# Moscow Mathematical Journal

Volume 15, Issue 2, April–June 2015 pp. 353–372.

Chiral de Rham Complex over Locally Complete Intersections

**Authors**:
Fyodor Malikov (1) and Vadim Schechtman (2)

**Author institution:**(1) Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA

(2) Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062 Toulouse, France

**Summary: **

Given a locally complete intersection X ↪ Y we define a version of a derived chiral De Rham complex, thereby “chiralizing” a result by Illusie and Bhatt. A similar construction attaches to a graded ring a dg vertex algebra, which we prove to be Morita equivalent to a dg algebra of differential operators. For example, the dg vertex algebra associated to a fat point, which also arises in the Landau–Ginzburg model, is shown to be derived rational.

2010 Math. Subj. Class. 14, 18.

**Keywords:**Vertex algebra, chiral differential operator, dga resolution.

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