Algebraic Groups Connected and Reduced?
Lastly I was a bit surprised about a statement regarding the difference of group schemes to algebraic groups at wiki
https://en.wikipedia.org/wiki/Group_scheme
Let me quote it: "... Group schemes arise
naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme
structure, but group schemes are not necessarily connected, smooth, or defined over a field...."
My question is simply why is this provides a distinguishing criterion? Are algebraic groups allways connected? Up to now I never heard that without some extra assumtions (e.g. irreducibility) all algebraic groups are connected. Or do I oversee here a detail?
Futher question: What about beeing reduced? Should algebraic groups always be reduced as varieties? If yes, where does this in their definition flows in?
group-theory algebraic-geometry algebraic-groups
asked Dec 9 at 0:04
KarlPeter
5881314
add a comment |
Lastly I was a bit surprised about a statement regarding the difference of group schemes to algebraic groups at wiki
https://en.wikipedia.org/wiki/Group_scheme
Let me quote it: "... Group schemes arise
naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme
structure, but group schemes are not necessarily connected, smooth, or defined over a field...."
My question is simply why is this provides a distinguishing criterion? Are algebraic groups allways connected? Up to now I never heard that without some extra assumtions (e.g. irreducibility) all algebraic groups are connected. Or do I oversee here a detail?
Futher question: What about beeing reduced? Should algebraic groups always be reduced as varieties? If yes, where does this in their definition flows in?
group-theory algebraic-geometry algebraic-groups
asked Dec 9 at 0:04
KarlPeter
5881314
2
You're going to find different definitions in every textbook. Algebraic groups are usually defined as groups that are also varieties, and it's common for a variety to always be irreducible (thus connected) and reduced. I'm a little surprised to see "smooth" on the list. I think "not always defined over a field" is the most important distinction listed, though I'd say "not always finite type over a field" instead, since even group schemes over fields can be quite far from being algebraic groups.
– Slade
Dec 9 at 0:11
...and a variety is for you a scheme over $k$ such that it is integral (so irred and reduced) and the structure morphism $X to Spec(k)$ is separated and of finite typ? Sorry, for this definition question, but variety is another candidate which is in defined in quite every ag book in a way far away from uniqueness...
– KarlPeter
Dec 9 at 0:25
My point was that almost everybody agrees "algebraic group" = "group object in category of algebraic varieties", but there's some room for disagreement when defining varieties. The definition you gave is pretty close to my preferred definition, though I might add "geometrically integral" (i.e. the base change to the algebraic closure is integral).
– Slade
Dec 9 at 1:25
add a comment |
Lastly I was a bit surprised about a statement regarding the difference of group schemes to algebraic groups at wiki
https://en.wikipedia.org/wiki/Group_scheme
Let me quote it: "... Group schemes arise
naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme
structure, but group schemes are not necessarily connected, smooth, or defined over a field...."
My question is simply why is this provides a distinguishing criterion? Are algebraic groups allways connected? Up to now I never heard that without some extra assumtions (e.g. irreducibility) all algebraic groups are connected. Or do I oversee here a detail?
Futher question: What about beeing reduced? Should algebraic groups always be reduced as varieties? If yes, where does this in their definition flows in?
group-theory algebraic-geometry algebraic-groups
asked Dec 9 at 0:04
KarlPeter
5881314
Lastly I was a bit surprised about a statement regarding the difference of group schemes to algebraic groups at wiki
https://en.wikipedia.org/wiki/Group_scheme
Let me quote it: "... Group schemes arise
naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme
structure, but group schemes are not necessarily connected, smooth, or defined over a field...."
My question is simply why is this provides a distinguishing criterion? Are algebraic groups allways connected? Up to now I never heard that without some extra assumtions (e.g. irreducibility) all algebraic groups are connected. Or do I oversee here a detail?
Futher question: What about beeing reduced? Should algebraic groups always be reduced as varieties? If yes, where does this in their definition flows in?
group-theory algebraic-geometry algebraic-groups
group-theory algebraic-geometry algebraic-groups
asked Dec 9 at 0:04
KarlPeter
5881314
asked Dec 9 at 0:04
KarlPeter
5881314
asked Dec 9 at 0:04
KarlPeter
5881314
asked Dec 9 at 0:04
KarlPeter
5881314
asked Dec 9 at 0:04
KarlPeter
5881314
5881314
2
You're going to find different definitions in every textbook. Algebraic groups are usually defined as groups that are also varieties, and it's common for a variety to always be irreducible (thus connected) and reduced. I'm a little surprised to see "smooth" on the list. I think "not always defined over a field" is the most important distinction listed, though I'd say "not always finite type over a field" instead, since even group schemes over fields can be quite far from being algebraic groups.
