-Zariski topologies on a prime spectrum
Some authors (notably M. Hochster) consider topologies on prime spectra other than the Zariski topology.

First, there is the notion of constructible topology: given a ring A, the subsets of $$\operatorname{Spec}(A)$$ of the form $$\varphi^*(\operatorname{Spec} B), \varphi: A \to B$$ satisfy the axioms for closed sets in a topological space. This topology on $$\operatorname{Spec}(A)$$ is called the constructible topology.

In, Hochster considers what he calls the patch topology on a prime spectrum. By definition, the patch topology is the smallest topology in which the sets of the forms $$V(I)$$ and $$\operatorname{Spec}(A) - V(f)$$ are closed.

Global or relative Spec
There is a relative version of the functor $$\operatorname{Spec}$$ called global $$\operatorname{Spec}$$, or relative $$\operatorname{Spec}$$. If $$S$$ is a scheme, then relative $$\operatorname{Spec}$$ is denoted by $$\underline{\operatorname{Spec}}_S$$ or $$\mathbf{Spec}_S$$. If $$S$$ is clear from the context, then relative Spec may be denoted by $$\underline{\operatorname{Spec}}$$ or $$\mathbf{Spec}$$. For a scheme $$S$$ and a quasi-coherent sheaf of $\mathcal{O}_S$-algebras $$\mathcal{A}$$, there is a scheme $$\underline{\operatorname{Spec}}_S(\mathcal{A})$$ and a morphism $$f : \underline{\operatorname{Spec}}_S(\mathcal{A}) \to S$$ such that for every open affine $$U \subseteq S$$, there is an isomorphism $$f^{-1}(U) \cong \operatorname{Spec}(\mathcal{A}(U))$$, and such that for open affines $$V \subseteq U$$, the inclusion $$f^{-1}(V) \to f^{-1}(U)$$ is induced by the restriction map $$\mathcal{A}(U) \to \mathcal{A}(V)$$. That is, as ring homomorphisms induce opposite maps of spectra, the restriction maps of a sheaf of algebras induce the inclusion maps of the spectra that make up the Spec of the sheaf.

Global Spec has a universal property similar to the universal property for ordinary Spec. More precisely, just as Spec and the global section functor are contravariant right adjoints between the category of commutative rings and schemes, global Spec and the direct image functor for the structure map are contravariant right adjoints between the category of commutative $$\mathcal{O}_S$$-algebras and schemes over $$S$$. In formulas,
 * $$\operatorname{Hom}_{\mathcal{O}_S\text{-alg}}(\mathcal{A}, \pi_*\mathcal{O}_X) \cong \operatorname{Hom}_{\text{Sch}/S}(X, \mathbf{Spec}(\mathcal{A})),$$

where $$\pi \colon X \to S$$ is a morphism of schemes.

Example of a relative Spec
The relative spec is the correct tool for parameterizing the family of lines through the origin of $$\mathbb{A}^2_\mathbb{C}$$ over $$X = \mathbb{P}^1_{a,b}.$$ Consider the sheaf of algebras $$\mathcal{A} = \mathcal{O}_X[x,y],$$ and let $$\mathcal{I} = (ay-bx)$$ be a sheaf of ideals of $$\mathcal{A}.$$ Then the relative spec $$\underline{\operatorname{Spec}}_X(\mathcal{A}/\mathcal{I}) \to \mathbb{P}^1_{a,b}$$ parameterizes the desired family. In fact, the fiber over $$[\alpha:\beta]$$ is the line through the origin of $$\mathbb{A}^2$$ containing the point $$(\alpha,\beta).$$ Assuming $$\alpha \neq 0,$$ the fiber can be computed by looking at the composition of pullback diagrams
 * $$\begin{matrix}

\operatorname{Spec}\left( \frac{\mathbb{C}[x,y]}{\left(y-\frac{\beta}{\alpha}x\right)} \right) & \to & \operatorname{Spec}\left( \frac{\mathbb{C}\left[\frac{b}{a}\right] [x,y]}{\left(y-\frac{b}{a}x\right)} \right) & \to & \underline{\operatorname{Spec}}_X\left( \frac{\mathcal{O}_X[x,y]}{\left(ay-bx\right)} \right)\\ \downarrow & & \downarrow & & \downarrow \\ \operatorname{Spec}(\mathbb{C})& \to & \operatorname{Spec}\left(\mathbb{C}\left[\frac{b}{a}\right]\right)=U_a & \to & \mathbb{P}^1_{a,b} \end{matrix}$$ where the composition of the bottom arrows
 * $$\operatorname{Spec}(\mathbb{C})\xrightarrow{[\alpha:\beta]} \mathbb{P}^1_{a,b}$$

gives the line containing the point $$(\alpha,\beta)$$ and the origin. This example can be generalized to parameterize the family of lines through the origin of $$\mathbb{A}^{n+1}_\mathbb{C}$$ over $$X = \mathbb{P}^n_{a_0,...,a_n}$$ by letting $$\mathcal{A} = \mathcal{O}_X[x_0,...,x_n]$$ and $$\mathcal{I} = \left( 2\times 2 \text{ minors of } \begin{pmatrix}a_0 & \cdots & a_n \\ x_0 & \cdots & x_n\end{pmatrix} \right).$$

