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For a full treatment of the foundations it's best to consult Part I of the book Representations of Algebraic Groups by J.C. Jantzen (2nd ed., AMS, 2003) even though it's not easily available online. Rational (or Hochschild) cohomology has been well developed, including the broader scheme framework (Demazure-Gabriel book and Jantzen). What CPS and van der Kallen do in their important paper is to relate indirectly the algebraic group cohomology with finite group cohomology for related finite groups of Lie type. This theme has been much further developed in many later papers, but is subtle.

For the algebraic groups themselves, this kind of cohomology theory has also been studied in many papers; but relating it to abstract group cohomology for the algebraic (rather than finite) groups such as the special linear group is not at all obvious.

By the way, the Inventiones paper and some others by CPS et al. are freely available online through http://gdz.sub.uni-goettingen.de (just do a quick search for Parshall).

ADDED: Maybe I can answer the original question in more detail and respond to Ralph's further question. For an affine group scheme $G$ over a field $k$, rational (Hochschild) cohomology is defined as usual in terms of derived functors of the fixed point functor. But everything is done in the category of rational $G$-modules; for an affine algebraic group over an algebraically closed field and finite dimensional modules this means that representing matrices have coordinate functions in $k[G]$.

Hochschild realized that for groups with added structure, one must use injective resolutions (there are usually not enough projectives). in any case, rational cohomology tends to diverge a lot from the usual group cohomology. In characteristic 0, you are essentially getting Lie algebra cohomology. Studying rational $G$-modules is equivalent to studying modules for the Hopf dual of $k[G]$ (hyperalgebra, or algebra of distributions). So the answer to Ralph's question is yes: the notions of cohomology agree.

Jantzen's main focus is on prime characteristic and reductive algebraic groups, where powers of the Frobenius map yield kernels which are finite group schemes. Roughly speaking, injectives for $G$ are direct limits of injectives for finite dimensional hyperalgebras, starting with the restricted enveloping algebra of the Lie algebra of $G$ (whose cohomology usually differs from the ordinary Lie algebra cohomology). Relating rational cohomology of $G$ to ordinary cohomology of finite subgroups gets even more subtle, as discussed above. By now there is a lot of literature on these matters but many unanswered questions.

For a full treatment of the foundations it's best to consult Part I of the book Representations of Algebraic Groups by J.C. Jantzen (2nd ed., AMS, 2003) even though it's not easily available online. Rational (or Hochschild) cohomology has been well developed, including the broader scheme framework (Demazure-Gabriel book and Jantzen). What CPS and van der Kallen do in their important paper here is to relate indirectly the algebraic group cohomology with finite group cohomology for related finite groups of Lie type. This theme has been much further developed in many later papers, but is subtle.

For the algebraic groups themselves, this kind of cohomology theory has also been studied in many papers; but relating it to abstract group cohomology for the algebraic (rather than finite) groups such as the special linear group is not at all obvious.

By the way, the Inventiones paper and some others by CPS et al. are freely available online through http://gdz.sub.uni-goettingen.de (just do a quick search for Parshall).

ADDED: Maybe I can answer the original question in more detail and respond to Ralph's further question. For an affine group scheme $G$ over a field $k$, rational (Hochschild) cohomology is defined as usual in terms of derived functors of the fixed point functor. But everything is done in the category of rational $G$-modules; for an affine algebraic group over an algebraically closed field and finite dimensional modules this means that representing matrices have coordinate functions in $k[G]$.

Hochschild realized that for groups with added structure, one must use injective resolutions (there are usually not enough projectives). in any case, rational cohomology tends to diverge a lot from the usual group cohomology. In characteristic 0, you are essentially getting Lie algebra cohomology. Studying rational $G$-modules is equivalent to studying modules for the Hopf dual of $k[G]$ (hyperalgebra, or algebra of distributions). So the answer to Ralph's question is yes: the notions of cohomology agree.

Jantzen's main focus is on prime characteristic and reductive algebraic groups, where powers of the Frobenius map yield kernels which are finite group schemes. Roughly speaking, injectives for $G$ are direct limits of injectives for finite dimensional hyperalgebras, starting with the restricted enveloping algebra of the Lie algebra of $G$ (whose cohomology usually differs from the ordinary Lie algebra cohomology). Relating rational cohomology of $G$ to ordinary cohomology of finite subgroups gets even more subtle, as discussed above. By now there is a lot of literature on these matters but many unanswered questions.

