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We construct an algebraic-cycle based model for the motivic cohomology on thecategory of schemes of finite type over a field, where schemes may admitarbitrary singularities and may be non-reduced.  We show that our theory is functorial on the category, that it detectsnilpotence, and that its restriction to the subcategory of smooth schemesagrees with the pre-existing motivic cohomology theory, which is the higherChow theory of S. Bloch (Adv. Math., 1986). A few structures and applicationsare discussed.

On extension of the motivic cohomology beyond smooth schemes