The subject of this monograph lies on the border of algebraic geometry with modern homological algebra. A scheme (in the sense of Grothendieck) is a basic notion in contemporary algebraic geometry, while other basic notions involved (cotorsion modules, and more generally complete cotorsion pairs) are fundamental to homological ring and module theory of the last quarter century. Contraherent cosheaves have been previously introduced by the author and are further presented and developed in this book.

Leonid Positselski did his undergraduate studies in Moscow, Russia, and received his Ph.D. from Harvard University in 1998.  After several years of postdocs in the U.S. and Europe, Positselski returned to Moscow for ten years from 2003-2014, where his research program solidified.  For several years, Positselski taught at the newly established Faculty of Mathematics of the Higher School of Economics in Moscow.  In spring 2014, he left Russia for Israel, where he lived in Haifa.  Since autumn 2018, Positselski is working at the Institute of Mathematics of the Czech Academy of Sciences in Prague.

Positselski is an expert in Homological Algebra, Category Theory, and algebraic aspects of Algebraic Geometry.  His specific topics include quadratic and Koszul algebras, absolute Galois groups and Galois cohomology, derived nonhomogeneous Koszul duality, coderived and contraderived categories, matrix factorization categories, semi-infinite homological algebra and algebraic geometry, Matlis-Greenlees-May duality, and infinitely generated tilting theory.  Positselski is the author of four monographs published by AMS and Birkhäuser/Springer, an AMS Memoir, an SMF Mémoire, two long surveys, and more than sixty research articles.