3 deleted 1 characters in body

In his paper "resolution of unbounded complexes" Compositio Math. (1988) N. Spaltenstein introduces the concept of K-projectives K-projective and K-injectives K-injective chain complexes. He proves (theorem C) that any unbounded chain complex $C.$ has a K-projective resolution or a K-injective resolution. For example if you work with chain complexes of modules over a commutative $R$. And if you take a chain complex $C.$ there exists a quasi-isomorphism from a K-projective resolution $F.$ of $C.$ into $C.$.

In his introduction N. Spalsenstein gives a very good example that motivates the introduction of such a notion (p. 124).

Let me just say that the point is that when you want to compute derived functors you do need to replace your chain complex by a "nicer" guy, this nicer guy should have nice chain homotopic properties. Because you want your computations to be chain homotopic invariant. Over a field all chain complexes are K-projective. But over a ring this is not the case even if you take a chain complexes of projective $R$-modules so you need an additional asumption assumption (see definition p.124).

If you know closed model categories, then a K-projective chain complex is a cofibrant object for a CMC structure on chain complexes of $R$-modules.

In his paper "resolution of unbounded complexes" Compositio Math. (1988) N. Spaltenstein introduces the concept of K-projectives and K-injectives chain complexes. He proves (theorem C) that any unbounded chain complex $C.$ has a K-projective resolution or a K-injective resolution. For example if you work with chain complexes of modules over a commutative $R$. And if you take a chain complex $C.$ there exists a quasi-isomorphism from a K-projective resolution $F.$ of $C.$ into $C.$.

In his introduction N. Spalsenstein gives a very good example that motivates the introduction of such a notion (p. 124).

Let me just say that the point is that when you want to compute derived functors you do need to replace your chain complex by a "nicer" guy, this nicer guy should have nice chain homotopic properties. Because you want your computations to be chain homotopic invariant. Over a field all chain complexes are K-projective. But over a ring this is not the case even if you take a chain complexes of projective $R$-modules so you need an additional asumption (see definition p.124).

If you know closed model categories, then a K-projective chain complex is a cofibrant object for a CMC structure on chain complexes of $R$-modules.

1

In his "resolution of unbounded complexes" Compositio Math. (1988) N. Spaltenstein introduces the concept of K-projectives and K-injectives chain complexes. He proves (theorem C) that any unbounded chain complex $C.$ has a K-projective resolution or a K-injective resolution. For example if you work with chain complexes of modules over a commutative $R$. And if you take a chain complex $C.$ there exists a quasi-isomorphism from a K-projective resolution $F.$ of $C.$ into $C.$.

In his introduction N. Spalsenstein gives a very good example that motivates the introduction of such a notion (p. 124).

Let me just say that the point is that when you want to compute derived functors you do need to replace your chain complex by a "nicer" guy, this nicer guy should have nice chain homotopic properties. Because you want your computations to be chain homotopic invariant. Over a field all chain complexes are K-projective. But over a ring this is not the case even if you take a chain complexes of projective $R$-modules so you need an additional asumption (see definition p.124).

If you know closed model categories a K-projective chain complex is a cofibrant object for a CMC structure on chain complexes of $R$-modules.