Journal of Mathematical Sciences, Vol. 142, No. 4, 2007 GRADED ALGEBRAS AND THEIR DIFFERENTIAL GRADED EXTENSIONS D. Piontkovski UDC 512.55 Abstract. In the survey, we deal with the following situation. Let A be a graded algebra or a differential graded algebra. Adjoining a set x of free (in any sense) indeterminates, we make a new differential graded algebra A〈x〉 by setting the differential values d : x → A on x. In the general case, such a construction is called the Shafarevich complex. Beginning with classical examples like the bar-complex, Koszul complex, and Tate resolution, we discuss noncommutative (and sometimes even nonassociative) versions of these no- tions. The comparison with the Koszul complex leads to noncommutative regular sequences and complete intersections; Tate’s process of killing cycles gives noncommutative DG resolutions and minimal models. The applications include the Golod–Shafarevich theorem, growth measures for graded algebras, character- izations of algebras of low homological dimension, and a homological description of Gr¨obner bases. The same constructions for categories of algebras with identities (like Lie or Jordan algebras) allow one to give a homological description of extensions and deformations of PI-algebras. CONTENTS 1. Introduction .............................................. 2268 1.1. Classical Examples ....................................... 2268 1.2. Recent Developments ...................................... 2269 1.3. A Note on Bibliography .................................... 2270 1.4. Acknowledgment ........................................ 2270 2. Shafarevich Complex for Associative Algebras ........................... 2270 2.1. Assumptions and Notation ................................... 2270 2.2. Shafarevich Complex for Graded Algebras .......................... 2271 2.3. Noncommutative Regular Sequences: Strongly Free Sets .................. 2272 2.4. Gr¨ obner Bases and Homology ................................. 2274 2.5. Shafarevich Complex of a Free Algebra ............................ 2275 2.6. Noncommutative Complete Intersections Are Algebras of Global Dimension 2 ...... 2277 2.7. Golod–Shafarevich Theorem .................................. 2279 2.8. Noncommutative Complete Intersections and Generic Algebras ............... 2281 2.9. Algebras of Exponential Growth ................................ 2283 2.10. Algebras of Global Dimension Three ............................. 2287 2.11. Shafarevich Complex for Differential Graded Algebras. Noncommutative Tate Resolution 2288 2.12. Minimal Models ......................................... 2289 3. Shafarevich Complex for the Varieties of Algebras ........................ 2290 3.1. Varieties of PI-Algebras and Superalgebras .......................... 2290 3.2. Shafarevich Complex in the Varieties of Superalgebras ................... 2292 3.3. Complete-Intersection Criterion for Varieties ......................... 2292 3.4. Hilbert Series of Free Products in the Varieties of Associative Algebras .......... 2294 3.5. Deformations and Extensions ................................. 2295 3.6. Shafarevich Complex for Lie Algebras ............................. 2295 3.7. Lie Strongly Free (Inert) Sets and Rational Homotopies ................... 2297 References ............................................... 2298 Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 30, Algebra, 2005. 1072–3374/07/1424–2267 c  2007 Springer Science+Business Media, Inc. 2267