A classic example of a projective resolution is given by the Koszul complex of a regular sequence, which is a free resolution of the ideal generated by the sequence.

Let x < sub > 1 </ sub >, …, x < sub > d </ sub > be a system of parameters for R, let F < sub >•</ sub > be a free R-resolution of the residue field of R with F < sub > 0 </ sub > = R, and let K < sub >•</ sub > denote the Koszul complex of R with respect to x < sub > 1 </ sub >, …, x < sub > d </ sub >.

In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul ( see Lie algebra cohomology ).

This chain complex K < sub >•</ sub >( x ) is called the Koszul complex of R with respect to x.

Now, if x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub > are elements of R, the Koszul complex of R with respect to x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub >, usually denoted K < sub >•</ sub >( x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub >), is the tensor product in the category of R-complexes of the Koszul complexes defined above individually for each i.

The Koszul complex is a free chain complex.

For the case of two elements x and y, the Koszul complex can then be written down quite succinctly as

In the case that the elements x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub > form a regular sequence, the higher homology modules of the Koszul complex are all zero.

If k is a field and X < sub > 1 </ sub >, X < sub > 2 </ sub >, ..., X < sub > d </ sub > are indeterminates and R is the polynomial ring kX < sub > 2 </ sub >, ..., X < sub > d </ sub >, the Koszul complex K < sub >•</ sub >( X < sub > i </ sub >) on the X < sub > i </ sub >' s forms a concrete free R-resolution of k.

* Koszul – Tate complex

In the paper above, a specific complex, called the Koszul complex, is defined for a module over a Lie algebra, and its cohomology is taken in the normal sense.

* The Koszul complex is a DGA.

* The Tensor algebra is a DGA with differential similar to that of the Koszul complex.

A Koszul connection is a connection generalizing the derivative in a vector bundle.

In 1950, Jean-Louis Koszul unified these new ideas of covariant differentiation in a vector bundle by means of what is known today as a Koszul connection or a connection on a vector bundle.

Henri Dutilleux is the great-grandson of painter Constant Dutilleux and of composer Julien Koszul.

where is a permutation, and is the Koszul sign of the permutation

On the other hand, the notion of covariant differentiation was abstracted by Jean-Louis Koszul, who defined ( linear or Koszul ) connections on vector bundles.

Other notable participants in later days were Hyman Bass, Laurent Schwartz, Jean-Pierre Serre, Alexander Grothendieck, Jean-Louis Koszul, Samuel Eilenberg, Serge Lang and Roger Godement.

For instance, a Koszul connection in a vector bundle also allows for the parallel transport of vectors in much the same way as with a covariant derivative.

In 1950, Jean-Louis Koszul gave an algebraic framework for regarding a connection as a differential operator by means of the Koszul connection.

The Koszul connection was both more general than that of Levi-Civita, and was easier to work with because it finally was able to eliminate ( or at least to hide ) the awkward Christoffel symbols from the connection formalism.

The number of his official students was small, but includes Adrien Douady, Roger Godement, Max Karoubi, Jean-Louis Koszul, Jean-Pierre Serre and René Thom.

Connections on vector bundles are also sometimes called Koszul connections after Jean-Louis Koszul, who gave an algebraic framework for describing them.

The covariant derivative generalizes straightforwardly to a notion of differentiation associated to a connection on a vector bundle, also known as a Koszul connection.

Using ideas from Lie algebra cohomology, Koszul successfully converted many of the analytic features of covariant differentiation into algebraic ones.

In particular, Koszul connections eliminated the need for awkward manipulations of Christoffel symbols ( and other analogous non-tensorial ) objects in differential geometry.

The first Koszul homology H < sub > 1 </ sub >( K < sub >•</ sub >( x, y )) therefore measures exactly the relations mod the trivial relations.

With more elements the higher-dimensional Koszul homologies measure the higher-level versions of this.

Accidental war is so sensitive a subject that most of the people who could become directly involved in one are told just enough so they can perform their portions of incredibly complex tasks.

I am suggesting that a case-history approach to the Oedipus complex is a blind alley for a storyteller.

It is most probable that Freud and the Oedipus complex never entered his head in the writing of this story.

The board's action shows what free enterprise is up against in our complex maze of regulatory laws.

The transportation system which serves the National Forests is a complex of highways and access roads and trails under various ownerships and jurisdictions.

But there is still the sometimes complex problem of helping campers choose the best equipment for their individual needs.

Perhaps the best way to indicate the versatility of design that characterizes the use of plastics in signs and displays would be to look at what is happening in only one of the areas in this complex field -- changeable signs.

Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.

Hence the totality of singular lines is the T order complex of lines which meet Aj.

Since the complex of singular lines is of order K and since there is no complex of invariant lines, it follows from the formula Af that the order of the involution is Af.

The most obvious of these is the quadratic complex of tangents to Q, each line of which is transformed into the entire pencil of lines tangent to Q at the image of the point of tangency of the given line.

We now observe that the case in which **zg is a Af curve on a quadric is impossible if the complex of singular lines consists exclusively of the lines which meet Aj.

However, if there is no additional complex of singular lines, the order of the image regulus of a pencil is precisely Af.

In societies like ours, however, its place is less clear and more complex.

This behavior is more `` veridical '' -- or true -- than other testing behavior for some types of evaluation, and so can give quick and accurate estimates of complex functioning.

Of all the possible forms of nonverbal expression, that which seems best to give release, and communicational expression, to complex and undifferentiated feelings is laughter.