For a full treatment of the foundations it's best to consult Part I of the book Representations of Algebraic Groups by J.C. Jantzen (2nd ed., AMS, 2003) even though it's not easily available online. Rational (or Hochschild) cohomology has been well developed, including the broader scheme framework (Demazure-Gabriel book and Jantzen). What CPS and van der Kallen do in their important paper is to relate indirectly the algebraic group cohomology with finite group cohomology for related finite groups of Lie type. This theme has been much further developed in many later papers, but is subtle.

For the algebraic groups themselves, this kind of cohomology theory has also been studied in many papers; but relating it to abstract group cohomology for the algebraic (rather than finite) groups such as the special linear group is not at all obvious.

By the way, the Inventiones paper and others by CPS et al. are freely available online through http://gdz.sub.uni-goettingen.de (just do a quick search for Parshall).

For a full treatment of the foundations it's best to consult Part I of the book Representations of Algebraic Groups by J.C. Jantzen (2nd ed., AMS, 2003) even though it's not easily available online. Rational (or Hochschild) cohomology has been well developed, including the broader scheme framework (Demazure-Gabriel book and Jantzen). What CPS and van der Kallen do in their important paper is to relate indirectly the algebraic group cohomology with finite group cohomology for related finite groups of Lie type. This theme has been much further developed in many later papers, but is subtle.

For the algebraic groups themselves, this kind of cohomology theory has also been studied in many papers; but relating it to abstract group cohomology for the algebraic (rather than finite) groups such as the special linear group is not at all obvious.

By the way, the Inventiones paper and some others by CPS et al. are freely available online through http://gdz.sub.uni-goettingen.de (just do a quick search for Parshall).

ADDED: Maybe I can answer the original question in more detail and respond to Ralph's further question. For an affine group scheme $G$ over a field $k$, rational (Hochschild) cohomology is defined as usual in terms of derived functors of the fixed point functor. But everything is done in the category of rational $G$-modules; for an affine algebraic group over an algebraically closed field and finite dimensional modules this means that representing matrices have coordinate functions in $k[G]$.

Hochschild realized that for groups with added structure, one must use injective resolutions (there are usually not enough projectives). in any case, rational cohomology tends to diverge a lot from the usual group cohomology. In characteristic 0, you are essentially getting Lie algebra cohomology. Studying rational $G$-modules is equivalent to studying modules for the Hopf dual of $k[G]$ (hyperalgebra, or algebra of distributions). So the answer to Ralph's question is yes: the notions of cohomology agree.

Jantzen's main focus is on prime characteristic and reductive algebraic groups, where powers of the Frobenius map yield kernels which are finite group schemes. Roughly speaking, injectives for $G$ are direct limits of injectives for finite dimensional hyperalgebras, starting with the restricted enveloping algebra of the Lie algebra of $G$ (whose cohomology usually differs from the ordinary Lie algebra cohomology). Relating rational cohomology of $G$ to ordinary cohomology of finite subgroups gets even more subtle, as discussed above. By now there is a lot of literature on these matters but many unanswered questions.

For a full treatment of the foundations it's best to consult Part I of the book Representations of Algebraic Groups by J.C. Jantzen (2nd ed., AMS, 2003) even though it's not easily available online. Rational (or Hochschild) cohomology has been well developed, including the broader scheme framework (Demazure-Gabriel book and Jantzen). What CPS and van der Kallen do in their important paper is to relate indirectly the algebraic group cohomology with finite group cohomology for related finite groups of Lie type. This theme has been much further developed in many later papers, but is subtle.

For the algebraic groups themselves, this kind of cohomology theory has also been studied in many papers; but relating it to abstract group cohomology for the algebraic (rather than finite) groups such as the special linear group is not at all obvious.

For a full treatment of the foundations it's best to consult Part I of the book Representations of Algebraic Groups by J.C. Jantzen (2nd ed., AMS, 2003) even though it's not easily available online. Rational (or Hochschild) cohomology has been well developed, including the broader scheme framework (Demazure-Gabriel book and Jantzen). What CPS and van der Kallen do in their important paper is to relate indirectly the algebraic group cohomology with finite group cohomology for related finite groups of Lie type. This theme has been much further developed in many later papers, but is subtle.

For the algebraic groups themselves, this kind of cohomology theory has also been studied in many papers; but relating it to abstract group cohomology for the algebraic (rather than finite) groups such as the special linear group is not at all obvious.

By the way, the Inventiones paper and others by CPS et al. are freely available online through http://gdz.sub.uni-goettingen.de (just do a quick search for Parshall).