Representation theory perspective
From the perspective of representation theory, a prime ideal I corresponds to a module R/I, and the spectrum of a ring corresponds to irreducible cyclic representations of R, while more general subvarieties correspond to possibly reducible representations that need not be cyclic. Recall that abstractly, the representation theory of a group is the study of modules over its group algebra.

The connection to representation theory is clearer if one considers the polynomial ring $$R=K[x_1,\dots,x_n]$$ or, without a basis, $$R=K[V].$$ As the latter formulation makes clear, a polynomial ring is the group algebra over a vector space, and writing in terms of $$x_i$$ corresponds to choosing a basis for the vector space. Then an ideal I, or equivalently a module $$R/I,$$ is a cyclic representation of R (cyclic meaning generated by 1 element as an R-module; this generalizes 1-dimensional representations).

In the case that the field is algebraically closed (say, the complex numbers), every maximal ideal corresponds to a point in n-space, by the Nullstellensatz (the maximal ideal generated by $$(x_1-a_1), (x_2-a_2),\ldots,(x_n-a_n)$$ corresponds to the point $$(a_1,\ldots,a_n)$$). These representations of $$K[V]$$ are then parametrized by the dual space $$V^*,$$ the covector being given by sending each $$x_i$$ to the corresponding $$a_i$$. Thus a representation of $$K^n$$ (K-linear maps $$K^n \to K$$) is given by a set of n numbers, or equivalently a covector $$K^n \to K.$$

Thus, points in n-space, thought of as the max spec of $$R=K[x_1,\dots,x_n],$$ correspond precisely to 1-dimensional representations of R, while finite sets of points correspond to finite-dimensional representations (which are reducible, corresponding geometrically to being a union, and algebraically to not being a prime ideal). The non-maximal ideals then correspond to infinite-dimensional representations.

Functional analysis perspective
The term "spectrum" comes from the use in operator theory. Given a linear operator T on a finite-dimensional vector space V, one can consider the vector space with operator as a module over the polynomial ring in one variable R = K[T], as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of K[T] (as a ring) equals the spectrum of T (as an operator).

Further, the geometric structure of the spectrum of the ring (equivalently, the algebraic structure of the module) captures the behavior of the spectrum of the operator, such as algebraic multiplicity and geometric multiplicity. For instance, for the 2×2 identity matrix has corresponding module:
 * $$K[T]/(T-1) \oplus K[T]/(T-1)$$

the 2×2 zero matrix has module
 * $$K[T]/(T-0) \oplus K[T]/(T-0),$$

showing geometric multiplicity 2 for the zero eigenvalue, while a non-trivial 2×2 nilpotent matrix has module
 * $$K[T]/T^2,$$

showing algebraic multiplicity 2 but geometric multiplicity 1.

In more detail:
 * the eigenvalues (with geometric multiplicity) of the operator correspond to the (reduced) points of the variety, with multiplicity;
 * the primary decomposition of the module corresponds to the unreduced points of the variety;
 * a diagonalizable (semisimple) operator corresponds to a reduced variety;
 * a cyclic module (one generator) corresponds to the operator having a cyclic vector (a vector whose orbit under T spans the space);
 * the last invariant factor of the module equals the minimal polynomial of the operator, and the product of the invariant factors equals the characteristic polynomial.

Generalizations
The spectrum can be generalized from rings to C*-algebras in operator theory, yielding the notion of the spectrum of a C*-algebra. Notably, for a Hausdorff space, the algebra of scalars (the bounded continuous functions on the space, being analogous to regular functions) is a commutative C*-algebra, with the space being recovered as a topological space from $$\operatorname{MaxSpec}$$ of the algebra of scalars, indeed functorially so; this is the content of the Banach–Stone theorem. Indeed, any commutative C*-algebra can be realized as the algebra of scalars of a Hausdorff space in this way, yielding the same correspondence as between a ring and its spectrum. Generalizing to non-commutative C*-algebras yields noncommutative topology.

title: Spectrum of a ring