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An algebraic construction of the generic singularities of Boardman-Thom

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Abstract

In this paper we study some of the functorial properties of the infinite jet space in order to give a coordinate free algebraic definition of the generic singularities of Boardman-Thom. More precisely, suppose thatk is a commutative ring with an identity and suppose that A is a commutative ring with an identity which is ak-algebra. An A-k-Lie algebra L is ak-Lie algebra with ak-Lie algebra map ϕ from L to the algebra ofk-derivations of A to itself such that ford, d′εL anda, a′εA, then

$$[ad'],a 'd'] = a(\varphi (d')ad' - a'(\varphi (d')a)d' + aa'[d',\;d'].$$

. There is a universal enveloping algebra for such Lie algebras which we denote by E(L). Denote byL-alg the category of A-algebras B which have L and hence E(L) acting as left operators such that foraεA,dεL, (da)i B=d(a.i B). If F is the forgetful functor fromL-alg to the category of A-algebras, we show that F has a left adjoint J(L, ·) which is the natural algebraic translation of the infinite jet space.

In the third section of this paper we construct a theory of singularities for a derivation from a ring to a module and then we apply this construction to J(L, C) where C is an A-algebra. These singularities are subschemas with defining sheaf of ideals given by Fitting invariants of appropriately chosen modules when A and B are polynomial rings over a fieldk and C=A⊗ k B; these are the generic singularities of Boardman-Thom.

Finally we show that, under some rather general conditions on the structure of C as an A-algebra, the generic singularities are regular immersions in the sense of Berthelot.

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Authors
  1. K. R. Mount
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  2. O. E. Villamayor
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Additional information

This work was supported in part by NSF Grant GP 28915. Also part of the work was carried out during this author’s visit to Buenos Aires under the auspices of the Organization of American States.

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Mount, K.R., Villamayor, O.E. An algebraic construction of the generic singularities of Boardman-Thom. Publications Mathématiques de L’Institut des Hautes Scientifiques 43, 205–244 (1974). https://doi.org/10.1007/BF02684370

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  • DOI: https://doi.org/10.1007/BF02684370

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