– Slade
Dec 9 at 0:11
...and a variety is for you a scheme over $k$ such that it is integral (so irred and reduced) and the structure morphism $X to Spec(k)$ is separated and of finite typ? Sorry, for this definition question, but variety is another candidate which is in defined in quite every ag book in a way far away from uniqueness...
– KarlPeter
Dec 9 at 0:25
My point was that almost everybody agrees "algebraic group" = "group object in category of algebraic varieties", but there's some room for disagreement when defining varieties. The definition you gave is pretty close to my preferred definition, though I might add "geometrically integral" (i.e. the base change to the algebraic closure is integral).
– Slade
Dec 9 at 1:25
add a comment |
2
You're going to find different definitions in every textbook. Algebraic groups are usually defined as groups that are also varieties, and it's common for a variety to always be irreducible (thus connected) and reduced. I'm a little surprised to see "smooth" on the list. I think "not always defined over a field" is the most important distinction listed, though I'd say "not always finite type over a field" instead, since even group schemes over fields can be quite far from being algebraic groups.
– Slade
Dec 9 at 0:11
...and a variety is for you a scheme over $k$ such that it is integral (so irred and reduced) and the structure morphism $X to Spec(k)$ is separated and of finite typ? Sorry, for this definition question, but variety is another candidate which is in defined in quite every ag book in a way far away from uniqueness...
– KarlPeter
Dec 9 at 0:25
My point was that almost everybody agrees "algebraic group" = "group object in category of algebraic varieties", but there's some room for disagreement when defining varieties. The definition you gave is pretty close to my preferred definition, though I might add "geometrically integral" (i.e. the base change to the algebraic closure is integral).
– Slade
Dec 9 at 1:25
2
2
You're going to find different definitions in every textbook. Algebraic groups are usually defined as groups that are also varieties, and it's common for a variety to always be irreducible (thus connected) and reduced. I'm a little surprised to see "smooth" on the list. I think "not always defined over a field" is the most important distinction listed, though I'd say "not always finite type over a field" instead, since even group schemes over fields can be quite far from being algebraic groups.
– Slade
Dec 9 at 0:11
You're going to find different definitions in every textbook. Algebraic groups are usually defined as groups that are also varieties, and it's common for a variety to always be irreducible (thus connected) and reduced. I'm a little surprised to see "smooth" on the list. I think "not always defined over a field" is the most important distinction listed, though I'd say "not always finite type over a field" instead, since even group schemes over fields can be quite far from being algebraic groups.
– Slade
Dec 9 at 0:11
...and a variety is for you a scheme over $k$ such that it is integral (so irred and reduced) and the structure morphism $X to Spec(k)$ is separated and of finite typ? Sorry, for this definition question, but variety is another candidate which is in defined in quite every ag book in a way far away from uniqueness...
– KarlPeter
Dec 9 at 0:25
...and a variety is for you a scheme over $k$ such that it is integral (so irred and reduced) and the structure morphism $X to Spec(k)$ is separated and of finite typ? Sorry, for this definition question, but variety is another candidate which is in defined in quite every ag book in a way far away from uniqueness...
– KarlPeter
Dec 9 at 0:25
My point was that almost everybody agrees "algebraic group" = "group object in category of algebraic varieties", but there's some room for disagreement when defining varieties. The definition you gave is pretty close to my preferred definition, though I might add "geometrically integral" (i.e. the base change to the algebraic closure is integral).
– Slade
Dec 9 at 1:25
My point was that almost everybody agrees "algebraic group" = "group object in category of algebraic varieties", but there's some room for disagreement when defining varieties. The definition you gave is pretty close to my preferred definition, though I might add "geometrically integral" (i.e. the base change to the algebraic closure is integral).
– Slade
Dec 9 at 1:25
add a comment |
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2
You're going to find different definitions in every textbook. Algebraic groups are usually defined as groups that are also varieties, and it's common for a variety to always be irreducible (thus connected) and reduced. I'm a little surprised to see "smooth" on the list. I think "not always defined over a field" is the most important distinction listed, though I'd say "not always finite type over a field" instead, since even group schemes over fields can be quite far from being algebraic groups.
– Slade
Dec 9 at 0:11
...and a variety is for you a scheme over $k$ such that it is integral (so irred and reduced) and the structure morphism $X to Spec(k)$ is separated and of finite typ? Sorry, for this definition question, but variety is another candidate which is in defined in quite every ag book in a way far away from uniqueness...
– KarlPeter
Dec 9 at 0:25
My point was that almost everybody agrees "algebraic group" = "group object in category of algebraic varieties", but there's some room for disagreement when defining varieties. The definition you gave is pretty close to my preferred definition, though I might add "geometrically integral" (i.e. the base change to the algebraic closure is integral).
– Slade
Dec 9 at 1:25