Koszul duality

 ( Algebras ) 

In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of , abstract algebras (semisimple algebra)Ben Webster, Koszul algebras and Koszul duality. November 1, 2007 and topology (e.g., equivariant cohomologyMark Goresky, Robert Kottwitz, and Robert MacPherson. Equivariant cohomology, Koszul duality, and the localization theorem. Inventiones Mathematicae 131 (1998).). The prototypical example of Koszul duality was introduced by Joseph Bernstein, Israel Gelfand, and Sergei Gelfand,.Joseph Bernstein, Israel Gelfand, and Sergei Gelfand. Algebraic bundles over $P^n$ and problems of linear algebra. Funkts. Anal. Prilozh. 12 (1978); English translation in Functional Analysis and its Applications 12 (1978), 212-214 It establishes a duality between the derived category of a symmetric algebra and that of an exterior algebra, as well as the BGG correspondence, which links the stable category of finite-dimensional graded modules over an exterior algebra to the bounded derived category of coherent sheaves on projective space. The importance of the notion rests on the suspicion that Koszul duality seems quite ubiquitous in nature.

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Koszul duality for graded modules over Koszul algebras The simplest, and in a sense prototypical case of Koszul duality arises as follows: for a 1-dimensional vector space V over a field k, with dual vector space $V^*$, the exterior algebra of V has two non-trivial components, namely

$\backslash bigwedge^1\; V=V,\; \backslash quad\; \backslash bigwedge^0\; V\; =\; k.$

This exterior algebra and the symmetric algebra of $V^*$, $\backslash operatorname\{Sym\}(V^*)$, serve to build a two-step chain complex

$V\; \backslash otimes\_k\; \backslash operatorname\{Sym\}(V^*)\; \backslash to\; k\; \backslash otimes\_k\; \backslash operatorname\{Sym\}(V^*)$

whose differential is induced by natural evaluation map

$V\; \backslash otimes\_k\; V^*\; \backslash to\; k,\; \backslash quad\; v\; \backslash otimes\_k\; \backslash varphi\; \backslash mapsto\; \backslash varphi(v).$

Choosing a basis of V, $\backslash operatorname\{Sym\}(V^*)$ can be identified with the polynomial ring in one variable, $kt$, and the previous chain complex becomes isomorphic to the complex

$kt\; \backslash stackrel\{t\}\{\backslash longrightarrow\}\; kt$

whose differential is multiplication by t. This computation shows that the cohomology of the above complex is 0 at the left hand term, and is k at the right hand term. In other words, k (regarded as a chain complex concentrated in a single degree) is quasi-isomorphic to the above complex, which provides a close link between the exterior algebra of V and the symmetric algebra of its dual.

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Koszul dual of a Koszul algebra Koszul duality, as treated by Alexander Beilinson, Victor Ginzburg, and Wolfgang SoergelAlexander Beilinson, Victor Ginzburg, Wolfgang Soergel. Koszul duality patterns in representation theory, Journal of the American Mathematical Society 9 (1996), no. 2, 473-527. can be formulated using the notion of Koszul algebra. An example of such a Koszul algebra A is the symmetric algebra $S(V)$ on a finite-dimensional vector space. More generally, any Koszul algebra can be shown to be a quadratic algebra, i.e., of the form

$A\; =\; T(V)\; /\; R,$

where $T(V)$ is the tensor algebra on a finite-dimensional vector space, and $R$ is a submodule of $T^2(V)\; =\; V\; \backslash otimes\; V$. The Koszul dual then coincides with the quadratic dual

$A^!\; :=\; T(V^*)\; /\; R\text{'}$

where $V^*$ is the ( k-linear) dual and $R\text{'}\; \backslash subset\; V^*\; \backslash otimes\; V^*$ consists of those elements on which the elements of R (i.e., the relations in A) vanish. The Koszul dual of $A=S(V)$ is given by $A^!\; =\; \backslash Lambda(V^*)$, the exterior algebra on the dual of V. In general, the dual of a Koszul algebra is again a Koszul algebra. Its opposite ring is given by the graded ring of self-Ext functor of the underlying field k, thought of as an A-module:

$(A^!)^\{\backslash text\{opp\}\}\; =\; \backslash operatorname\{Ext\}^*\_A(k,\; k).$

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Koszul duality If an algebra $A$ is Koszul, there is an equivalence between certain subcategories of the derived category of graded module $A$- and $A^!$-modules. These subcategories are defined by certain boundedness conditions on the grading vs. the cohomological degree of a complex.

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Variants As an alternative to passing to certain subcategories of the derived categories of $A$ and $A^!$ to obtain equivalences, it is possible instead to obtain equivalences between certain quotients of the homotopy categories. Usually these quotients are larger than the derived category, as they are obtained by factoring out some subcategory of the category of acyclic complexes, but they have the advantage that every complex of modules determines some element of the category, without needing to impose boundedness conditions. A different reformulation gives an equivalence between the derived category of $A$ and the 'coderived' category of the coalgebra $(A^!)^*$.

An extension of Koszul duality to D-modules states a similar equivalence of derived categories between dg-modules over the dg-algebra $\backslash Omega\_X$ of Kähler differentials on a smooth algebraic variety X and the $D\_X$-modules. Kapranov, Mikhail M. On DG-modules over the de Rham complex and the vanishing cycles functor. Algebraic geometry (Chicago, IL, 1989), 57–86, Lecture Notes in Math., 1479, Springer, Berlin, 1991.Positselski, Leonid: Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence., Mem. Amer. Math. Soc. 212 (2011), no. 996, vi+133 pp. , see Appendix BGerd Faltings; Chai, Ching-Li. Degeneration of abelian varieties. With an appendix by David Mumford. Springer-Verlag, Berlin, 1990. xii+316 pp. . Section VI.3

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Koszul duality for operads An extension of the above concept of Koszul duality was formulated by Ginzburg and Kapranov who introduced the notion of a quadratic operad and defined the quadratic dual of such an operad.Ginzburg, Victor; Kapranov, Mikhail. Koszul duality for operads. Duke Math. J. 76 (1994), no. 1, 203–272. Very roughly, an operad is an algebraic structure consisting of an object of n-ary operations for all n. An algebra over an operad is an object on which these n-ary operations act. For example, there is an operad called the associative operad whose algebras are associative algebras, i.e., depending on the precise context, non-commutative rings (or, depending on the context, non-commutative graded rings, differential graded rings). Algebras over the so-called commutative operad are commutative algebras, i.e., commutative (possibly graded, differential graded) rings. Yet another example is the Lie operad whose algebras are . The quadratic duality mentioned above is such that the associative operad is self-dual, while the commutative and the Lie operad correspond to each other under this duality.

Koszul duality for operads states an equivalence between algebras over dual operads. The special case of associative algebras gives back the functor $A\; \backslash mapsto\; A^!$ mentioned above.

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See also

• Zinbiel algebra
• BGG correspondence

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Notes

• Priddy, Stewart B. Koszul resolutions. Transactions of the American Mathematical Society 152 (1970), 39–60.

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External links

• http://www.math.harvard.edu/~lurie/282ynotes/LectureXXIII-Koszul.pdf
• http://people.mpim-bonn.mpg.de/geordie/Soergel.pdf
• http://arxiv.org/pdf/1109.6117v1.pdf

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